Branes, topology, and perturbation theory in two-dimensional quantum gravity

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Content BRANES, TOPOLOGY, AND PERTURBATION THEORY IN TWO-DIMENSIONAL QUANTUM GRAVITY by Ashton Camill Lowenstein A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (PHYSICS) August 2024 Copyright 2024 Ashton Camill Lowenstein Dedication To my wife, Adeline. ii Acknowledgements I’d first like to express my deepest gratitude to my advisor Clifford Johnson for encouraging my passion for low-dimensional physics, and for pointing me toward matrix models. I have benefited greatly from his experience and guidance over the last several years. I am also grateful for Krzyzstof Pilch and Itzhak Bars for their guidance at the beginning of my graduate school career, and for helping me get started in the High Energy Theory research group. Many thanks to Stephan Haas for agreeing to be the chair of my defense committee. I owe a great deal to Felipe, Robert, and Avik, who were very welcoming and nice to me when I joined the group. I would also like to thank Wasif Ahmed for his collaboration over the last couple years and for the many productive conversations we’ve had. I am very grateful to have made so many amazing friends at USC, most importantly Aaron, Jason, and John. I will always be thankful for the camaraderie we had when we were taking classes and for the moral support they’ve given me over the years, especially before and at my wedding. I would not have made it through my degree without the support of my family, in particular my wife Addie, my parents, and my parents-in-law. I must give special thanks to Chris and Dede for their incredible generosity, and for making my transition to living in LA so wonderful. I am also extremely grateful to my grandparents for inspiring me to get an advanced degree and to follow my passion. Finally, thank you to my pets Billie, Jo, Eddie, and Isabelle. iii Table of Contents Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Canonical Quantization Of The Bosonic String . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Path Integral Quantization Of The Bosonic String . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Conformal Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.5 Liouville Theory And Minimal String Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.6 Superstrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.6.1 Minimal Superstrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.7 Branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.7.1 General Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.7.2 Branes In Non-Critical String Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Chapter 2: The Kortewig-de Vries Hierarchy And Intersection Theory . . . . . . . . . . . . . . . . . . 35 2.1 Integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.2 The KdV Hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.3 The Lax Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.3.1 Gelfand-Dikii Resolvent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.4 Tau And Baker Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.5 Intersection Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.6 Weil-Petersson Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Chapter 3: Random Matrix Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.1 Motivation And Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.2 Saddle Point Analysis And The Large-N Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.3 Matrix Models And Families Of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.3.1 β = 2 Wigner-Dyson Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.3.2 β = 1 Wigner-Dyson Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.3.3 (α, β) = (1 + 2Γ, 2) Altland-Zirnbauer Theories . . . . . . . . . . . . . . . . . . . . . 75 iv 3.4 A Prelude To The Double Scaling Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Chapter 4: Double Scaled Hermitian Matrix Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.1 Double Scaled Wigner-Dyson Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.2 Double Scaled Altland-Zirnbauer Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.2.1 Non-Perturbative Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.3 Notable Example Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Chapter 5: String Equation Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.1 Novikov Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.2 The DJM Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.2.1 Open String Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.2.2 0A Closed String Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.3 N = 2 Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.3.1 Closed String Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.3.2 Open String Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.3.3 Interpreting The Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Chapter 6: Open And Closed String Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . 124 6.1 Macroscopic Loop Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6.1.1 General Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 6.1.2.1 Non-Supersymmetric Theories . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.1.2.2 Supersymmetric Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 6.1.3 KdV Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 6.2 Geodesic Loop Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6.2.1 Boundary Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 6.2.2 Trumpet Partition Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 6.2.3 FZZT Brane Partition Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 6.2.4 The Matrix Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 6.2.5 Eigenbranes And Microstates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 6.3 Generalized Weil-Petersson Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 6.3.1 Non-Supersymmetric Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 6.3.2 N = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 6.3.3 N = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 6.4 Gluing Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 6.5 Topological Recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 Chapter 7: Unoriented Two-Dimensional Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 7.1 Orthogonal Polynomial Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 7.2 Previous Double Scaling Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 7.3 New Results in Matrix Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 7.3.1 Double Scaled Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 7.3.2 New String Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 7.4 Unoriented Gravity And Unoriented Minimal Strings . . . . . . . . . . . . . . . . . . . . . . . 183 7.4.1 The GOE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 7.4.2 The k = 1 To k = 2 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 v 7.4.3 Pure Unoriented Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 7.4.4 The (2, 3) Unoriented Minimal String . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 7.5 Unoriented JT Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 Chapter 8: Summary And Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 The Wentzel–Kramers–Brillouin Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 A.1 Computing The Eigenvalue Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 New Results Regarding The Gelfand-Dikii Resolvent . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 vi List of Tables 1.1 The different boundary conditions giving the two types of strings. . . . . . . . . . . . . . . . 7 1.2 The different boundary conditions for the worldsheet fermions. . . . . . . . . . . . . . . . . . 24 1.3 Minimal string D-branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.1 A list of important example matrix models and their coupling constants. . . . . . . . . . . . 96 5.1 Locations of open and closed string physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 vii List of Figures 3.1 The propagator for a β = 2 matrix model. The lines carry arrows representing the flow from barred to unbarred indices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.2 Two generic diagrams drawn on different topologies. This would contribute at different overall powers of N even if they have the same number of vertices. The matrix propagator is shown in blue, and the dual diagram is shown in red in each case. . . . . . . . . . . . . . . 56 3.3 The propagator for a β = 1 matrix model. The lines carry no arrows because there is no consistent way to define flow between indices. The first term is functionally identical to the β = 2 propagator. The second term can be thought of as being twisted with respect to the first term. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.4 A diagram drawn on an unorientable surface. Notice that the tessellation is necessarily self-intersection in this number of dimensions due to the cross cap. . . . . . . . . . . . . . . . 58 3.5 The Wigner semi-circle densities for the Gaussian Orthogonal Ensemble (GOE) and Gaussian Unitary Ensemble (GUE). In both plots the eigenvalue histograms are shown in blue, while the corresponding density functions are shown in red. Close inspection reveals that the spectra actually leak out past the endpoints predicted by the saddle point analysis. 61 4.1 A pre-double scaling diagram with an insertion of tr e −lM. . . . . . . . . . . . . . . . . . . . . 86 4.2 The left panel shows the full solution on the desired range of x values. The right panel is a zoomed in portion around the origin. The nonperturbative solution develops a well for Γ = 0, which disappears as Γ is increased. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.3 The left panel shows the full solution on the desired range of x values. The right panel is a zoomed in portion around the origin. The nonperturbative solution develops a well for Γ = 0, which disappears as Γ is increased. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.4 Gluing procedure for a surface with g = 2 and one asymptotic boundary. The Schwarzian mode is shown in green near the asymptotic boundary on the trumpet. . . . . . . . . . . . . 99 5.1 The leading order (left) and first open string sector correction (right) to the potential u for the (2, 3) minimal string. As the energy grows, the nonzero vertical asymptote in u0,1 gets pushed toward x → −∞. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 viii 5.2 The left panel shows a contribution at O(Γ 2 ) that involves only one brane. The right panel shows a contribution at O(Γ 2 ) that involves two different branes. . . . . . . . . . . . . 113 5.3 The combined leading and first subleading contributions to u in the k = 1 0A multicritical model for different values of E, for Γ = 0 (left) and Γ = 1 (right), all with h̵ = 0.1. We can see as E is increased the function is translated up and to the right. The potential switches concavity at Γ = 1 2 . Only the positive branch of each solution is displayed. . . . . . . . . . . 117 6.1 The disk geometry as it is usually presented in nAdS2 with an asymptotic boundary of renormalized length β, but without the wiggling Schwarzian mode. Even though a matrix model with arbitrary tk may not have an explicit geometric interpretation, it is useful to keep this picture in mind in the topological expansion. . . . . . . . . . . . . . . . . . . . . . . 128 6.2 The web of connections forming in the string equation formalism, which includes important matrix model objects and the Weil-Petersson volumes. . . . . . . . . . . . . . . . . 143 6.3 The trumpet geometry with one asymptotic boundary of renormalized length β, and one geodesic boundary (shown in red) of length b. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 6.4 The two-brane density of states ρ0(E) h̵ + h̵Γ 2ρ0,2(E; 5) with h̵ = 0.1 and Γ 2 = .1 is represented by the solid purple line. The disk density ρ0(E) = √ E/π is represented by the dashed green line. An opening has appeared at E = 5, representing the fact that the eigenvalue has been “frozen” there. The width of the opening is directly related to the size of h̵Γ. The density actually asymptotes to −∞ both as E → 0 and E → 5. Divergences like this are common in the perturbative expansion of the density. . . . . . . . . . . . . . . . 153 6.5 The two-brane density of states ρ0(E) h̵ + h̵Γ 2ρ0,2(E; 5) with h̵ = 0.1 and Γ 2 = .1 is represented by the solid purple line. The disk density ρ0(E) = √ E π + 16πE 3 2 27 is represented by the dashed green line. The scale of the y-axis is different that the k = 1 model due to the presence of the k = 2 model and the fact that the extra coupling constant t2 > 1. . . . . . 154 6.6 The two-brane density of states in the simplest 0A model with h̵ = 0.1 and Γ 2 = .1 is represented by the solid purple line. The disk density ρ0(E) = √ E π + µ 2π √ E is represented by the dashed green line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 7.1 Our result from (7.20) exactly overlaps with the standard result (7.7), as can be seen from (c). Two plots are indistinguishable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 7.2 ρ0 vs E curve. We can see that ρ0 perfectly wraps around E 3 2 in this energy window. . . . . 188 7.3 ρ0 vs E curve. We can see the deviation at higher energies. . . . . . . . . . . . . . . . . . . . . 189 7.4 For lower energies all the curves with different cutoffs merge together. The curve deviates earlier as the cutoff is decreased. If we zoom in, we can see that for a particular energy the curve with a lower cutoff is more off from the E 3 2 curve. . . . . . . . . . . . . . . . . . . . 189 ix 7.5 The first unoriented contribution to the eigenvalue density is small compared to the disk level contribution, even with h̵ = 1. The combined contribution to the total density still has fictitious oscillations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 7.6 ρ0 vs E curve. We can see that ρ0 perfectly wraps around the expected result in this energy window. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 7.7 The first unoriented contribution to the eigenvalue density in the interpolated matrix model. 191 x Abstract Random matrix theory offers a powerful toolset to study a variety of two-dimensional quantum gravity and string theories both perturbatively and nonperturbatively. In this thesis we study three classes of matrix models describing general two-dimensional oriented gravity theories, supergravity theories, and unoriented gravity theories. Chapters 1 through 4 contain a review of necessary background material, including string theory, integrable hierarchies and intersection theory, and random matrix theory. In Chapter 5 we study perturbative solutions of two classes of differential equations, known as string equations, describing β = 2 Wigner-Dyson class matrix models and (1 + 2Γ, 2) Altland-Zirnbauer class matrix models. In Chapter 6 we apply these results to operator correlation functions in the open and closed string sectors of the theories. We consider a duality between the open and closed string sectors of these models consisting of a relationship between correlation functions in one sector and the free energy in the other. We use this duality to deepen the understanding of the connection between D-branes, generalized Weil-Petersson volumes, and the statistical properties of certain gravity theories. Specific examples are considered and compared to existing results in the literature. In Chapter 7 we develop a double scaling procedure in β = 1 Wigner-Dyson class matrix models, which are dual to certain unoriented gravity theories. We compute the leading density of states in two models, and form a conjecture for how to define unoriented JT gravity in a minimal model decomposition. xi Chapter 1 Introduction All things must pass None of life’s strings can last - “All Things Must Pass” by George Harrison 1.1 Overview One of the greatest remaining puzzles in theoretical physics is the quantization of gravity. Einstein’s general theory of relativity, which is now nearly 110 years old, is still being validated by experiments deep into its nonlinear regime. However, general relativity is a classical theory of gravity, and lacks a satisfactory description of certain properties of black holes, as well as the Big Bang, that are expected to become important at quantum time- and length-scales. Moreover, black holes suffer from a number of paradoxes centered around their event horizons and singularities that should be resolved by a consistent quantum theory of gravity. Efforts to quantize general relativity began immediately following the success of quantum field theory in unifying the Standard Model of particle physics in the mid-20th century. The tools developed to describe the electromagnetic, weak nuclear, and strong nuclear forces at the quantum level are remarkably powerful 1 and have yielded some of the most astonishing agreements between theoretical predictions and precision measurements ever. Yet, when these tools are naively applied to gravity in four dimensions they are met with ultraviolet (UV) divergences that they were not designed to deal with. There has, however, been some success in quantizing gravity in two and three dimensions. Such models are simplified by having no bulk propagating degrees of freedom (i.e. gravitons), yet still contain entities like black holes that are fundamentally important to quantum gravity. One such theory, Jackiw-Teitelboim (JT) gravity [1–3], is both a model of two-dimensional black holes as well as a theory describing the dynamics of a certain near-extremal four-dimensional black holes. Other low-dimensional gravity systems are often obtained from string theory via dimensional reduction or the worldsheet formalism. String theory is currently the most-explored and most viable option for quantizing gravity. Its fundamental degree of freedom, the string, is one-dimensional and therefore evades many of the aforementioned point-like UV divergences one obtains from standard quantum field theory techniques. The different sectors of the theory naturally describe both gravity and the gauge theories that make up the Standard Model, making it a promising avenue toward a unified theory of all four fundamental forces. This thesis will be concerned with general sets of string theories that exist in two dimensions1 and that have varying amounts of supersymmetry. Such string theories are often deemed noncritical, and although they are simpler than their 10- and 26-dimensional cousins they still contain interesting, non-trivial information. In particular a large subset of the models considered here will fall under the classification of topological (super)gravity, which are dual to general minimal (super)string theories. The last 10 years has seen a resurgence in the study of two-dimensional quantum gravity. Developments in the 0+1-dimensional SYK theory [4–6] led to an interest in Jackiw-Teitelboim (JT) gravity because both theories are described by Schwarzian dynamics in a certain regime. The landmark discovery made by Mirzakhani regarding volumes of moduli spaces of bordered hyperbolic Riemann surfaces [7] and the 1The dimensionality of the models we consider is subtle, since there are two notions of dimension in string theory: the target space and the worldsheet. Strictly speaking the target space dimension of the models considered here will be less than 1. 2 subsequent connection to topological recursion in random matrix theory made by Eynard and Orantin [8] provided some of the tools necessary for Saad, Shenker, and Stanford to show that JT gravity is described by a double scaled matrix model [9]. Such results reignited an interest in matrix models and topological gravity that started nearly 40 years ago, as borne out by applications to non-perturbative quantum gravity [10, 11], flat space holography [12, 13], and gauge-string duality [14]. The use of random matrix techniques in quantum field theory began in the late 70s (e.g. [15]) and saw its first golden age for applications to quantum gravity in the late 80s and early 90s with works like [16–27]. A review and summary of the state of low-dimensional string theory and random matrix theory at that time can be found in [28]. The study of continuum physics via random matrix theory often involves the use of the double scaling limit2 , a process in which the size of the matrix N is taken to infinity while tuning the parameters gi of V to their critical values [17, 18, 20, 29]. This can be interpreted geometrically as taking the number of faces in a tessellation to infinty while sending the average size of a face to 0, producing a smooth closed surface with non-vanishing area. The symmetry properties of the random matrices come into play here: Hermitian matrices produce oriented surfaces, while real symmetric matrices produces unoriented surfaces. The expansion in powers of 1/N becomes an expansion in a new small parameter h̵, which is a renormalized version of the former. When the random matrix is Hermitian, the resulting surfaces with g handles come weighted by a factor h̵2g−2 , and when the random matrix is real symmetric, the resulting surfaces with g handles and c crosscaps come weighted by a factor of h̵2g−2+c . It is in this sense that expansions in powers of h̵ are deemed topological expansions. This dissertation will be organized as follows. The rest of Chapter 1 will contain a cursory review of string theory, with an emphasis on its two-dimensional aspects. The general goal of this review is to introduce the theories whose matrix model descriptions we will study later, as well as to introduce 2There are interesting cases where the finite or large-N matrix model has an interpretation in terms of string theory constructions, see for example [14]. 3 key concepts like open and closed string sectors, as well as branes. In particular we will focus on the involvement of conformal field theory (CFT) in string theory, which includes general formalism as well as key example theories like (super) Liouville theory and the minimal (super) CFTs. We will also discuss quantization and the spectra of the different sectors of the theory. This will lead naturally to considering branes, which we will consider primarily at a qualitative level. This review will draw from a number of sources [28, 30–35]. Chapter 2 will review integrable systems and the Kortewig-de Vries (KdV) hierarchy, as well as intersection theory and topological gravity. This will include a general review of integrability, the KdV equation and its related hierarchy of differential equations, and important technology like the Lax formalism. We will then pivot to a brief discussion of algebraic geometry in the form of intersection theory, and its connection to the KdV hierarchy via the Witten-Kontsevich theorem. We will conclude with a discussion of the Mirzakhani recursion relation for the Weil-Petersson volumes of the moduli spaces of stable bordered Riemann surfaces. In Chapters 3 and 4 we will review the various classes of random matrix models relevant to the models of gravity studied later on. Even though matrix models have a long history in physics, not everybody is well-acquainted with them. Because the majority of the original work presented in this thesis is rooted in matrix model calculations, it is worth spending a good deal of time on this background material in particular. This will include a review of the basic definitions, the applications of various families of polynomials, and the double scaling limit. This will connect us to the discussion in Chapter 2 regarding integrable systems and the KdV hierarchy. With this general framework in mind we will review the connections between matrix models and two-dimensional gravity and string theory via the string equation formalism. Chapters 5, 6, and 7 will contain the bulk of the original work presented in this thesis, which is primarily drawn from [36, 37] but will also include new results. Chapter 5 contains an in-depth and systematic study of the perturbative solutions to several families of differential equations, known as string equations, that 4 determine key properties of their associated random matrix theories and string theories. We pay particular attention to the interpretations of the solutions to these equations, emphasizing when they describe open versus closed strings. The results obtained are applicable to general matrix models in their respective classes, and we will look at examples of recent interest. We also apply the techniques we have developed to study perturbation theory in the string equation formalism of N = 2 matrix models for the first time. In Chapter 6 we present the perturbative analysis of correlation functions in Hermitian matrix models. We focus on two important composite operators, which represent the creation of asymptotic and geodesic boundaries in the geometric interpretation of the theories. Using the framework we develop, we study an open-closed duality present in non-supersymmetric and supersymmetric matrix models, and consider applications of the duality to the perturbative expansion of interesting correlation functions. The duality will also be used to calculate brane-related quantities in two-dimensional string theory, and further expand upon the role that branes play in the microstate physics of quantum gravity. Chapter 7 will present new results in two dimensional unoriented gravity. In particular, a formalism for studying arbitrary superpositions of unoriented bosonic minimal models will be explored, with an aim at studying unoriented JT gravity. A method of obtaining “new string equations” for the added functional degrees of freedom in the model will be described. Several interesting examples are used as a testing ground for the methods we develop. Considering unoriented spacetimes in Euclidean quantum gravity is interesting for several reasons. First, an unoriented bulk spacetime is appropriate for describing a boundary theory with time reversal symmetry in the context of the AdS/CFT correspondence. Second, although the Gaussian Orthogonal Ensemble (GOE) and Gaussian Symplectic Ensemble (GSE) have been studied extensively in the math literature (see for example [38]), the double scaling limits of the more general Wigner-Dyson β-ensembles is not as investigated. Other descriptions of unoriented gravity in two dimensions can be found in [39–41]. Since these models describe unoriented minimal strings any effort in this direction provides nice contact between math and physics. 5 There are two appendices whose purposes are to collect results, some of which are new, that are related to the body of the dissertation but are slightly orthogonal or ancillary to the main narrative. In Appendix A we describe the Wentzel–Kramers–Brillouin (WKB) Approximation, a method for solving certain linear differential equations. In particular we present the standard application of the approximation to the Schrodinger equation, its subsequent application in the statistical analysis of matrix models, and some new results incorporating branes. This allows us to calculate the density of states corresponding to the presence of eigenbranes inserted in an arbitrary theory, which connects to a discussion in Chapter 6. In Appendix B we present some new results concerning the off-diagonal representation of the Gelfand-Dikii resolvent and compare it to known results. This will include a generalization of some of the Gelfand-Dikii polynomials, which are key objects in the string equation formalism of matrix models. 1.2 Canonical Quantization Of The Bosonic String Just as a relativistic particle traces out a one-dimensional worldline in spacetime, described by an intrinsic coordinate τ , a string will trace out a two-dimensional worldsheet in spacetime with intrinsic coordinates σ i = (τ, σ). The direct generalization of the first-order action of a free relativistic particle to a string is S = − T 2 ∫ d 2 σ √ hhij(σ)gµν(X)∂iX µ ∂jX ν . (1.1) The intrinsic geometry of the worldsheet is described by the metric hij , while the geometry of the string in the target space, the ambient spacetime in which the string propagates, is described by the metric gµν. The embedding of the worldsheet in the target space is given by the function Xµ (σ). For now we will leave the dimension of spacetime D unfixed. The parameter T has mass dimension 2 and is interpreted as the tension of the string. 6 The action (1.1) is diffeomorphism invariant and thus enjoys the local reparametrization invariance of the worldsheet σ i → σ ′i (σ). It is also invariant under the local Weyl rescaling hij → Λ(σ)hij . These three transformations provide enough power to fix each independent element of the metric hij . A simple choice is the flat metric hij = ηij in Lorentz signature or hij = δij in Euclidean signature. We must also keep in mind that the equation of motion for the worldsheet metric is δS δhij = 0, which implies the vanishing of the energy-momentum tensor Tij = 0. The choice to utilize Weyl scaling as a gauge symmetry will have important ramifications in the quantum theory. With the flat worldsheet gauge choice, the classical equation of motion obtained from (1.1) is the wave equation (∂ 2 σ − ∂ 2 τ )X µ = 0. (1.2) However, this equation is not sufficient to ensure that the action is stationary: if the solution is not taken to be periodic in σ, a variation of the action yields a surface term that requires the Neumann conditions ∂σX µ (τ, 0) = ∂σX µ (τ, π) = 0. (1.3) Thus we arrive at two different types of strings, determined by their respective boundary conditions, which are listed in Table (1.1). They represent closed strings (with the topology of a circle) and open strings (with the topology of a line segment). Table 1.1: The different boundary conditions giving the two types of strings. String Type Boundary Condition Closed Xµ (τ, 0) = Xµ (τ, π) Open ∂σXµ (τ, 0) = ∂σXµ (τ, π) = 0 7 The next step toward performing canonical quantization is to introduce the coordinates σ ± = τ ± σ. In these coordinates the Minkowski metric has components η+− = η−+ = − 1 2 and η++ = η−− = 0. The general solution to the wave equation is Xµ (σ) = X µ R (σ − ) + X µ L (σ + ), where X µ R/L are so far arbitrary. As is standard in quantum field theory, it is convenient to Fourier decompose the specific solutions. In the open string case it is X µ (τ, σ) = x µ + 2α ′ p µ τ + i √ 2α′ ∑ n≠0 1 n α µ n e −inτ cosnσ, (1.4) where x µ is interpreted as the center of mass position and p µ the center of mass momentum of the string, and we have defined α ′ = 1/2πT. The open string boundary conditions allow the separate functions X µ R/L to combine into standing waves. This is unlike the closed string case, where the two functions remain independent, and thus have independent Fourier modes X µ L (σ + ) = 1 2 x µ + α ′ p µ σ+ + i √ α′/2 ∑ n≠0 1 n α˜ µ n e −2inσ+ , X µ R (σ − ) = 1 2 x µ + α ′ p µ σ− + i √ α′/2 ∑ n≠0 1 n α µ n e −2inσ− . (1.5) In both cases the functions are required to be real, which necessitates α µ −n = (α µ n ) † , α˜ µ −n = (α˜ µ n ) † . (1.6) 8 The Fourier modes α µ n and α˜ µ n, and the center of mass quantities obey the Poisson brackets familiar from quantum field theory. Upon quantization we obtain the commutation relations [x µ , pν ] = iηµν , [α µ m, αν n ] = mδm+nη µν , [α µ m,α˜ ν n ] = 0, [α˜ µ m,α˜ ν n ] = mδm+nη µν . (1.7) The modes therefore have the interpretation of (independent) raising and lowering operators, related to conventional ones by a change of normalization. In the closed string sector, where α and α˜ are distinct, they are usually differentiated from one another by calling one of the right-moving and the other leftmoving. These modes are used to build a Fock space representation of the string Hilbert space. Introduce a Fock vacuum ∣0⟩ that is annihilated by α µ n for n > 0. The Hilbert space should also include eigenstates of either x µ or p µ , and the conventional choice is to choose the center of mass momentum. The energy-momentum tensor components can be mode-expanded as well. It is convenient to define the generators Lm = 1 2 ∑ n∈Z αm−n ⋅ αn, (1.8) where the dot denotes contraction of the target space indices, and with a similar expression for the other modes in the closed string sector. The specific form of these generators arises from classical considerations, meaning there is an ordering ambiguity. The modes αm−n and αn commute unless m = 0, so the only actual problem is with L0. The natural solution is to define L0 by its normal ordered form L0 = 1 2 α 2 0 + ∞ ∑ n=1 α−n ⋅ αn. (1.9) 9 The operators Ln obey the commutation relations [Lm, Ln] = (m − n)Lm+n + D 12 (m3 − m)δm+n, (1.10) which is known as the Virasoro algebra with central charge c = D. The analogs L˜ n satisfy an identical algebra, and commute with Ln. The vanishing of the energy-momentum tensor translates into the Hilbert space formalism in the following way. The physical subspace {∣ψ⟩} of the total Fock space is defined to satisfy Lm ∣ψ⟩ = 0 for all m > 0 and (L0 −a) ∣ψ⟩ = 0 for some number a 3 . These are called Virasoro constraints, and play a large role in the representation theory of the Virasoro algebra, and hence in conformal field theory. We conclude the review of canonical quantization with a brief look at the spectrum of the string in both the open and closed sectors. To do so, we switch to lightcone quantization. This entails a coordinate transformation not on the worldsheet (we already defined lightcone coordinates there), but in target space. The primary outcome of interest here is that string excitations are generated only by the transverse oscillators α i n , where i = 2, . . . , D −1. The simplest state in the open string sector is the Fock ground state with some momentum k, denoted ∣0; k⟩. This state has the squared mass m2 = 2−D 24α′ , which is negative if D > 2. These states are called tachyons and represent an instability in the vacuum of the string theory. Such states are undesirable and provide part of the motivation to appeal to supersymmetry later on. The next natural open string state comes from acting on ∣0; k⟩ with the first raising operator once, α i −1 ∣0; k⟩. This state has squared mass m2 = 26−D 24α′ . Here we see the first sign that D = 26 is a special dimension. In D dimensions the Lorentz group is SO(D − 1, 1). Massive vectors transform under the SO(D −1) subgroup, corresponding to the fact that they have D −1 degrees of freedom. Massless vectors on the other hand transform under the SO(D − 2) subgroup. The state we constructed has only D − 2 degrees of freedom. Therefore if the theory is to be Lorentz invariant, we must have this state be massless, 3When the dimension is chosen to be critical, D = 26, the value of a is set to 1. 10 which requires D = 26. Thus we can see the open string contains a massless vector field in its spectrum (with no other internal degrees of freedom, so far), making it viable to obtain QED out of string theory. Irrespective of any phenomenological interpretations, this state is interpreted as a gauge field Aµ(X) living in the target space, making it a functional of the worldsheet fields Xν . More complicated gauge groups come from attaching internal degrees of freedom to the endpoints of the open string, called Chan-Paton factors. Switching to lightcone gauge in the closed string sector follows in exactly the same way, and its simplest state is also a tachyon. Its next natural state is4 α i −1α˜ j −1 ∣0, 0; k⟩. It has the same squared mass as the corresponding open string state, making it massless in D = 26. Since α and α˜ commute, this should transform in the two-index tensor representation of SO(D − 2). This representation is reducible, and can be decomposed into a scalar, symmetric tensor, and anti-symmetric tensor. The symmetric part will be a spin-2 field, or in other words the graviton. Hence the closed string sector gives us gravity in the form of fluctuations of the target space geometry. The scalar is called the dilaton and is often denoted Φ. The anti-symmetric field, denoted Bij , is called the Calb-Ramond field. 1.3 Path Integral Quantization Of The Bosonic String In order to quantize the free string we use the un-gauge-fixed form of the action (1.1) and define the path integral Z = ∫ DhDXe−S[h,X] , (1.11) where we have made the convenient swap to a Euclidean signature on the worldsheet with coordinates σ α = (σ 1 , σ2 ). The path integral measure Dh ostensibly represents an integral over the three independent components of the worldsheet metric h++, h−−, and h+−. However it is convenient to use the three gauge 4The closed string sector has a requirement called level matching, meaning that for every αn applied to the ground state, we must also apply an α˜n. 11 symmetries (reparametrization and Weyl rescaling) to put the metric in the form hαβ = e φ ηαβ. This is obtained by utilizing the standard Fadeev-Popov method, introducing the appropriate ghost and anti-ghost fields c and b, respectively, with the right action. The exact form of the ghost action is not necessary for this discussion. The path integral is then Z = ∫ Dφ∫ DbDcDXe−S[X,b,c] . (1.12) There is a more general action that can be considered, which makes the most sense to consider in Euclidean signature. To S we add the term χ = 1 4π ∫ Σ d 2 σ √ hR + 1 2π ∫ ∂Σ dsK, (1.13) where Σ is the worldsheet, R is the Ricci scalar of the worldsheet, and K is the extrinsic curvature of the boundary. This is the most general term that is quadratic in derivatives and that maintains the desired symmetries of the theory. But, it is purely topological in two dimensions and is in fact the Euler characteristic of Σ, given by χ(Σ) = 2g − 2 + n, where g is the number of handles and n is the number of boundaries. The new action is S = SX + λχ, where SX denotes the explicitly X-dependent part from before. Previously we had not considered the topology of the worldsheet because it did not make as much sense in Lorentzian signature. However now it is clear that the metric part of the path integral should include some information about what topology the worldsheet has. In the context of the path integral, we see that each worldsheet topology is weighted by an amount e −λχ. For open strings, worldsheets can only change by increasing or decreasing the number of boundaries, and so χ can only change in increments of 1. Therefore the coupling constant for string splitting and joining is actually e −λ/2 . A closed string worldsheet can change by changing the number of handles, for which χ changes in increments of 2. Therefore the coupling constant in the closed string sector is e −λ . This is the 12 basis of string perturbation theory in the form of worldsheet Feynman diagrams, which we will see again later in the identification of the open and closed string couplings in matrix models. As we will see in the following section, the Virasoro operators Ln generate local conformal transformations on the worldsheet. If the worldsheet theory is to remain conformally invariant at the quantum level, then the Virasoro algebra should remain a symmetry algebra of the theory, and provides a way to carry out the quantization in the path integral formulation. This is an application of BRST quantization to string theory, the general point of which is to use the cohomology of a symmetry algebra to figure out the Hilbert space of the quantum theory. 1.4 Conformal Field Theory One of the reasons why so much progress can be made in string theory is that the action (1.1) defines a two-dimensional conformal field theory (CFT). Such theories have larger symmetry groups than generic quantum field theories, making them easier to solve. Some CFTs are actually easier to solve than others — in particular there is a class of CFTs referred to as minimal models which are simpler than the CFT defining the bosonic string, and thus provide an interesting opportunity to construct a toy string theory model. We continue to work in Euclidean signature. Define the complex coordinates z = σ 1 + iσ2 and z = σ 1 − iσ2 , with the corresponding derivatives ∂ = 1 2 (∂1 − i∂2), ∂ = 1 2 (∂1 + i∂2). (1.14) The metric has components gzz = gzz = 1 2 and gzz = gzz = 0, giving the invariant line element ds2 = dzdz. The (local) conformal group in two dimensions consists of holomorphic reparametrizations of the coordinates z → f(z), z → f(z), (1.15) 13 leaving the metric invariant up to an overall Weyl factor ds2 → ∣f ′ (z)∣2dzdz. In this coordinate system the string action in Minkowski space is S = 1 2πα′ ∫ d 2 z ∂Xµ ∂Xµ, (1.16) and the classical equation of motion is ∂∂Xµ (z, z) = 0. Conformal field theories are most commonly quantized via the path integral. As usual, we can think of Xµ as an operator. Using standard arguments, one can show that an operator equation corresponding to the equation of motion is 1 πα′ ∂∂Xµ (z, z)X ν (z ′ , z ′ ) = −η µνδ 2 (z − z ′ , z − z ′ ). (1.17) There is an apparent tension here created by the fact that the portion ∂∂Xµ (z, z) on the left hand side is the classical equation of motion, which we would naively expect to vanish. But as it turns out, the expectation value ⟨∂∂Xµ (z, z)Xν (z ′ , z ′ )⟩, defined with respect to the path integral, only vanishes if the second X insertion is not at coincident points. This introduces the need for field normal ordering, or the operator product expansion (OPE) ∶ X µ (z1, z1)X ν (z2, z2) ∶ = X µ (z1, z1)X ν (z2, z2) + α ′ 2 η µν ln ∣z12∣ 2 , (1.18) where z12 ≡ z1 − z2. The more general statement of the operator product expansion is that any two operators evaluated close to one another can be approximated by a sum of other local operators. If Oi represents some set of operators in the theory, then Oi(z1)Oj(z2) = ∑ k cijk(z12)Ok(z2) (1.19) 14 The interactions amongst the operators of the theory are strongly constrained by conformal invariance. In order to further discuss these constraints, we must look at the how the operator content is organized. A primary field is a local operator O(z, z) with the following behavior under a general conformal transformation of the form (1.15) O ′ (z ′ , z ′ ) = f ′ (z) −h f(z) −h˜ O(z, z). (1.20) The numbers (h, h˜) are the conformal weights of O. The sum ∆ = h + h˜ is the scaling dimension (also called conformal dimension) and the difference j = h − h˜ is the spin. When focusing on just the holomorphic or anti-holomorphic sector of the theory it is common to use weight and dimension interchangeably. Conformal invariance severely restricts the two-point function of primary operators. For example, take two primary operators O1 and O2. If their conformal dimensions are not the same they have vanishing two-point function. If they are the same, let their weights be h and h˜. Then ⟨O1(z1, z1)O2(z2, z2)⟩ = C12 z 2h 12 z 2h˜ 12 , (1.21) where C12 is a scalar that depends on the two operators, which is typically set to 1 by normalizing the operators. The three point function does not require the operators to have the same weights, but it is still constrained by to be a function of the separations of the insertions raised to powers determined by the weights of the operators. The vanishing of the trace of the energy-momentum tensor translates to complex coordinates as Tzz = 0, and the its conservation implies ∂Tzz = ∂Tzz = 0. It is convenient to drop the subscripts on T and T˜ since each function is holomorphic or anti-holomorphic. The OPE of T(z) with a primary operator O with holomorphic weight h is T(z)O(0, 0) = h z 2 O(0, 0) + 1 z ∂O(0, 0) + . . . , (1.22) 15 where the dots represent terms analytic in z. Evidently the energy-momentum tensor is able to sense the conformal properties of primary operators. This is borne out more fully by conservation relations called Ward identities, which indicate that the generator of an infinitesimal (holomorphic) conformal transformation with parameter ϵ(z) is Qϵ = 1 2πi ∮ dz ϵ(z)T(z). (1.23) In complex coordinates the Fourier mode decomposition of the energy-momentum tensor is written T(z) = ∑ n∈Z Ln z n+2 , T˜(z) = ∑ n∈Z L˜ n z n+2 , (1.24) where Ln and L˜ n are the Virasoro generators. This implies that the generator Qϵ can be written Qϵ = ∑ n∈Z ϵnLn, (1.25) where ϵ(z) = ∑n∈Z ϵnz n+1 . Hence the Virasoro algebra is actually intimately connected to the conformal invariance of the theory. Primary operators transform under the Virasoro generators as [Ln,O(z, z)] = h(n + 1)z nO(z, z) + z n+1 ∂O(z, z), (1.26) with a similar relation between L˜ n and anti-holomorphic quantities. In the same way that we introduced a Hilbert space for the string by starting with a vacuum or ground state ∣0⟩, we can do the same here. The operator O with weights (h, h˜) creates an asymptotic state in the standard quantum field theoretic sense, which in complex coordinates is ∣Oin⟩ = lim z,z→0 O(z, z) ∣0⟩ ≡ ∣h, h˜⟩ (1.27) 16 It can be shown that L0 ∣h, h˜⟩ = h ∣h, h˜⟩ and Ln ∣h, h˜⟩ = 0 for n > 0. Using the Virasoro algebra [L0, L−n] = nL−n, we can see then that L−k1L−k2⋯L−km ∣h, h˜⟩ is also an eigenstate of L0 with eigenvalue h ′ = h + ∑ m i=1 ki . Such a state is called a descendant of the state ∣h, h˜⟩ and the sum ∑ m i=1 ki is called its level5 . The set of states obtained by acting on a state ∣h, h˜⟩ created by a primary operator with the Virasoro generators is a representation of the Virasoro algebra called a Verma module, and is a subspace of the full Hilbert space. It is for this reason that studying the representations of the Virasoro algebra is so important in conformal field theory: the physical Hilbert space is built out of them. Denote the holomorphic Verma module generated by ∣h⟩ as V (c, h), where c is the central charge of the CFT. The basis states of V (c, h) are the descendants of ∣h⟩. There is no guarantee that the norms of these basis states are positive, and such states are called ghosts6 in the string theory context. A unitary physical theory is required to have no negative norm (ghost) states. For example ⟨h∣LnL−n∣h⟩ = [2nh + cn(n 2 − 1) 12 ] ⟨h∣h⟩. (1.28) If c < 0 the norm becomes negative for large enough n, which means that modules with negative central charge are non-unitary. Similarly, modules that have h < 0 are also non-unitary. Label the basis states by ∣i⟩. Then we introduce the Gram matrix of basis state inner products with matrix elements Mij = ⟨i∣j⟩. A general state in V (c, h) will have negative norm if and only if M has at least one negative eigenvalue. The Gram matrix is block diagonal, with blocks corresponding to basis states with the same level. Denote the block with level N by M(N) . Then the determinant of this block is given by the Kac determinant detM(N) = αN ∏ r,s≥1 rs≤N [h − hr,s(c)] p(N−rs) , (1.29) 5This is the ‘level’ in ‘level matching’ referenced in the closed string discussion above. 6Not to be confused with Fadeev-Popov ghosts. 17 where p(N −rs) is the number of partitions of the integer N −rs and αN is a positive constant independent of h and c whose exact form is not necessary here. While we were initially motivated by unitarity to study the Gram matrix, we have gained something else as well. Notice that if h = hr,s(c) for some r, s there will be at least one null state, i.e. a nonzero state with zero norm, at level rs. A Verma module corresponding to such a value of h is called a reducible module. The impact of this requires us to zoom out a bit. Recall that the operator content of a CFT is not limited to the states in a single Verma module. It turns out that the OPE of any operator with an operator from a reducible module is truncated, making it simpler than it otherwise would have been. This is to say that if the CFT has null states, the operator algebra is constrained. Moreover, one finds that the set of conformal families associated with reducible modules is closed under the OPE. However, there can still be an infinite number of such conformal families in the theory for arbitrary central charge. This truncation is escalated if the central charge is given by c = 1 − 6 (p − q) 2 pq , (1.30) where p, q are relatively prime integers. There are special weights given by hr,s = (pr − qs) 2 − (p − q) 2 4pq , (1.31) which yield an infinite number of null vectors in V (hr,s, c). The operator algebra is consequently truncated much more severely than before. The result is that there is only a finite number of conformal families — corresponding to a finite number of conformal weights restricted by 1 ≤ r < q and 1 ≤ s < p — that close under the OPE. There ends up being only (p−1)(q −1)/2 distinct primaries due to the reflection property Op−r,q−s = Or,s. 18 A CFT with central charge given by (1.30) and with conformal weights (1.31) is called the (p, q) minimal model. The operator content is dramatically limited compared to most other CFTs. For example, despite the simplicity of the free boson, which we have seen is used to define the free string in flat space, it still has an infinite number of primary fields. 1.5 Liouville Theory And Minimal String Theories The goal of this section is to review a toy model construction called minimal string theory, which roughly consists of coupling a minimal CFT to Liouville theory. We’ll begin with a general justification for the presence of Liouville, followed by a brief discussion of its key details. Minimal string theory, despite being dramatically simplified by the finite number of matter CFT primaries, still contains many of the interesting features of critical string theory like branes. They will be of central importance to the matrix model results presented later. This discussion follows [28, 35]. Consider a path integral Z = ∫ DXDg e−SM[X;g]− µ0 8π ∫Σ √g , (1.32) where SM is a conformally invariant action for matter coupled to a two-dimensional surface Σ with fluctuating metric g. The volume term in the action is coupled to a bare cosmological constant µ0. This is the general blueprint for a ‘string theory’: a conformally invariant matter theory coupled to two-dimensional quantum gravity. In the bosonic string the matter theory was given by the action (1.1), and the matter fields represented the embedding of the worldsheet in the target spacetime. Recall that this theory has an enormous amount of gauge symmetry in the form of diffeomorphism invariance and Weyl rescaling. Recall also the comment made before about how using Weyl rescaling to 19 put the metric in a desirable form would have ramifications later on. The issue introduced is that the measures DX and Dg are not invariant under the transformation g → e φ g. In fact DeφgX = e cmatter 48π SL(φ)DgX, (1.33) where cmatter is the central charge of the matter theory and SL(φ) is the Liouville action SL(φ) = ∫ Σ √ g ( 1 2 ∂φ ⋅ ∂φ + Rφ + µeφ ) . (1.34) The anomaly for the metric measure Dg is handled by introducing the Fadeev-Popov ghosts (b, c) mentioned in the description of the bosonic string, as well as an integral over the moduli of Σ and the Liouville mode φ. The ghosts’ path integral measure is also anomalous, given by Deφg(b, c) = e − 26 48π SL(φ)D(b, c). (1.35) This is not yet the end of the story, because not even the measure Dφ is invariant. The overall effect is to introduce a Liouville action with different coefficients on each term. In addition to the matter and ghost actions, we have the following term in the path integral e −∫Σ √g(a∂φ ˜ ⋅∂φ+˜bRφ+µecφ˜ ) , (1.36) where the constants a, ˜ ˜b, and c˜ are determined to maintain diffeomorphism invariance. The result is ˜b ∝ 25 − cmatter, a˜ = 1 2 ˜b. (1.37) 20 The crucial outcome here is that the action defining the path integral is still conformally invariant. The total central charge of the CFT is ctotal = cmatter +cL +cghosts, where cL = 26−cmatter and cghosts = −26. Thus the total central charge is vanishing, which ensures overall conformal invariance. What we have just seen is that coupling CFT matter to two-dimensional quantum gravity and gauge fixing produces another CFT consisting of Liouville plus the matter and ghosts. The normalization we will choose for the Liouville action makes it SL = 1 4π ∫ Σ √ g (∂φ ⋅ ∂φ + QRφ + µe2bφ) , (1.38) where Q = b+b −1 is interpreted as a background charge when we set g to be flat. In this normalization the Liouville central charge is cL = 1 + 6Q2 . The only primary operators in this theory are vertex operators Vα = e 2αφ which have dimension hα = −( Q 2 − α) 2 + Q2 4 . Comparing this to the discussion of reducible module, when αr,s = (1 − r)b −1 + (1 − s)b there are null vectors in the Verma module V (hα, cL). By the same mechanism as before, if we choose these values of α we can constrain the operator algebra, which allows for the theory to be solved. In order to construct minimal string theory, we choose the matter theory to be the (p, q) minimal model and set the Liouville parameter b 2 = p/q to ensure that ctotal = 0. The physical operators of the theory are built out of minimal CFT, ghost, and Liouville operators. The process of combining the operators of the matter CFT with Liouville operators is called gravitational dressing. Due to the distinguished role the letter b plays in Liouville theory, we will rename the ghosts to be c and c, which are of course not to be confused with the central charge c. An interesting distinction between this model and critical string theory is that minimal string theory has physical operators at all values of the ghost number7 . At ghost number 0, a general operator has the form Oˆ r,s = Lr,s ⋅ Or,se 2αr,sφ , (1.39) 7The ghost number of an operator roughly counts how many appearances of c and c there are. 21 where Lr,s is a polynomial in ghost operators and Virasoro generators, and r and s have the same constrained values as in the minimal CFT: 1 ≤ r ≤ p−1 and 1 ≤ s ≤ q−1. In terms of X = 1 2Oˆ 2,1 and Y = 1 2Oˆ 1,2, these operators are Oˆ r,s = Us−1(X)Ur−1(Y ), (1.40) where Un is the n th Chebyshev polynomial of the second kind. The operators X and Y are referred to as the generators of the ghost number 0 operators. At ghost number 1, an important class of operators are constructed similarly by taking a combination Or,se 2βr,sφ and multiplying by cc. The Liouville dimension βr,s = (p + q − ∣rq − sp∣)/√pq is chosen so that the conglomerate operator has dimension (1, 1). This combination, denoted Tr,s, is called a tachyon operator. They are generated via Tr,s = Oˆ r,sT1,1. The tachyon operators thus form a module of the ring of ghost number 0 operators, however they do not form a faithful representation. There is an additional constraint on the generators X and Y that arises from looking at the tachyons: Tp(X) − Tq(Y ) = 0, (1.41) where Tn is the n th Chebyshev polynomial of the first kind. We will return to this relation in the context of the branes in minimal string theory. 1.6 Superstrings From a phenomenological point of view there are two major shortcomings of bosonic string theory. First, although it does contain gauge fields and a spin-2 particle in its massless spectrum, it cannot fully contain the standard model. Unfortunately the standard model contains fermions like quarks and the electron, without which string theory has no hope of forming a realistic model of the universe. Second, we found that the spectrum of open and closed bosonic strings contained tachyons. We remarked at the time 22 that such states could indicate an instability in the proposed vacuum, and while that could be true a simpler task is to formulate a consistent theory whose spectrum does not include tachyons. From a practical point of view, sometimes calculations are impossible to do without supersymmetry, since the inclusion of the additional symmetry imposes extra constraints on the system. A natural place to begin on the worldsheet is to introduce a set of fermionic fields ψ µ and ψ˜µ . These fields can be combined into a two-component Majorana spinor ψ. The new action in conformal gauge with coordinates σ i = (τ, σ) is S = − 1 2π ∫ Σ d 2 σ [∂iX µ ∂ iXµ − iψ µ ρ i ∂iψµ] , (1.42) where once again we leave the range of µ = 0, . . . , D unfixed, ρ i represents the two-dimensional gamma matrices, and ψ = ψ †ρ 0 . In addition to the symmetries discussed for the bosonic string, this theory is also invariant under the infinitesimal transformations δXµ = ϵψµ δψµ = −iρi ∂iX µ ϵ, (1.43) where ϵ is a constant two-component fermionic spinor. This is an example of a supersymmetry transformation; loosely speaking8 this is an N = 1 supersymmetry. There are two boundary conditions for the fermions that are consistent with Lorentz invariance. They are called Ramond (R) and Neveu-Schwarz (NS) boundary conditions, and are displayed in Table (1.2). The two boundary conditions mandate different mode expansions. For example, in complex coordinates ψ µ (z) = ∑ r ψ µ r z r+ 1 2 , (1.44) 8There are different levels of specificity one can choose when counting supersymmetry. Since there is one two-dimensional spinor parameter ϵ here, it is fair to call this N = 1. However spinors in even dimensions are reducible, and so there are two independent Weyl spinor parameters. Therefore this can also be called N = (1, 1) supersymmetry. 23 Table 1.2: The different boundary conditions for the worldsheet fermions. Type Boundary Condition Ramond ψ µ (τ, 0) = ψ˜(τ, 0) ψ µ (τ, π) = ψ˜µ (τ, π) Neveu-Schwarz ψ µ (τ, 0) = −ψ˜µ (τ, 0) ψ µ (τ, π) = ψ˜µ (τ, π) where r ∈ Z + 1 2 for the R condition, and r ∈ Z for the NS condition. In canonical quantization, the modes will obey the anti-commutation relation {ψ µ r , ψν s } = {ψ˜µ r ,ψ˜ν , s} = δr+sη µν . (1.45) The modes in each sector will have separate and unique impacts on the spectrum of the string. In particular, the open string ground state for each type of mode will be dramatically different. In both cases we may again choose the ground state to satisfy ψ µ r ∣0⟩ = 0 for all r > 0. There is no NS mode with r = 0, and the ones with r < 0 will be fermionic raising operators and hence can only be applied once. The R sector has D modes with r = 0, all satisfying {ψ µ r , ψν 0 } = 0. Hence each mode with r = 0 does not change the mass/energy of the state, making ψ µ 0 ∣0⟩ a ground state for each value of µ. Moreover, the modes ψ µ 0 form a representation of the Dirac algebra, and thus the R ground states must as well. Recall that the spectrum of the closed string theory roughly involves taking a tensor product of two open string spectra, and therefore we must have two copies of the fermion boundary conditions. Thus we get different possible closed string spectra corresponding to pairs of R and NS. We have another way of classifying the states, which involves the worldsheet fermion number operator e iπF . Each time a mode ψ µ r is applied it changes the value of F by 1, which switches the exponential between 0 and 1. We denote the part of the spectrum with F even by NS+ and R+, and the part with F odd with a minus subscript. In 24 order to gauge fix, we will once again need to introduce a family of ghosts to the theory, some of which will be fermionic, which will necessarily change the character of the ground states. In particular, it turns out that e iπF ∣0⟩NS = −∣0⟩NS when the ghosts are taken into account, meaning the full NS ground state is in NS−. One can also show that the NS− ground state provides the only tachyon contribution. The closed string spectrum can be described by choosing combinations of NS± and R± states, a process referred to as the GSO projection. In order to preserve desirable symmetries and features of the theory, it will not be possible to include all possible combinations, and the choices made will lead to distinct theories. One class of theories include mixed R-NS (or NS-R) sectors, which include spacetime fermions, and are referred to as type II. Their spectra are described by IIA ∶ (NS+,NS+) ⊕ (NS+, R−) ⊕ (R+,NS+) ⊕ (R+, R−), IIB ∶ (NS+,NS+) ⊕ (NS+, R+) ⊕ (R+,NS+) ⊕ (R+, R+). (1.46) These theories both exclude the tachyon and are two of the most widely studied types of superstring theories. The worldsheet fermion number has an interpretation in terms of spacetime chirality, which can be seen by constructing the target space Lorentz generators on the worldsheet. Examination of the spectra shows that the IIA theory is non-chiral in target space, while the IIB theory is chiral. There is another class of theories that preserve certain desirable symmetries. They have no mixed R-NS or NS-R sectors, and thus no spacetime fermions, and are referred to as type 0. Their spectra are summarized as 0A ∶ (NS+,NS+) ⊕ (NS−,NS−) ⊕ (R+, R−) ⊕ (R−, R+), 0B ∶ (NS+,NS+) ⊕ (NS−,NS−) ⊕ (R+, R+) ⊕ (R−, R−). (1.47) One major difference between the type 0 and type II string theories is that the type 0 ones have a tachyon in the spectrum, which makes them undesirable in target space dimension D > 2. The fact that the tachyon 25 is massless when D = 2 is one reason why the type 0 theories are most commonly encountered in twodimensional string theories. No matter what choice is made for the boundary conditions, the energy-momentum tensor must be modified to accommodate the presence of the fermions. In particular, the new fermionic piece can be mode-expanded similarly to how the bosonic piece was expanded previously. In complex coordinates the overall energy-momentum tensor will still have Tzz = T(z) and Tzz = T˜(z). Denoting the z-dependent fermionic piece by TF , TF (z) = ∑ r Gr z r+ 3 2 , (1.48) where r takes the same values in the R and NS sectors as for ψ µ . The modes Gr are the fermionic analog of the Virasoro generators Ln. Together, they form two related superalgebras consisting of the Virasoro algebra (1.10) supplemented with {Gr, Gs} = 2Lr+s + c 12 (4r 2 − 1)δr+s, [Ln, Gr] = n − 2r 2 Gn+r. (1.49) The algebra with r, s ∈ Z is called the Ramond algebra and the one with r, s ∈ Z + 1 2 is the Neveu-Schwarz algebra. There are two analogous algebras for the z modes. These algebras are generically referred to as the N = 1 super Virasoro algebra. 1.6.1 Minimal Superstrings The family of N = 1 superconformal minimal models is parametrized by two integers p, q subject to a number of constraints: p, q ≥ 2 and one of the following: – p and q are odd and coprime, or 26 – p and q are even, p 2 and q 2 are coprime, and (p − q)/2 is odd. The central charge is cˆ = 1 − 2(p − q) 2 pq . (1.50) The primary operators are still labelled by two integers r and s, with 1 ≤ r ≤ p − 1 and 1 ≤ s ≤ q − 1. The primaries are also still subject to the reflection property Op−r,q−s = Or,s. The distinction between the R and NS versions of the super Virasoro algebra leads to differentiation between R and NS sector operators. The R operators have r − s odd, while the NS ones have r − s even. The operator dimensions are given by hr,s = (pr − qs) 2 − (p − q) 2 8pq + 1 − (−1) r−s 32 . (1.51) One operator of note is Op 2 , q 2 , which only exists in the theories with (p, q) even. It corresponds to the Ramond ground state. Recall that the ground state in the Ramond sector is special because it furnishes a representation of the Dirac algebra and is supersymmetric. Hence the lack of its existence in the (p, q) odd theories means that they break the supersymmetry. The operators obey the same algebraic relations and constraints as in the non-supersymmetric theory, and so the matter algebra is once again generated by the operators O1,2 and O2,1. Note that both of these operators are in the Ramond sector. Liouville theory has an N = 1 supersymmetric generalization, which is also a superconformal theory. The key bosonic field will remain φ. The central charge is cˆ = 1 + 2Q 2 , (1.52) where again Q = b + b −1 . The basic primaries of super Liouville are the NS operators Nα = e αφ and the R operators R ± α = σ ± e αφ. The operators σ ± are known as spin fields and have dimension 1/16, and the ± 27 superscript denotes the corresponding state’s fermion number. The dimensions of the NS and R primaries are h(Nα) = 1 2 α(Q − α), h(R ± α) = 1 2 α(Q − α) + 1 16 . (1.53) The primaries become degenerate when 2αr,s = (1 − r)b −1 + (1 − s)b, and r − s is even for the NS sector and odd for the R sector. The Ramond ground state is obtained by choosing α = Q 2 and has dimension h = cˆ 16 . An important thing to note is that when performing the GSO projection described above, we will keep only one of the Ramond ground state operators, say R − Q 2 . Considering super Liouville on its own, it turns out that this implies the Ramond ground state only exists when the Liouville cosmological constant µ is positive. Another way of saying this is that the Ramond ground state operator in the superconformal minimal model can only be supergravitationally dressed to produce the minimal superstring ground state when µ > 0. In order to couple the superconformal minimal model to super Liouville theory we once again set b = √ p/q. The tachyon operators Tr,s are once again built out of the matter operators Or,s and the Liouville primaries with weights 2βr,s = (p + q − ∣rq − sp∣)/√pq, as well as appropriate superghosts. In addition to the Liouville operators, the matter operators will also carry a ± superscript to indicate its fermion number. The GSO projection in the combined theory will involve a correlation between the matter and Liouville fermion numbers, and the outcome implies a shift in interpretation of the theory when the bulk cosmological constant µ changes sign. To preview what lies ahead, in the 0A theory there can be D-branes for µ > 0 and RR flux for µ < 0. The opposite is true in the 0B theory. We conclude this discussion by noting the relationship between the 0A and 0B theories. Recall that a major difference between the A and B theories in type II superstrings was that A wasn’t chiral and B was. There is a sense in which, at least schematically, the two theories are related by ‘forgetting’ about chirality in the B theory to obtain the A one. That is precisely what happens for type 0 superstrings. There is a 28 left-moving spacetime fermion number (−1) FL that acts in the 0B theory; if we orbifold by this operator, we obtain the 0A theory. This will have a dramatic effect on the matrix model interpretation in the 0A case. 1.7 Branes 1.7.1 General Context Although we will not be concerned with the branes of critical string theory, they offer the most intuitive insights into higher dimensional extended objects in string theory. By studying a special behavior under dimensional reduction, called T-duality, in both the open and closed string sectors of a critical string theory one finds that the open strings act rather unexpectedly. In a certain limit, the closed strings actually behave like the dimensional reduction didn’t happen, but with modified properties. That is, the apparent number of dimensions remains unchanged. The same limit applied in the open string sector decreases the apparent number of dimensions. The interpretation is that when p coordinates are dimensionally reduced, the endpoints of the open strings are actually constrained to a codimension-p hyperplane, called a D(p + 1)-brane9 . Clearly D-branes are intimately connected to open string physics, and offer a different perspective on string theory as a whole. Let us be more specific, returning once again to the critical bosonic string. Suppose that the coordinates Xm = {X25, X24, . . . , Xp+1 } are affected by the T-duality transformation. The index m will be taken to run through these directions. Then the massless vector states of the open string are split into α µ −1 ∣0; k⟩ ↔ A µ (X a ) & α m −1 ∣0; k⟩ ↔ Φ m(X a ), (1.54) 9The general class of Dp-branes is often abbreviated to just D-branes. 29 where µ, a = 0, 1, . . . , p. The first portion is interpreted as a gauge field A living in the worldvolume of the brane. The second portion is interpreted as a family of scalar fields Φ that describe the directions transverse to the brane, i.e. its embedding in the target space. The total description of the brane involves a gauge theory living in its worldvolume and some scalars describing its shape in the ambient spacetime. This discussion ignored the closed string sector of the theory, but recall that the first non-tachyon massless states included the graviton, which is interpreted as a fluctuation of the target space geometry. A string theory with open strings will have closed strings as well, and so we expect that there should be metric fluctuations that impact the brane. The overall effect is to turn the scalars Φ into fluctuating fields, instead of just fixed values. Recall that the open and closed string sectors of the theory came originally from boundary conditions imposed on the worldsheet theory. The physics of the ‘interior’ portions of open and closed strings should therefore be the same. An alternate approach to string perturbation theory involves setting up a target space action for the massless closed string fields discussed before (which includes the target space metric). But as we just discovered, D-branes are extended objects that exist in the target spacetime. Hence there should be an interplay between the closed strings and the D-branes naturally associated to the open strings. In very special circumstances one can establish a duality between open string and closed string descriptions of the same physics. Such a duality will be explored in the context of matrix models in the body of this thesis. A more famous example of an open-closed duality is the AdS/CFT correspondence [42]. The original construction relates the very things described in the preceding paragraph, in the context of type IIB superstring theory. When N D3-branes are placed on top of one another in the target space the gauge symmetry of the gauge field A is enhanced, in this case from U(1) N to SU(N). This particular brane construction preserves a certain amount of the starting supersymmetry, such that the full gauge theory describing the massless modes in the open string sector is 4-dimensional N = 4 super Yang-Mills (SYM) 30 theory10. This theory has a superconformal symmetry, making it much easier to analyze than ordinary 4-dimensional Yang-Mills. On the other hand, one can use the background field approach in the closed string sector to describe the same setup. The low-energy physics of this theory is described by IIB supergravity. The metric solution to this supergravity very near the branes is AdS5×S 5 , with certain geometric information like the AdS radius being determined by N. Since the two approaches are describing the same stack of D-branes, one can establish a dictionary that maps quantities like correlation functions in the supergravity theory to ones in the gauge theory, and vice versa. 1.7.2 Branes In Non-Critical String Theory There are branes in two-dimensional string theory, although they often lack an intuitive target space interpretation. The D-branes in minimal string theory are best described using the boundary state formalism: the open string boundary conditions that describe the brane are translated into operator statements in the closed string sector, which are subsequently turned into states via the operator-state correspondence in CFT. The state describing a D-brane in minimal string theory will be built out of a state from Liouville and a state from the minimal model, much like how the CFT operators were (super)gravitationally dressed to produce the minimal string operators. First, consider the non-supersymmetric case. D-branes in Liouville theory come in two varieties. The first, called Fateev-Zomolodchikov-Zomolodchikov-Teschner (FZZT) branes, comes from the straightforward inclusion of a boundary in the surface Σ on which the Liouville action is defined. One then includes the boundary interaction SB = µB ∮ ∂Σ e bφ , (1.55) where µB is the boundary cosmological constant. This introduces a Neumann condition on the Liouville field: calling the ‘spatial’ variable σ, one requires ∂σφ = −2πbµBe bφ. These branes are extended in the φ 10This is typically abbreviated to just N = 4 SYM. 31 direction, starting at φ = −∞ and dissolving at some finite position φ ∼ −b −1 log µB. The corresponding boundary state for an FZZT brane with cosmological constant µB is a Cardy state11 ∣σ⟩ labelled by a nondegenerate Verma module with dimension hα, where α = Q 2 + iσ 2 and σ is related to the cosmological constant by µB = cosh πbσ. The other variety of D-brane in Liouville theory are the Zomolodchikov-Zomolodchikov (ZZ) branes. They form a discrete family parametrized by integers m, n ≥ 1 and have boundary states∣m, n⟩ corresponding to degenerate Verma modules with weight hα, where αm,n = Q 2 − 1 2 (mb−1 + nb). The simplification caused by the fact that the Verma modules are degenerate allows us to relate the ZZ branes to the FZZT branes via ∣m, n⟩ = ∣σ(m, n)⟩ − ∣σ(m,−n)⟩. Geometrically, they are localized in the region φ = +∞. Although there is in general no Lagrangian formulation for the minimal CFTs, we can still consider the analog of including a boundary. The Cardy states in minimal models are in one-to-one correspondence with the Verma modules in the theory. Thus they are also labelled by two integers k, l and are constrained to take a finite number of values. These states are conventionally called ∣k, l⟩. Note that there is only one variety of ‘brane’ in the minimal CFT. The D-brane states of minimal string theory will be given by the tensor product of an FZZT or ZZ brane state from Liouville with a Cardy state in the minimal model. Due to additional constraints coming from BRST cohomology, analysis of the one-point functions of the theory, and the fact that the Liouville parameter b 2 = p/q is rational, the total set of independent D-branes in minimal string theory is actually smaller than one would naively expect. They are displayed in table 1.3. The ZZ branes obey the same reflection property that the primaries do, meaning there are only (p−1)(q−1)/2 of them. The independent ZZ branes are called the principal branes. Although a target space interpretation is not as readily accessible here as it was in the critical bosonic or supersymmetric cases, some sense can be made of the present situation, and it will lead us toward 11The exact definition of a Cardy state is not important here, just that it is a state arising in treating conformal boundary conditions carefully. 32 Table 1.3: Minimal string D-branes Type State Parameter Values FZZT ∣z⟩ ⊗ ∣1, 1⟩ z = cosh √πσ pq ∈ C ZZ ∣m, n⟩ ⊗ ∣1, 1⟩ 1 ≤ m ≤ p − 1, 1 ≤ n ≤ q − 1, qm − pn > 0 considering double scaled matrix models. The insight comes from the disk amplitude Z(µB) of the FZZT brane. Defining x = µB and y = ∂µB Z(µB), one finds that x and y obey the algebraic relation Tp(y) − Tq(x) = 0, (1.56) which is precisely the relationship between the generators of the ghost number 0 operators in the theory. This relationship also defines a Riemann surface Mp,q, the properties of which determine the amplitude Z. Hence there is a sense in which the algebraic properties of the operators of the theory inform the geometric properties of the brane. The specific relationship between µB and b allows us to use the parameter z that labels the FZZT brane states in an interesting way. Notice that setting x = Tp(z) and y = Tq(z) solves the algebraic relation defining the surface. Hence z is a uniformizing coordinate for the Riemann surface Mp,q. As we will come to find out, these relationships will match ones related to fundamental quantities in a double scaled Hermitian matrix model, for which p = 2 and q is an odd integer. The quantity y will be a complexified version of the spectral density of the matrix model. The quantity x will be related to the eigenvalues of the random matrix. The branes of type 0 minimal superstrings share many similarities with their non-supersymmetric counterparts, but ultimately are more complicated due to the differences between the NS and R sectors, and the differences in the GSO projections. There are still the two types of brane (FZZT and ZZ) with the 33 same spatial interpretations in terms of the Liouville coordinate φ. The brane states are also still labeled in terms of the boundary cosmological constant µB and the matter indices r, s, in addition to three new Z2-valued labels whose exact details are not relevant to our discussion. There is an analogous story in the type 0 theories involving the algebraic relation (1.56). The 0B theories are most naturally identified with the same type double scaled Hermitian matrix models as the non-supersymmetric case12. The orbifolding procedure that maps 0B → 0A implies that xA = x 2 B . This means that the eigenvalues in the matrix model related to 0A theories must be non-negative — indeed this will prove to be the case. 12We are being purposefully vague here without having done a discussion of double scaled matrix models. The 0B theories are actually different from their non-supersymmetric counterparts, though fundamentally they share a very similar description. 34 Chapter 2 The Kortewig-de Vries Hierarchy And Intersection Theory In this chapter we will review the Kortewig-de Vries (KdV) hierarchy of differential equations, with a focus on aspects that will be useful later. While at face value this is a stark departure from the preceding discussion of string theory, it turns out that our minimal string theories are intimately related to KdV, which will be borne out most clearly via the connection between matrix models and the KdV hierarchy. The review of integrability and KdV will draw from [43, 44]. After the review of the KdV hierarchy, we will pivot slightly to a brief overview of intersection theory on the moduli space of stable Riemann surfaces. Although the specific mathematical details will not be particularly useful in the rest of the dissertation, we will encounter several very important concepts that connect this topic in algebraic geometry to random matrix theory. The first is the celebrated WittenKontsevich theorem [45, 46], which relates the generating function of certain intersection numbers to the KdV hierarchy. The second is Mirzakhani’s recursion relation for the Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces [7]. Generalizations of these intersection numbers and volumes, as they pertain to certain classes of double scaled matrix models, will be discussed in detail in Chapter 6. 35 2.1 Integrability Consider a Hamiltonian system with n generalized positions qi , n conjugate momenta pj , and Hamiltonian H(qi , pj). Hamilton’s equations of motion are expressed via the Poisson bracket as q˙i = {qi , H} = ∂H ∂pi , p˙i = {pi , H} = − ∂H ∂qi , (2.1) with the canoncial bracket {qi , pj} = δij . A function f(qi , pj) is called an integral of motion if {H, f} = 0 when the equations of motion are satisfied. Two functions f and g on the phase space are in involution if {f, g} = 0. This system is completely integrable if there exist n independent integrals of motion Ii that are in involution with one another. The functions Ii are also often referred to as conserved quantities. Given n integrals of motion it is possible to perform a canonical transformation to action-angle variables (q, p) ↦ (I, α) such that all of the time evolution is linear and is done by the angles αi . Solving the original system then amounts to inverting the canonical transformation to get q(I, α, t) and p(I, α, t). There is no general way of telling if a given Hamiltonian is integrable, and even if it is, there is no guarantee that the canonical transformations can be explicitly determined, let alone inverted. An alternate route to solving an integrable system is via the Lax formalism. In the finite dimensional case, a Lax pair is a pair of matrix valued functions Q(q, p) and P(q, p) satisfying the Lax equation Q˙ = [Q, P], such that the Lax equation is equivalent to Hamilton’s equations. The size of the matrix is not guaranteed to be n × n, and a priori there is no way to determine it. Once a Lax pair has been obtained, the integrals of motion are obtained from ˜Ik = trQk . Not all of the ˜Ii will be independent, as guaranteed by Cayley’s theorem, but they will all be conserved. The set of n independent integrals of motion Ii can be chosen from the ˜Ii . We will return to the Lax formalism later in the specific contexts of the KdV hierarchy and double scaled matrix models. 36 2.2 The KdV Hierarchy The KdV equation is a nonlinear partial differential equation for a function of two variables u(t, x), given in one normalization by ∂u ∂t = 2u ∂u ∂x − 1 3 ∂ 3u ∂x3 . (2.2) This equation was originally derived in 1895 to describe shallow water waves in a channel. Perhaps surprisingly, the equation admits soliton solutions representing waves whose shape do not change as they propagate, i.e. non-dispersive solutions. Moreover, two solitons in this system can scatter through each other without changing shapes. This implies that there must be an infinite number of conserved quantities, and indeed this turns out to be true. The KdV equation (2.2) has several useful symmetries: 1. Translations: x → x + c1 and t → t + c2 where c1 and c2 are constants, 2. Scaling: x → cx, t → c 3 t, and u → c −2u, where c is a constant, and 3. Galilean boost: x → x + vt, t → t, and u → u + v, with v a constant. In fact using these symmetries one can change the normalization in (2.2). In the sense of a mechanical system, the KdV equation has an infinite dimensional phase space. However, this system is still Hamiltonian, and with some modification we can invoke the finite-dimensional formalism laid out above. To begin, the generalized coordinate will be the function u(x, t), and derivatives with respect to it will become functional derivatives. The next step is to find a Poisson bracket and Hamiltonian such that the KdV equation is Hamilton’s equation of motion. If {u(x), u(y)} = f(x, y) for some distribution f, then the Poisson bracket of two functionals A[u] and B[u] is {A[u], B[u]} = ∫ dxdy δA δu(x) f(x, y) δB δu(y) . (2.3) 37 It turns out that there are two pairs of Hamiltonian and Poisson bracket that reproduce the KdV equation. They are {u(x), u(y)} = ∂ ∂xδ(x − y), H2 = ∫ ∞ −∞ dx [ 1 3!u 3 − ( ∂u ∂x) 2 ] {u(x), u(y)} = [ ∂ 3 ∂x3 + 1 3 ( ∂ ∂xu(x) + u(x) ∂ ∂x)] , H1 = ∫ ∞ −∞ dx 1 2 u(x) 2 . (2.4) The naming of the two Hamiltonians will be justified shortly. Systems with two choices of Poisson bracket and Hamiltonian are called Bi-Hamiltonian. That the KdV equation has an infinite number of conserved quantities can be proven by appealing to a related differential equation, called the modified KdV (mKdV) equation. The specifics of the proof and of the mKdV equation are ancillary to this discussion, so the interested reader may find the details1 in [43]. By recognizing that the KdV equation takes the form of a continuity equation, we immediately know that I0 = ∫ ∞ −∞ dx u(x), (2.5) is a conserved quantity. Moreover, I1 = H1 and I2 = H2 are conserved quantities as well2 . The rest of the integrals of motion Ik are generated by the recursion relation D δIk δu(x) = (D 3 + 1 3 (Du + uD)) δIk−1 δu(x) , (2.6) where D ≡ ∂/∂x. One can show that the conserved quantities are in involution with each other with respect to both of the Poisson brackets of the KdV equation. 1As well as a fun historical account of the conserved quantities of the KdV equation. 2 Judging based the complexity of the Poisson brackets and their corresponding Hamiltonians in (2.4), either pair could rightfully be labelled 1 and the other 2. For all intents and purposes, we are choosing to label the Hamiltonians by their increasing complexity. 38 Since two of the explicitly constructed integrals of motion are valid Hamiltonians, it is no stretch of the imagination to suppose that the other conserved quantities Ik are Hamiltonians as well, with their own evolution equations. This supposition leads to a hierarchy of differential equations called the KdV hierarchy. Since the Ik are in involution, each equation shares the same conserved quantities and is therefore integrable. Further, the evolution in each equation will commute with all of the other evolutions, so we may think of u(x, t0, t1, . . . ) as a function of x and infinitely many evolution parameters called KdV times, or times for short. The evolution equation in the k th time is given by ∂u ∂tk = D δHk+1 δu(x) . (2.7) A powerful formalism for determining the functional derivative in (2.7) was developed by Gelfand and Dikii, and makes use of their eponymous differential polynomials [47–49]. The k th Gelfand-Dikii polynomial R˜ k[u] is a polynomial in u and its x-derivatives up to the k th order, satisfying the recursion relation R˜′ k+1 = 1 4 R˜′′′ k − uR˜′ k − 1 2 u ′R˜ k, (2.8) where we have switched to denoting x-derivatives with primes3 . The first three Gelfand-Dikii polynomials are R˜ 0 = 1 2 , R˜ 1 = − 1 4 u, R˜ 2 = 1 16 (3u 2 − u ′′). (2.9) The tk-evolution equation is subsequently written ∂u ∂tk = R˜′ k+1 . (2.10) This form of the evolution equation will be referred to as a KdV flow, or flow for short. 3The tilde is used to denote a particular normalization used in [47]. This normalization will not be used throughout the entire thesis, but is still presented here as part of the historical development. 39 The KdV flows are generated by an infinite number of commuting vector fields defined by ˜ξk = ∞ ∑ l=0 R˜ (l+1) k ∂ ∂u(l) . (2.11) The superscripts denoted the number of x-derivatives. The derivatives can be thought of as being functional derivatives, but the dependence on x is not specifically important in the flows, so the partial derivative notation is more common in the algebraic approach to the KdV hierarchy. The k th flow can also be written ∂u ∂tk = ˜ξk+1 ⋅ u. (2.12) 2.3 The Lax Formalism As mentioned previously, finding the Lax pair corresponding to a mechanical system can be crucial for establishing and utilizing integrability when other methods fail. The Lax formalism can also be generalized to the infinite-dimensional formalism with u as the generalized coordinate, in which Q and P will be represented by differential operators. Part of the underlying motivation in the Lax formalism is finding a linear operator that is linearly dependent on the function u, and whose eigenvalues are constant under the nonlinear time evolution generated by one of the Hamiltonians in the KdV hierarchy. An operator whose eigenvalues are constant under time evolution is called isospectral in this context. Denote this operator by Q(u(x, t)) = Q(t). Then in analogy with the Heisenberg formalism in quantum mechanics, the operator Q(t) should be unitarily related to Q(0). Call this unitary evolution operator U(t). Then we denote its time derivative by 40 ∂U ∂t = P(t)U(t), where P is Hermitian or anti-Hermitian depending on convention4 . Then the condition for Q(t) to be isospectral under the KdV time evolution is the Lax equation ∂Q ∂t = [P(t), Q(t)]. (2.13) Since Q is linearly dependent on u (and no other functions), its partial time derivative will be a multiplicative operator proportional to the flow of u. Therefore in the k th system in the KdV hierarchy, the isospectral condition is [P, Q] = R˜′ k . (2.14) In particular, one can show that the appropriate choice for Q in any system in the hierarchy is Q = − ∂ 2 ∂x2 + u. (2.15) Since this operator is k-independent it must be the case that P depends on k, and so we will append a subscript k to keep track of that. One can show that Pk = (Q k+ 1 2 ) + satisfies the Lax equation. To understand this, we will make a small detour into the realm of pseudo-differential operators. Since Q is a second-order differential operator there is a natural way raise it to a half-integer power. For ease of notation, we will denote ∂/∂x ≡ ∂. Begin by setting Q 1 2 = ∂ + ∞ ∑ n=0 q−n∂ −n , (2.16) 4The operator P is the analog of the Hamiltonian in quantum mechanics, which is exponentiated to form the evolution operator. However in this case it will turn out that Q actually takes the form of a Schrodinger Hamiltonian. 41 where a negative power of ∂ represents an antiderivative, and q−n will be a polynomial in u and its derivatives. The generalization of the Leibniz rule to antiderivatives is given by ∂ −n f = ∞ ∑ i=0 ( −n i )f (i) ∂ −n−i , (2.17) where f is a function of x. Then by utilizing this generalized Leibniz rule, the coefficient functions q−n are determined term-by-term by imposing (Q 1 2 ) 2 = Q. The first several coefficient functions are q0 = 0, q−1 = − i 2 u, q−2 = i 4 u ′ , q−3 = − i 8 (u ′′ + u 2 ) . (2.18) Once we have Q 1 2 , the higher half-integer powers are obtained by Q k+ 1 2 = QkQ 1 2 . The + subscript denotes keeping only the non-negative powers of ∂. The general form of Pk is Pk = a ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ ∂ 2k+1 + k ∑ j=1 (bj(u)∂ 2j−1 + ∂ 2j−1 bj(u)) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , (2.19) where a is a constant overall normalization. The functions bj will be differential polynomials in u, and are obtained explicitly by solving the Lax equation. The exact form of Pk is not often used in Hermitian matrix models, but as we will show in Chapter 7 it is crucial to solving matrix models describing unoriented gravity. 2.3.1 Gelfand-Dikii Resolvent An important object that arises as an outcropping of the Lax formalism is the resolvent of the operator Q [47]. It has a representation in terms of a kernel function Rˆ given in bra-ket notation by Rˆ(x, y; ζ) = ⟨x∣(Q + ζ) −1 ∣y⟩. (2.20) 42 Explicit results for an asymptotic expansion of Rˆ in ζ are easiest to obtain along the diagonal x = y. See Appendix B for some calculations of the expansion on the off-diagonal. One finds Rˆ(x; ζ) = ∞ ∑ k=0 R˜ k ζ k+ 1 2 , (2.21) where once again R˜ k is the k th Gelfand-Dikii polynomial. Notice that Rˆ is implicitly a function of x through the function u. For the remainder of this discussion, we will exclusively consider the diagonal kernel function The function Rˆ(x; ζ) satisfies a nonlinear differential equation 4(u + ζ)Rˆ2 − 2RˆRˆ′′ + (Rˆ′ ) 2 = 1, (2.22) called the Gelfand-Dikii differential equation. The recursion relation (5.2) for the Gelfand-Dikii polynomials can be obtained by inserting the asymptotic expansion (2.21) into the Gelfand-Dikii equation. Define the vector field δ δu = ∞ ∑ k=0 (−1) k ( d dx) k ∂ ∂u(k) . (2.23) It can be shown that Rˆ and R˜ k satisfy the following identities involving δ/δu δ δu Rˆ(x; ζ) = ∂ ∂ζ Rˆ(x; ζ) δ δu R˜ k = −(k − 1 2 )R˜ k−1. (2.24) In fact these objects satisfy the simpler identities obtained by replacing δ/δu with ∂/∂u. The latter version of the identity ∂Rˆ/∂u = ∂Rˆ/∂ζ makes sense because, by definition, the resolvent depends on the sum (u + ζ). 43 The product of two resolvents, evaluated at different values of the spectral parameter ζ1, ζ2 combine to form a composite object Rˆ(x; ζ1)Rˆ(x; ζ2) = Pˆ′ (x; ζ1, ζ2), (2.25) which is directly related to the conserved quantities of a set of equations related to the KdV hierarchy, and as we will see in the following chapters, double scaled matrix models. A differential equation for the function u of the form n ∑ k=0 ckR˜ k = 0, (2.26) where ck are constants, is called a Novikov equation. The function Pˆ has a simultaneous asymptotic expansion in ζ1, ζ2 analogous to (2.21), in terms of differential polynomials Pk,l satisfying RkR ′ l = P ′ k,l. The objects Cl = n ∑ k=0 ckPk,l, (2.27) where l = 1, . . . , n, are conserved quantities for the Novikov equation defined by the constants ck. There is a natural way to associate vector fields to conserved quantities, described in detail in [47]. The vector field associated to Cl turns out to be precisely the KdV vector field ˜ξl . 2.4 Tau And Baker Functions The unconstrained operator L ≡ Q 1 2 given in (2.16) is related to a more general hierarchy of differential equations called the Kadomtsev–Petviashvili (KP) hierarchy. By imposing constraints on powers of L, for instance demanding (L 2 )− = 0 as we did for KdV, one obtains specific sub-hierarchies. It is possible to relate L to ∂ by a sort of similarity transformation L = ϕ∂ϕ−1 , ϕ = ∞ ∑ i=0 wi∂ −i . (2.28) 44 Each of the coefficient functions wi will generically be a differential polynomial in u. Introduce an auxiliary variable z and define ψ(t, z) = e ∑ ∞ k=1 tkz k ∞ ∑ i=0 wiz i . (2.29) The function ψ is called the formal Baker-Akhiezer function or wavefunction. It is convenient to define wˆ(t, z) = ∑ ∞ i=0 wiz i . The function ψ has a dual, called ψ ∗ , that arises from considering the formal conjugate of the dressing operator ϕ, called ϕ ∗ . Together, the functions ψ and ψ ∗ satisfy a bilinear identity that holds if and only if they are related to the distinguished operator L that defines a KP hierarchy. That is, knowing the wavefunction is tantamount to having solved the integrable hierarchy. Let us try to understand that statement. Introduce the shift operator acting on functions of the times tk and z by G(ζ)f(t, z) = f (t1 − 1 ζ , t2 − 1 2ζ 2 , t3 − 1 3ζ 3 , . . . , z) . (2.30) The function wˆ obeys the identity ∂ log ˆw(t, z) = (1 − G(z))w1(t). (2.31) The single function w1 contains a remarkable amount of information about the system, given that w1 is only one of infinitely many coefficient functions contained in wˆ, and hence in ψ. Moreover, the dressing operator ϕ facilitates a coordinate transformation between the algebra formed by u and its derivatives and the functions wi . Therefore understanding (2.31) will prove useful, as knowing the functions contained in wˆ tells us a lot about our integrable hierarchy via the wavefunction. 45 Naively, one would like to integrate (2.31) to obtain wˆ, but the logarithm leaves for the possibility of an indefinite factor depending on (t2, t3, . . . ). Introduce a function τ such that w1 = −∂ log τ . Then a landmark theorem by Sato states that log ˆw = (G(z) − 1) log τ , or wˆ(t, z) = τ (t1 − 1 z , t2 − 1 2z 2 , . . . ) τ (t1, t2, . . . ) . (2.32) The function τ is known as the τ -function related to the hierarchy. It satisfies an identity known as Hirota substitution, which states that the coefficient of ∂ −1 in L is ∂ 2 log τ . This identity will become very important in the following discussions of intersection theory and matrix models. In particular, we saw before that the coefficient of ∂ −1 for the KdV hierarchy was proportional to u. This implies that τ ∼ e∬ u , which we will see is the form of the partition function for a double scaled matrix model. The Baker-Akheizer function will also make an appearance later on. 2.5 Intersection Theory Let Mg,n be the moduli space of Riemann surfaces of genus g and n ordered punctures. Let Mg,n be the compactification of the moduli space obtained by adjoining surfaces with double points, called the Deligne-Mumford compactification. The points in Mg,n correspond to stable curves, or surfaces whose only singularities occur at double points and whose automorphism group is finite. Let Σ be a stable curve with n marked points (x1, . . . , xn), which are always chosen so that they do not coincide with double points. Thus, each xi has a complex cotangent space T ∗Σ∣xi that varies holomorphically with xi , giving a holomorphic line bundle L(i) over Mg,n. Let d1, . . . , dn be nonnegative integers such that ∑ n i=1 di = 3g − 3 + n. This is the dimensionality condition under which the intersection number ⟨ n ⋀ i=1 c1(L(i)) di ,Mg,n⟩ , (2.33) 46 where c1 is the first Chern class, may be nonzero. This is to say that the moduli space Mg,n has complex dimension 3g − 3 + n. We denote these intersection numbers by ⟨τd1⋯τdn ⟩, and define ψi = c1(L(i)) so that ⟨τd1⋯τdn ⟩ = ∫Mg,n ψ d1 1 ⋯ψ dn n , (2.34) where the wedge product is implicit. If r0 of the di are 0, r1 of them are 1, etc., it is conventional to write ⟨τd1⋯τd1 ⟩ = ⟨τ r0 0 τ r1 1 ⋯⟩. It proves useful to think about intersection numbers by reimagining them as correlation functions of some operators ψi in an abstract QFT. Consider a Lagrangian L0 dependent on some arbitrary set of fields, and a family of deformations defined by integrating ψi over some surface. The full Lagrangian is L = L0 − ∑ i ti ∫ ψi , (2.35) and the Euclidean path integral in this theory is written as Fg(ti) = ∫ D(Fields) e −L0 e ∑i ti ∫ ψi . (2.36) Expanding the deformation part of exponential prompts us to define the generating function of stable intersection theory on moduli space at genus g Fg(t0, t1,⋯) = ∑ {ni} ∞ ∏ i=0 t ni i ni ! ⟨τ n0 0 τ n1 1 ⋯⟩ g , (2.37) where the sequence of numbers ni is chosen so that 3g−3 = ∑i ni(i−1). The The full generating function is defined by F(t0, t1,⋯) = ∞ ∑ g=0 Fg(t0, t1,⋯). (2.38) 47 This leads us to the content of the Witten-Kontsevich theorem [45, 46]. Introduce the notation ⟪τd1 τd2⋯τdn ⟫g = ∂ ∂td1 ∂ ∂td2 ∂ ∂tdn Fg(t0, t1,⋯), (2.39) and let u = ⟪ψ 2 0⟫. Then u obeys the KdV flow equations given in (2.10), as well as the string equation u = t0 + ∞ ∑ i=1 tiRi . (2.40) These conditions uniquely fix the generating function F. Finally, the “partition function” Z ∼ e F is a tau-function of the KdV hierarchy, which can be seen from the Hirota substitution identity at the end of the previous section. Historically, this began as a conjecture by Witten and was proven by Kontsevich. Witten’s conjecture was motivated by aspects of certain double scaled Hermitian matrix models5 . Further, Kontsevich devised his own matrix model in his proof of the theorem. Hence the content of the Witten-Kontsevich theorem is intimately connected, both in origin and proof, to matrix models, and will be important in Chapter 6. 2.6 Weil-Petersson Volumes As we have seen, there is a rich story to tell about the moduli space of stable curves with marked points. However, as we will see later on when we return to studying gravity, it is important to be able to describe surfaces with boundaries. For instance, recall from Section 1.5 that the process of gauge fixing in a string theory path integral involved integrating over the moduli of the worldsheet. In the open string sector, this can involve integrating over the moduli of Riemann surfaces with geodesic boundaries. 5 Such matrix models will be reviewed in the following chapters. 48 Let Mg,n(L) be the moduli space of Riemann surfaces of genus g and n ordered geodesic boundaries with lenghts L = (L1, L2,⋯, Ln), and let Mg,n(L) be the corresponding Deligne-Mumford compactification, defined by including surfaces with double points. The moduli space Mg,n(L) has a symplectic structure given by the Weil-Petersson form ωW P . This two-form is related to the kappa class κ1, a 2-form similar to the ψ classes, through a choice of normalization κ1 = 2π 2ωW P . (2.41) Denote the volume of the moduli space, the Weil-Petersson volume, by Vg,n(L) Vg,n(L) = ∫Mg,n(L) ω d W P d! (2.42) where d = 3g − 3 + n. We can consider more general intersection numbers than the ones defined purely in terms of ψi by including the Weil-Petersson form, or κ1: ⟨κ m 1 τd1⋯τdn ⟩g = ∫Mg,n κ m 1 ψ d1 1 ⋯ψ dn n , (2.43) where ∑ n i=1 di+m = d. Notice that these are intersection numbers on the moduli space with points in stead of boundaries. The Weil-Petersson form can be determined by other methods, which in principle allows one to compute such intersection numbers. However, the Weil-Petersson form on Mg,n(L) is not equal to the WP form on Mg,n, but is instead given by ωW P (L) = ωW P (0) + 1 2 n ∑ i=1 L 2 i ψi . (2.44) 49 This leads us to a pair of theorems due to Mirzakhani. The first one states that the WP volume is related to the intersection numbers via Vg,n(L) = ∑ ∣d∣+m=d (2π 2 ) m 2 ∣d∣d!m! ⟨κ m 1 τd1⋯τdn ⟩gL 2d1 1 ⋯L 2dn n , (2.45) where d = (d1, d2,⋯, dn), and we are using the notation ∣d∣ ≡ d1 + ⋯ + dn and d! ≡ d1!⋯dn! [7]. The second theorem states that the Weil-Petersson volumes satisfy the famous Mirzakhani recursion relations [7] Vg,n(L) = 1 2L1 ∑ A ∐ B ∑ g1+g2=g ∫ L1 0 ∫ ∞ 0 ∫ ∞ 0 xyH(t, x + y)Vg1,n1 (x,LA)Vg2,n2 (y,LB)dxdydt + 1 2L1 ∫ L1 0 ∫ ∞ 0 ∫ ∞ 0 xyH(t, x + y)Vg−1,n+1(x, y,Lˆ1 )dxdydt + 1 2L1 n ∑ j=2 ∫ L1 0 ∫ ∞ 0 x(H(x, L1 + Lj) + H(x, L1 − Lj))Vg,n−1(x,L1̂,j)dxdt, (2.46) where H(x, y) = 1 1 + e (x+y)/2 + 1 1 + e (x−y)/2 , (2.47) and A,B are subsets of {1,⋯, n}. The volumes V0,1 and V0,2 do not exist, and the two initial conditions for the recursion relation are V0,3(L) = 1, V1,1(L) = L 2 + 4π 2 48 . (2.48) Inspired by the Witten-Kontsevich theorem, a natural question to ask is whether or not there exists a generating function for these intersection numbers, which would contain all of the information necessary to determine the Weil-Petersson volumes. Moreover, if it is fixed uniquely by the KdV flows and a string equation it will be computable by different means, and could be related to a matrix model. 50 Define the rescaled Weil-Petersson volumes vg,n(L) ≡ Vg,n(L) 2 dπ 2d = 1 d! ∫Mg,n(L) (κ1 + n ∑ i=1 L 2 i ψi) d = ∑ d0+∣d∣=d 1 di ! ⟨κ d0 1 n ∏ i=0 τdi ⟩ g,n n ∏ j=1 L 2dj j . (2.49) The clear choice for a generating function is then G(s, t0, t1,⋯) = ∑ g ⟨e sκ1+∑i tiψi ⟩ g = ∑ g ∑ m,{ni} ⟨κ m 1 ψ n0 0 ψ n1 1 ⋯⟩ g s m m! ∞ ∏ i=0 t ni i ni ! . (2.50) In fact, this generating function is related to the generating function for the ordinary intersection numbers. This is the content of a theorem by Mulase and Safnuk [50]. In particular, G(s, t0, t1,⋯) = F(t0, t1, t2 + γ2, t3 + γ3,⋯), (2.51) where γk = (−1) k (2k+1)k! s k−1 . Moreover, for any fixed value of s, e G is a τ -function of the KdV hierarchy. An implication of this theorem is that the function u˜ = ∂ 2G ∂t2 0 satisfies the KdV hierarchy and the string equation with the shifted coefficients t˜k listed above. More directly, we can say u˜ = u(t˜). Using the properties of G one can show that, since e G is a τ function of the KdV hierarchy, the satisfaction of the Virasoro conditions by G implies the Mirzakhani recursion relation. It turns out that this relationship between the generating functions had been realized in a different context over a decade earlier by Johnson in [51]. This realization made in the context of double scaled matrix models and string theory in the earlier 90s will form the basis of much of the original work presented in Chapter 6. 51 Chapter 3 Random Matrix Theory 3.1 Motivation And Basic Definitions Despite the relative simplicity of the minimal string theories described in Chapter 1, as compared to critical string theories for instance, there remains the fact that path integrals and correlation functions are generally quite hard to compute in an interacting quantum theory, even at the level of perturbation theory. One of the methods proposed to sidestep many of the intricacies and difficulties of conventional string theory is to discretize the worldsheet. In fact it turns out that by starting with a discretized worldsheet, one is able to obtain continuum results by carefully taking certain limits in the theory. The process of connecting matrices to the discretized worldsheet takes a great deal of inspiration from the ’t Hooft ribbon diagrams one would draw in a gauge theory like SU(N) Yang-Mills [52]. The standard classification of a Yang-Mills diagram includes, among other things, the genus of the surface that the diagram could be drawn on without intersections. In this way, a gauge theory naturally contains an enumeration of all two-dimensional surfaces in a topological sense. The simplest gauge theory one can conceive of focuses on a “field” M taking values in some set of N ×N matrices, with no time or space dependence. The classical theory will have some “action” S and we can quantize the theory by introducing the “path integral” ∫ DMe−S . The ’t Hooft diagrams of this theory 52 will enumerate two-dimensional surfaces of arbitrary topology1 in the same way that they do in Yang-Mills theory. Gauge transformations consist of conjugation by some (constant) element of a matrix Lie group, like GL(N, C) or one of its subgroups. Observables will then be gauge invariant scalar functions of M. It turns out, though, that this type of theory has played a part in quantum physics for over half a century. The so-called path integral over M is merely the definition of an important object in a random matrix theory. Random matrices were popularized in physics by Wigner in an attempt to glean some universal behavior of heavy atomic nuclei [53, 54]. In his original set up, the field M represented the Hamiltonian of such a nucleus, and each configuration (specific value of M) in the ensemble represented a different system. The different configurations are weighted in the path integral, as they are in a standard quantum field theory, by the exponential of the action. Since M is actually just a matrix of numbers, and not fields, it is appropriate to consider it in the context of garden variety probability theory, albeit with random variables whose probability distributions can be coupled together in complicated ways. Hence we use the combined name “random matrix.” Excellent reviews of random matrix theory can be found in [38, 55]. A random matrix model is defined by a choice of the class of N ×N matrices M and a potential V (M), usually a polynomial defined by a set of coupling constants V (M) = 1 2 M2 + ∞ ∑ n=3 gnMn . (3.1) Different normalizations are sometimes used in the quadratic term, and we will restrict ourselves to V being an even function moving forward. The matrix elements Mij are random variables whose joint probability density function is p(M) = dMe−N tr V (M) , where dM is the flat, translation invariant measure 1This depends on which set of matrices the field M takes values in. For example, Hermitian matrices will correspond to closed oriented surfaces with any number of handles, while real symmetric matrices will correspond to closed unoriented surfaces with any number of handles or crosscaps. We will explore this in more depth shortly. 53 on the space of matrix elements. The matrix integral Z, the aforementioned analog of the path integral, is defined by Z = ∫ dMe−N tr V (M) . (3.2) It is conventional to not demand that Z = 1, which one might do in a more rigorous probability theoretic setting, but to instead normalize correlation functions using Z. Expectation values and correlation functions of gauge invariant observables are subsequently computed by inserting them into the matrix integral. Correlation functions have an expansion in powers of 1/N, à la ’t Hooft [52]. This expansion applied to the matrix integral itself has the interpretation as an enumeration of tessellations2 of closed surfaces [15, 56]. Thinking of this like a sum over geometries allows one to interpret Z like a sort of discrete quantum gravity path integral. However, it turns out that in order to reproduce the minimal string theories we are interested in we will need to constrain the ensemble of matrices we choose to consider. In the cases where M is diagonalizable by some matrix group G, it is often convenient to perform a change of variables from the matrix elements Mij to the eigenvalues λi . The apparent decrease in the number of degrees of freedom can be thought of as gauge fixing: the initial field M has a symmetry group G and therefore it has less than N2 independent degrees of freedom. There are two types of eigenvalue-based models that will be significant here. Introduce the function ∆ = ∏i<j(λj − λi), called the Vandermonde determinant. Then the two types of models are defined by Zβ = ∫ ∏ i dλi ∣∆∣ β e −N ∑j V (λj ) (3.3) Z(α,β) = ∫ ∏ i (dλiλ α i )∣∆∣ β e −N ∑j V (λj ) . (3.4) 2The numbers of sides that a face is allowed to have is determined by the rank of V . 54 Models described by Zβ are called Wigner-Dyson models [57], and ones described by Z(α,β) are called Altland-Zirnbauer models [58]. It is convenient to rescale the eigenvalues λi → λi/ √ N so that the Gaussian term has no overall factor of N, which will be useful in the large-N limit. The cases β = 1, 2, 4 are special in both types of model because they correspond to easily identifiable ensembles of matrices, with well-studied symmetry groups. They correspond to real symmetric, Hermitian, and symplectic matrices, respectively, with symmetry groups O(N), U(N), and Sp(2N), respectively. When the potential V is quadratic, the Wigner-Dyson models are commonly referred to as the Gaussian Orthogonal Ensemble (GOE), Gaussian Unitary Ensemble (GUE), and the Gaussian Symplectic Ensemble (GSE), respectively3 . A key difference between the Wigner-Dyson and Altland-Zirnbauer models, even for the same value of β, is that the Altand-Zirnbauer matrices are further constrained. For example, the case (α, β) = (1, 2) corresponds to non-negative Hermitian matrices given by M = H†H, where H is an N × N complex matrix. Different values of α can be obtained by making H rectangular. CruciallyM ≥ 0, and consequently so are the eigenvalues. By quickly studying the ’t Hooft diagrams of the different Wigner-Dyson ensembles we will be able to identify which ones will be of interest. In particular, we will focus on β = 1, 2. First, consider the GUE with potential V (M) = M2 . The hermitian matrix M is in the bi-fundamental representation N ⊗ N of U(N) ⊗ U(N), so its matrix elements are written with one lower and one upper index M j i . The “action” can be written in propagator language N tr(M2 ) = N N−1 ∑ i,j,m,n=0 Mi j δn i δj mMm n . (3.5) The propagator, 1 N δ i n δm j , can be represented as two lines with opposite orientation. The set up for the 3The corresponding Altland-Zirnbauer models are named by appending “chiral” to the front of each name. For example, β = 2 is the chiral Gaussian Unitary Ensemble, or chGUE. It is also sometimes referred to as the Wishart-Laguerre ensemble. 55 δ i n δm j ∶ m n j i Figure 3.1: The propagator for a β = 2 matrix model. The lines carry arrows representing the flow from barred to unbarred indices. (a) Sphere diagram (b) Torus diagram Figure 3.2: Two generic diagrams drawn on different topologies. This would contribute at different overall powers of N even if they have the same number of vertices. The matrix propagator is shown in blue, and the dual diagram is shown in red in each case. diagram, shown in Figure 3.1, is as follows: on either side of the lines the indices for the two matrices are stacked vertically, with contracted indices connected by a line. The arrows flow, by convention, from barred to unbarred indices. The directionality of the propagator is a consequence of the fact that Mi j = (Mj i ) ∗ . The order of indices is important because swapping them introduces complex conjugation. To get interesting tessellations, consider adding a quartic interaction term g4M4 to V (M). The matrix integral Z is then computed by expanding the exponential in a series in g4, which amounts to computing correlators of trM4 in the Gaussian theory. Each diagram will have a topology determined by which type of surface it can naturally be drawn on. This surface will also be tessellated by drawing the “dual diagram,” formed by connecting the centers of the faces in the ribbon diagram. Moreover, each diagram will be dual to an orientable closed surface (or surfaces in the disconnected cases) due to the directionality of the propagator and the trace, respectively. Generic diagrams on the sphere and torus topologies are drawn in Figure 3.2. 56 Due to differing numbers of index loops, which each contribute a factor of N to the diagram, each topology will be associated to a different power of N. As is standard in field theory, the quantity F˜ = −logZ, called the free energy, will be the generating function for connected correlation functions, and hence can also be written as a sum over diagrams that contribute with certain powers of N determined by their topologies. It is said that F˜ has the topological expansion F˜(β=2) = ∞ ∑ g=0 N 2−2gF˜(β=2) g , (3.6) where g represents the genus of the surface, and F˜ g contains all contributions at that genus. Given the interplay between the factors of 1/N introduced by the propagator and powers of N coming from index loops, there will be infinitely many surfaces drawn at each genus corresponding to having different numbers of faces in the tessellation, or overall area. To construct an enumeration of unoriented surfaces using a matrix integral, we restrict ourselves to real symmetric matrices H, which falls under the β = 1, O(N) symmetry class. The matrix H will be in the bi-fundamental representation N ⊗ N of O(N). The two fundamental representations are identical, making the indices of H indistinguishable. As such, it will be more convenient not to keep track of upper and lower indices. The Gaussian action can be cast in propagator language N tr(H 2 ) = N 2 N−1 ∑ i,j,m,n=0 Him(δinδmj + δijδmn)Hjn. (3.7) This implies that the propagator is 2 N (δinδmj +δijδmn) for β = 1 ensembles, with the second term coming from the indistinguishability of the matrix indices. The propagator drawn using double lines is displayed in Figure 3.3. As in the β = 2 case, the perturbative expansion of the matrix integral will continue to have an interpretation in terms of a sum over closed surfaces with different topologies. However in this case the 57 + i m n j i m j n δinδmj + δijδmn ∶ Figure 3.3: The propagator for a β = 1 matrix model. The lines carry no arrows because there is no consistent way to define flow between indices. The first term is functionally identical to the β = 2 propagator. The second term can be thought of as being twisted with respect to the first term. presence of the second “twisted” term in the propagator and the consequential lack of directionality of the diagrams means that the surfaces will be unoriented. The topological expansion will be F˜(β=1) = ∞ ∑ g,c=0 N 2−2g−cF˜(β=1) g,c , (3.8) where once again g counts genus and now c counts the number of crosscaps. A diagram drawn on the cross-cap topology in the quartic theory is shown in Figure 3.4. Figure 3.4: A diagram drawn on an unorientable surface. Notice that the tessellation is necessarily selfintersection in this number of dimensions due to the cross cap. 3.2 Saddle Point Analysis And The Large-N Limit A first pass at uncovering continuum physics comes from taking the large-N limit. In this limit we reparametrize the eigenvalues by defining i/N = X, with i = 0, 1, . . . , N −1. As N → ∞, the new variable X takes values in [0, 1). The eigenvalues can then be thought of as functions of X. 58 For any value of β and any matrix potential V , the matrix integral Zβ of a Wigner-Dyson model can be rewritten as Zβ = ∫ ∏ i dλi exp ⎧⎪⎪ ⎨ ⎪⎪⎩ −N ∑ i V (λi) + β 2 ∑ i≠j log ∣λi − λj ∣ ⎫⎪⎪ ⎬ ⎪⎪⎭ , (3.9) by turning the Vandermonde determinant into a logarithmic interaction potential. Introduce the spectral density ρ˜(λ) = dX/dλ to facilitate the transition to the large-N limit. The spectral density is the one-point correlation function of the eigenvalues themselves ρ˜(λ) = ⟨ 1 N N−1 ∑ i=0 δ(λ − λi)⟩ , (3.10) and has support on the spectrum of the ensemble of matrices. Ordinarily the spectrum is all of R, but due to the rescaling of the eigenvalues by √ N, the spectrum will be confined to some union of intervals containing 0. We will choose to only consider Wigner-Dyson models where the spectrum is one symmetric interval about the origin, denoted [−a, a]. The endpoints will depend on the value of β and the details of V . By swapping sums over i to integrals over X, with the appropriate factors of N included, the argument of the exponential in the integrand of Zβ can be written −N 2 [∫ ρ˜(λ)V (λ)dλ − β 2 ∬ ρ˜(λ)ρ˜(µ) log ∣λ − µ∣dλdµ] . (3.11) The goal now is to compute Zβ using the saddle point approximation. The large-N saddle results from the condition V ′ (λ) = βP ∫ a −a ρ˜(µ) λ − µ dµ, (3.12) 59 where P denotes the principal part of the integral. Introduce the function F(λ) = ∫ a −a ρ˜(µ) λ − µ dµ, (3.13) which is analytic everywhere in C except the portion on the real line where the spectrum is located. Via application of the Sokhotski–Plemelj formula, F has the discontinuity across the spectrum given by F(λ ± iϵ) = V ′ (λ) β ∓ iπρ˜(λ). (3.14) A solution for ρ˜ is found by using the ansatz F(λ) = V ′ (λ) β − Q(λ)π √ λ2 − a 2, (3.15) where both the polynomial Q and the endpoints ±a are determined by requiring that F → 1/λ as ∣λ∣ → ∞. The density is then given by ρ˜(λ) = Q(λ) π √ a 2 − λ2. (3.16) For the Gaussian case V (λ) = αλ2 with arbitrary normalization, one finds Q(λ) = 2α β and a 2 = β α , so that the density is ρ˜(λ) = 2 πa2 √ a 2 − λ2. (3.17) This is the famous Wigner semi-circle law. Given the canonical normalization α = 1 2 , we have the following results β = 1 ∶ ρ˜(λ) = √ 2 − λ2 π , β = 2 ∶ ρ˜(λ) = √ 4 − λ2 2π . (3.18) 60 The Wigner Semi-Circle in the GOE -1.5 -1 -0.5 0 0.5 1 1.5 The Wigner Semi-Circle in the GUE -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Figure 3.5: The Wigner semi-circle densities for the Gaussian Orthogonal Ensemble (GOE) and Gaussian Unitary Ensemble (GUE). In both plots the eigenvalue histograms are shown in blue, while the corresponding density functions are shown in red. Close inspection reveals that the spectra actually leak out past the endpoints predicted by the saddle point analysis. These results can be confirmed by doing “numerical experiments” (see for example [59]). Using a program like MATLAB, one can generate an ensemble of random matrices whose elements have Gaussian statistics4 . By considering 100,000 100 × 100 Gaussian-distributed matrices, we diagonalize each of them and then histogram the eigenvalues. The results are displayed in Figure 3.5. The function F is often referred to as the resolvent because it is related to the expectation value of the resolvent operator R(λ) = tr(M − λ) −1 . As we will see in upcoming chapters, the matrix model resolvent is a very important object. In particular, we can consider its correlation functions with respect to the matrix integral, which will have their own topological expansion like the free energy F˜. We write ⟨R(λ1)⋯R(λn)⟩ = ∞ ∑ g=0 N 2−2gRg,n(λ1,⋯, λn). (3.19) 4This is a special feature of Gaussian theories. Having a Gaussian matrix potential actually leads to each matrix element being an independent Gaussian distributed random variable. This is not the case, say, for the quartic case, where the matrix elements become correlated with each other. 61 It turns out that the resolvent functions Rg,n obey a recursion relation, which for general β and g = 0, 1 2 , 1,⋯, is [60] √ σ(λ)Rg,n(λ1,⋯, λn) = 1 2πi ∫ C dλ λ − λ1 Gg,n(λ, I) √ σ(λ) 2y(λ) , (3.20) where σ(λ) = λ 2 −a 2 , C is a contour surrounding the cut [−a, a], y(λ) is the spectral curve, I denotes the set {λ2,⋯, λn}, and Gg,n(λ, I) = (1 − 2 β ) ∂λRg− 1 2 ,n(λ, I) + Rg−1,n+1(λ, λ, I) + ∑ stable Rh,1+∣J∣(λ, J)Rh′ ,1+∣J′ ∣(λ, J′ ) + 2 n ∑ k=1 (R0,2(λ, λk) + 1 β 1 (λ − λk) 2 )Rg,∣I∣−1(λ, I ∖ λk). (3.21) The subscript ‘stable’ on the sum means we ignore terms where one of the factors is R0,1(λ) or R0,2(λ, λk). The spectral curve is a complexified version of the leading order eigenvalue density. 3.3 Matrix Models And Families Of Polynomials It is advantageous for several reasons to think of the matrix integral Z — in both Wigner-Dyson and Altland-Zirnbauer models — as being the exponential of a free energy function F˜. First, this fits nicely with the physical interpretation of the model describing some sort of statistical ensemble of Hamiltonians. Second, it makes direct contact with the study of integrable systems and topological gravity. The WittenKontsevich theorem casts the free energy F˜ of a (Hermitian) matrix model as the generating function of intersection numbers on the moduli space of Riemann surfaces [45, 46], and asserts that the matrix integral Z is related to a tau function of the KdV hierarchy. Third, the critical behavior of the free energy provides guidance for how to implement the double scaling limit and extract continuum physics from the models. Therefore devising a way to compute F˜ is of paramount importance. 62 The domain of integration for the eigenvalues in (3.3) is affected by the value of β. When N is finite, we can assume that the eigenvalues are ordered. The matrix integral can be defined with a domain of integration that respects the ordering so as not to overcount, and to eliminate the absolute value around ∆. However, for even values of β we can extend the integration to all of R N by dividing by a combinatorial factor to take into account relabelling the indices anytime λj − λi changes sign. Having each eigenvalue take any real value (prior to the large-N limit) is convenient, as will be explored shortly. We are not so fortunate when considering odd values of β though, and must use alternate methods5 . It can be shown that ∆ is the determinant of a matrix consisting of powers of the eigenvalues, ∆(λ) = det λ j i . By taking linear combinations of the rows with real coefficients, we can introduce a family of polynomials pi(x) so that ∆(λ) = det[pj(λi)]N−1 i,j=0 . The polynomials will be normalized so that pi(λ) = λ i + ⋯. We stress here that the introduction of these polynomials is possible for any value of β, and solely depends on the presence of the Vandermonde determinant. However, it will prove advantageous to bestow certain properties upon the polynomials depending on the context. 3.3.1 β = 2 Wigner-Dyson Theories A convenient choice for the polynomials pi in β = 2 theories is to make them orthogonal with respect to a power of the measure e −NV , for example ∫ ∞ −∞ dλe−NV (λ) pi(λ)pj(λ) = hiδij . (3.22) This is particularly helpful when the values of λ can be expanded to all of R. Note that the family of these polynomials is infinite, which will facilitate the large-N limit to be taken later. An immediate consequence of this is that we can compute Z2 in terms of the normalizations hi . First, notice that each term in the 5Actually the difficulty is not so straightforward. For example, the β = 4 theory shares many of the same complexities as the β = 1 theory, despite having an even power of β. We will be focused on β = 1, 2, and so for all intents and purposes their key difference can be attributed to being odd or even. 63 Vandermonde determinant factorizes into a product of terms involving each eigenvalue individually, with a generic factor looking like the left hand side of the orthogonality relation (3.22). There will be N! terms with pj(λi) paired (with matching values of i and j), which survive the evaluation of the integrals over the eigenvalues. All other terms vanish by orthogonality. Thus Z = N! N−1 ∏ i=0 hi . (3.23) Hence we have reduced the task of computing Z2 to finding the orthogonal polynomials pi and normalizing them. So, let us study more of their features. Define the oscillator wavefunctions ψi(λ) = h −1/2 i e −V (λ)/2 pi(λ). (3.24) The choice of name has a two-fold motivation. In the Gaussian matrix model, the orthogonal polynomials are the Hermite polynomials and thus the functions ψi are identical to the normalized wavefunctions of the harmonic oscillator. Additionally, we have the Fermi gas, or log-gas, perspective that arises from interpreting the Vandermonde determinant as a logarithmic interaction potential, as we saw in the saddlepoint analysis. The eigenvalues behave like fermions, and we can use a Fock space representation where each one is represented by a sort of “oscillator.” While formally the index i on ψi is the same as the index on λi , we know that even for a finite number of eigenvalues there are infinitely many wavefunctions. So, consider a many-body system comprised of N identical fermions. The wavefunction of a single fermion is represented by ψi , where the index i indicates the energy level occupied by that particular particle. Since the fermions are identical, only a single particle can occupy the i th energy level. By virtue of the fact that the Vandermonde determinant in the matrix model partition function can be written in terms of the orthogonal polynomials, so too can it be written in terms of the first N oscillator wavefunctions. 64 Hence the matrix model takes into account the first N energy levels of the fermions. In a fermionic manybody system, the energy at which all lower energy levels are occupied is called the Fermi level, and the underlying energy levels are called the Fermi sea. As such, the index value i = N −1 is given title of Fermi level6 , and all lower values are said to belong to the Fermi sea. The polynomials pi satisfy the following recursion relation7 λpi(λ) = pi+1(λ) + Ripi−1(λ), (3.25) with Ri = hi/hi−1. This yields another way to write Z2: Z2 = N!h N 0 N−1 ∏ i=0 R N−i i . (3.26) The corresponding recursion relation for the oscillator wavefunctions is λψi(λ) = √ Ri+1ψi+1(λ) + √ Riψi−1(λ). (3.27) The oscillator wavefunctions provide an orthonormal basis for the Hilbert space L 2 (R) with the standard inner product. Denote the basis as {∣ψi⟩}∞ i=0 . Operators on L 2 can thus be decomposed in terms of these functions in the usual way Oij ≡ ⟨ψi ∣Oˆ∣ψj ⟩ = ∫ ∞ −∞ dx ψi(x)Oˆψj(x). (3.28) The self-adjoint operator Λˆ that implements this transformation on L 2 has matrix elements Λij = √ Ri+1δj,i+1 + √ Riδj,i−1. (3.29) 6Note that since we begin counting at i = 0, the Nth level is at i = N − 1. 7There would be a third term if the potential were not an even function. 65 Notice that Λˆ is also the operator that implements multiplication by λ. In a quantum mechanical scenario, this would be the position operator. The recursion coefficients Ri are related to one another through their own recursion relation, which can be worked out by studying the action of the derivative d dλ on the polynomials. After integrating by parts N ∫ ∞ −∞ dλe−NV (λ)V ′ (λ)pn(λ)pn−1(λ) = nhn. (3.30) By invoking the recursion relation (3.25) to simplify, one arrives at an equation for the Ri . For example, consider the quartic potential V (λ) = 1 2 λ 2 + g4λ 4 . The equation obtained from (3.30) is Rn [1 + 4g4(Rn+1 + Rn + Rn−1)] = n N . (3.31) We will return to this relation in the context of the large-N limit later on. An additional outcome with great significance in this story is the ability to write the eigenvalue density ρ˜(λ) using the oscillator wavefunctions: ρ˜(x) N−1 ∑ i=0 ψi(x) 2 . (3.32) More generally, the n-level correlator ρ˜n(x1, . . . , xn) is given by ρ˜n(x1, . . . , xn) = det[KN (xi , xj)]n i,j=1 , KN (x, y) = N−1 ∑ i=0 ψi(x)ψi(y). (3.33) The function KN is the self-reproducing kernel of the matrix model, so-called because it has the property ∫ dz KN (x, z)KN (z, y) = KN (x, y). (3.34) 66 Evidently knowing the oscillator wavefunctions is incredibly powerful, because they allow us to compute Z and they give us KN , which can then be used to compute the level statistics of the theory. 3.3.2 β = 1 Wigner-Dyson Theories General n-point functions of the eigenvalues in a β = 1 Wigner-Dyson theory can be computed using an idea pioneered by Dyson [61] and applied to the GOE by Mehta and Mahoux [62], called integration over alternate variables. The analysis begins by introducing two arbitrary functions u and v, and considering the following expectation value with respect to the matrix ensemble GN (u, v) = ⟨ N 2 −1 ∏ α=0 u(λ2α) N 2 −1 ∏ β=0 v(λ2β+1)⟩ . (3.35) We must take care to specify a domain of integration that allows us to remove the absolute value on ∆. We write Z = ∫ I N−1 ∏ i=0 dλi∆(λ)e −N ∑ N−1 j=0 V (λj ) , (3.36) where I denotes the ordering −∞ < λN−1 ≤ λN−2 ≤ ⋯ ≤ λ1 ≤ λ0 < ∞. There are two choices we can make to compute GN . First, choose polynomials pi(λ) satisfying ∫ ∞ −∞ dλ e−2NV (λ) pi(λ)pj(λ) = hiδij . (3.37) 67 Notice that the integration measure includes a different power of e −NV than used previously. Unless stated otherwise, the polynomials pi used for β = 1 will be orthogonal with respect to this new measure. Consequently, the oscillator wavefunctions will be defined as ψi(λ) = 1 √ hi pi(λ)e −NV (λ) . (3.38) Introduce the functions Bi(v; λ2α) = ∫ λ2α −∞ dλ2α+1v(λ2α+1)ψi(λ2α+1). (3.39) By absorbing the factors of v into the determinant and taking linear combinations of the columns of the matrix, it can be shown that GN is given by GN (u, v) = CN ∫ ∞ −∞ dλ0e −NV (λ0) u(λ0)∫ λ0 −∞ dλ2e −NV (λ2) u(λ2)⋯ ⋯∫ λN−4 −∞ dλN−2e −NV (λN−2) u(λN−2) det(m(v; λ2α)), (3.40) where CN is a numerical factor depending on the normalizations hi and the matrix m is given by mi,2α = pi(λ2α), mi,2α+1 = Bi(v; λ2α). (3.41) The integrand is now symmetric under the swap of any two variables (due to the invariance of the determinant under swapping an even number of rows and columns), so we can enlarge the domain of integration to (−∞,∞) for each eigenvalue at the cost of a numerical prefactor, which we disregard. Subsequently, the integrals over each eigenvalue are decoupled and GN can be written GN (u, v) = C ′ N ∑ σ∈SN N 2 −1 ∏ α=0 Jσ(2α),σ(2α+1)(u, v), (3.42) 68 where Jij(u, v) = ∫ ∞ −∞ dx∫ ∞ −∞ dy u(x)v(y)ψ[i(x)ψj](y), (3.43) and the brackets denote anti-symmetrization. This determines the correlator GN as a Pfaffian, GN = Pf (J). Recall that for a Gaussian potential the orthogonal polynomials are the Hermite polynomials Hi . The derivative H′ i satisfies the recursion relation H′ i (x) = 2xHi(x)− Hi+1(x), while xHi(x) can be rewritten using (3.25). By taking linear combinations of the rows and columns of J, we can exploit these recursion relations to put J into a 2 × 2 block form that is amenable to computing the eigenvalue density. One finds Pf(J) → det1/2 ⎛ ⎜ ⎜ ⎜ ⎝ fαγ gαγ −gγα µαγ ⎞ ⎟ ⎟ ⎟ ⎠ , (3.44) where fαγ(u, v) = J2α,2γ(u, v), gαγ(u, v) = ∫ ∞ −∞ dx∫ ∞ −∞ dy u(x)v(y)(ψ2α(x)ψ ′ 2γ (y) − ψ2α(y)ψ ′ 2γ (x)), µαγ(u, v) = ∫ ∞ −∞ dx∫ ∞ −∞ dy u(x)v(y)(ψ ′ 2α(x)ψ ′ 2γ (y) − ψ ′ 2α(y)ψ ′ 2γ (x)). (3.45) If u, v are both even functions, the diagonal blocks of this new matrix are zero and the Pfaffian reduces to det(g). We now use the two functions u, v as sources and treat GN as a generating function for the eigenvalue correlators. These correlators are related to GN via [55] GN (1 + a, 1 + a) = N ∑ n=1 1 n! ∫ ∞ −∞ n−1 ∏ i=0 a(xi)dxiρn(x0, . . . , xn−1) (3.46) 69 In particular, this means that to compute the eigenvalue density ρ1(λ) ≡ ρ˜(λ), we take one functional derivative of GN and set the source to 0. Mehta shows that for the GOE ρ˜(λ) = N 2 −1 ∑ α=0 [ψ2α(λ) 2 − ψ ′ 2α(λ)∫ λ 0 dλ′ψ2α(λ ′ )] . (3.47) As we will discuss later, this method is not so easily adapted to more general matrix potentials. For this reason, we are motivated to make a second attempt at computing ρ˜ and Z1. In the second approach, we begin again with (3.36) and compute the correlator GN (u, u), setting u = v for convenience. We still arrive at the expression for GN in (3.40), where now we define mi,2α = qi(λ2α), mi,2α+1 = F˜ i(u; λ2α) = ∫ λ2α −∞ dλ′ e −NV (λ) qi(λ ′ ). (3.48) The polynomials qi(λ) are monic and chosen to obey the following relation [62–64] ⟨q2α, q2γ+1⟩s ≡ ∬ ∞ −∞ dλdλ′ e −NV (λ)−NV (λ ′ ) ε(λ − λ ′ )qi(λ)qj(λ ′ ) = −rαδαγ, (3.49) where ε(λ − λ ′ ) = sgn(λ − λ ′ ), and with all other ⟨qi , qj ⟩s = 0. The bilinear form ⟨⋅, ⋅⟩s is skew-symmetric, and hence is referred to as a skew inner product. The polynomials qi are called skew orthogonal polynomials. The correlator GN (u, u) is once again given by the pfaffian of a matrix J˜ ij = ⟨uqi , uqj ⟩s. If we set u = 1, this computes the full matrix integral. In analogy to the oscillator wavefunctions ψi defined above, introduce the skew oscillator wavefunctions ζi(λ) = r −1/2 i e −NV (λ) qi(λ). (3.50) 70 We identify the 2 × 2 block structure in the matrix J˜ as before: J˜[u] = ⎛ ⎜ ⎜ ⎜ ⎝ fαγ[u] gαγ[u] −gγα[u] µαγ[u] ⎞ ⎟ ⎟ ⎟ ⎠ , (3.51) where fαγ[u] ≡ J˜ 2α,2γ[u], gαγ[u] ≡ J˜ 2α,2γ+1[u], µαγ[u] ≡ J˜ 2α+1,2γ+1[u]. (3.52) The functions f, g and µ referenced here are not the same as the functions with the same names used in the orthogonal polynomial approach. Consider once again setting u = 1 + a where a is small. Then the 2 × 2 structure of J˜ can be written (schematically) as J˜ = ⎛ ⎜ ⎜ ⎜ ⎝ ϵf 1 + ϵν −(1 + ϵν) ϵµ ⎞ ⎟ ⎟ ⎟ ⎠ , (3.53) with ϵ small and related to the function a. The pfaffian pf(J˜) admits a straightforward expansion in ϵ (see Appendix A.7 in [55]). The density can once again be extracted from GN (1 + a) by taking a functional derivative with respect to a. The result is [62, 64] ρ˜(λ) = N 2 −1 ∑ α=0 (ζ2α(λ)[εˆ⋅ ζ2α+1](λ) − ζ2α+1(λ)[εˆ⋅ ζ2α](λ)). (3.54) The benefit of this formula is that it applies to models with more complicated matrix potentials than the Gaussian case. It will prove useful when we attempt to extract continuum results in β = 1 Wigner-Dyson theories to form a relationship between the two families of polynomials. The construction of skew orthogonal 71 polynomials and their properties are explored in detail in [63, 64]. A second operator of interest, in addition to Λˆ which implements multiplication by an eigenvalue on our functions, is the one that implements the derivative, which we denote by Lˆ in the finite-N regime8 . Let the rank of V be 2k. One can show that the matrix elements Lij are nonzero in general only when ∣i − j∣ ≤ 2k − 1, are determined by the recursion coefficients Ri , and are anti-symmetric because Lˆ is anti-self-adjoint. The operators Λˆ and Lˆ also satisfy the canonical commutation relation [Λˆ,Lˆ] = 1. Since V is an even function it can be shown that each polynomial pi and qi has definite parity, given by pi(−λ) = (−1) i pi(λ), and similarly for qi . This induces the Z2-grading L 2 = span{∣ψ2α⟩}⊕span{∣ψ2β+1⟩}. Thus we can conveniently decompose our operators into 2 × 2 block form. For example, both Λˆ and Lˆ are odd under the grading, so we designate Λˆ = ⎛ ⎜ ⎜ ⎜ ⎝ 0 ℓ ℓ 0 ⎞ ⎟ ⎟ ⎟ ⎠ , Lˆ = ⎛ ⎜ ⎜ ⎜ ⎝ 0 −cˆ † cˆ 0 ⎞ ⎟ ⎟ ⎟ ⎠ . (3.55) Objects that are labelled 0, . . . , N − 1 will carry a latin index i, j (but note that k is reserved for the order of the matrix potential). When considering Z2-grading we will decompose some of these objects into even and odd parts, and it will be convenient to write i = 2α or j = 2γ+1, respectively. Objects that have definite Z2 parity will thus carry a Greek index α, γ (but β is reserved to denote the type of matrix ensemble). For example, take the operator Lˆ. By definition the operator cˆ has matrix elements given by cγα = −L2α,2γ+1. Since the matrix representing Lˆ has 2k − 1 non-zero off-diagonals, the matrix representing cˆ 8This is not the same as the differential operator L mentioned briefly in the context of the KP hierarchy. 72 will have 2k − 1 non-zero off-diagonals (and the main diagonal will be nonzero as well, which is not true for Lˆ). The action of the derivative on the even wavefunctions ψ2α can be written ψ ′ 2α = − N 2 +k−1 ∑ γ=0 cγαψ2γ+1. (3.56) For this range of γ in the sum, some of the matrix elements cαγ will be zero depending on the value of α, but these bounds of summation are guaranteed to produce all necessary non-zero terms. The orthogonal polynomials pi introduced above form a basis for the ring of polynomials. The skew orthogonal polynomials qi can be expanded in terms of the pi : there exists a lower triangular matrix O with 1’s on the diagonal such that qi = i ∑ j=0 Oijpj . (3.57) Representing the skew oscillator wavefunctions with the kets∣ζi⟩, the defining relation for the polynomials qi is then given by ⟨qi , qj ⟩s = 2 √rirj ⟨ζi ∣εˆ∣ζj ⟩, where εˆ is the integral operator with kernel 1 2 ε(λ − λ ′ ) and the L 2 inner product is used. An implication of this is that the skew inner product can be implemented by an operator acting on L 2 with the measure e −2NV . This justifies the choice to modify the orthogonality relation for the orthogonal polynomials. By taking into account the normalizations, the skew oscillator wavefunctions can be expressed in terms of the oscillator wavefunctions in a way related to (3.57): ζi = i ∑ j=0 O˜ ijψj , O˜ ij = √ hj /riOij . (3.58) 73 The skew oscillator wavefunctions have the same Z2-grading as the oscillator wavefunctions, so the matrix O˜ must be even under the Z2-grading. Define the matrices a, b so that O˜ = ⎛ ⎜ ⎜ ⎜ ⎝ a 0 0 b ⎞ ⎟ ⎟ ⎟ ⎠ . In [63] the authors show that a, b are related to the derivative matrix c by c = b T a. (3.59) Hence the eigenvalue density is written (schematically) as ρ˜ ∼ (a ⋅ ψeven) ⋅ (εˆ⋅ b ⋅ ψodd) − (b ⋅ ψodd) ⋅ (εˆ⋅ b ⋅ ψeven). (3.60) This structure will guide us to double scaling the eigenvalue density in Chapter 7. To conclude, it should be noted that ρ˜ is the diagonal part of a larger function [64]. Define s(λ, λ′ ) = N 2 −1 ∑ α=0 (ζ2α(λ)[εˆ⋅ ζ2α+1](λ ′ ) − ζ2α+1(λ ′ )[εˆ⋅ ζ2α](λ)). (3.61) The analog of the self-reproducing kernel familiar from studies of β = 2 theories is given by f1(λ, λ′ ) = ⎛ ⎜ ⎜ ⎜ ⎝ s(λ, λ′ ) I(λ, λ′ ) ∂λs(λ, λ′ ) s(λ, λ′ ) ⎞ ⎟ ⎟ ⎟ ⎠ , (3.62) where I(λ, λ′ ) = 1 2 ∫ ∞ −∞ s(λ, z)ε(z − λ ′ )dz − 1 2 ε(λ − λ ′ ). (3.63) 74 The eigenvalue density is given by ρ˜(λ) = lim λ′→λ qdetf1(λ, λ′ ), (3.64) where qdet denotes the quaternionic determinant (see [62], for example). The result follows from the fact that I(λ, λ) = 0. 3.3.3 (α, β) = (1 + 2Γ, 2) Altland-Zirnbauer Theories The (α, β) = (α, 2) Altland-Zirnbauer models are theories of Hermitian matrices, and so we expect that fundamentally they should be described in a way that is similar to the β = 2 Wigner-Dyson models. Of particular interest here will be the cases (1, 2) and (1 + 2Γ, β) for some number Γ. Recall that they are defined in terms of the eigenvalues by integral Z(1+2Γ,2) = ∫ N ∏ i=1 [(dλiλi)(λ 2 i ) Γ ]∏ j<k ∣λ 2 k − λ 2 j ∣ 2 e −N ∑ N i=1 V (λi) , (3.65) which can be obtained by choosing the matrices of the form M = H†H, where H is a complex-valued rectangular matrix of size (N + Γ) × N, and diagonalizing M. The eigenvalues of M are non-negative by construction, and so the matrix integral Z(1+2Γ,2) can naturally be interpreted in terms of the β = 2 Wigner-Dyson matrix integral (3.3) by the change of coordinates yi = λ 2 i . The domain of integration for each variable yi can be extended to R+. Then Z(1+2Γ,2) = ∫ ∞ 0 N ∏ i=1 dyi y Γ i ∆(y) 2 e −N ∑ N i=1 V (yi) , (3.66) where we have recovered the Vandermonde determinant ∆ once again, and we have redefined the matrix potential V to account for the coordinate transformation. 75 The obvious first attempt at computing the matrix integral should once again involve introducing a family of polynomials to rewrite the Vandermonde determinant. If we choose the polynomials pi so that they satisfy ∫ ∞ 0 dy yΓ e −NV (y) pi(y)pj(y) = hiδij , (3.67) the result for Z(1+2Γ,2) has the exact same expression as Z2 written in terms of the normalizations hi . The polynomials will also still obey a recursion relation ypi(y) = pi+1(y) + Ripi−1(y). (3.68) The Gaussian model has the matrix potential V (y) ∝ y. In this case the orthogonal polynomials are related to the generalized Laguerre polynomials L (Γ) i (y). 3.4 A Prelude To The Double Scaling Limit We will end this chapter by resuming the discussion initiated in sections 3.1 and 3.2, in order to motivate the content of the following chapter. One ultimate goal of this endeavor is to reproduce string theory results, and so we need to adjust our matrix model technology to describe smooth surfaces. The double scaling limit is a combination of taking the size of the matrix N → ∞ while also focusing the objects of the theory to neighborhoods around specific critical values [17, 18, 20, 29]. In this limit the small parameter 1/N has a renormalized form, called h̵, which will be identified with the closed string coupling constant. The index i is replaced by a continuous variable x ∈ (−∞, µ], where µ is the double scaled Fermi level. Since the large-N limit confines the spectrum to a single interval, in the double scaling limit we zoom in on one of the edges of the interval, replacing λ with a new variable E ∈ [0,∞). In [16] Kazakov made the connection between a subset of double scaled matrix models and Liouville theory coupled to minimal CFT matter. When the matrix potential is the polynomial V (M) = 76 1 2M2 + ∑ p k=1 g2kM2k , the double scaled version is dual to the (2, 2p − 1) minimal string, once the coupling constants are tuned appropriately. By considering the double scaling limit of a matrix model with a polynomial potential and carefully tuning its coupling constants, we recover our desired minimal (closed) string theories! Although we have not considered any specific details about the β = 2 Altland-Zirnbauer theories, they prove to be invaluable in what is to come. Their double scaled versions are able to describe the open string sectors of both non-supersymmetric and supersymmetric theories, and also provide a viable non-perturbative completion of these models, which will be important for extracting results in unoriented gravity. 77 Chapter 4 Double Scaled Hermitian Matrix Models In the remainder of this thesis we will be entirely focused on the double scaling limit of various matrix models. As advertised, this will allow us to obtain continuum results about specific string theories with varying amounts of supersymmetry. We will also be able to make more general statements about generic gravity and matter theories described by double scaled matrix models. The goal of this chapter is to review the existing foundations of double scaled Hermitian — of both Wigner-Dyson and Altland-Zirnbauer class — and real symmetric matrix models, with a focus on how the orthogonal polynomial formalism morphs into a combination of a quantum mechanics problem and solving a family of differential equations. This will include uncovering how all of these theories are organized by the KdV hierarchy. 4.1 Double Scaled Wigner-Dyson Theories As we saw in Section 3.3.2, the β = 1 Wigner-Dyson models are controlled in part by elements from the β = 2 models, based on the relationship between our particular orthogonal and skew-orthogonal polynomials. In Chapter 7 we will see in depth that the β = 1 have more complicated double scaling limits, but nevertheless a large part of their formulation will still depend upon the procedures we will develop in β = 2 models. Additionally, although some aspects of (α, 2) Altland-Zirnbauer models will differ from 78 their simpler Wigner-Dyson counterparts, the dominating presence of the KdV hierarchy in both classes of models will allow us to uplift many results from β = 2 to (α, β) = (α, 2). Therefore we will begin the discussion of the double scaling limit by considering Hermitian Wigner-Dyson models. We will leave the discussion of the preexisting double scaling results in β = 1 theories for Chapter 7. When starting to study the double scaling limit of a Hermitian matrix model, one might as well start by mentioning the Gaussian case. In the large-N limit the equation obtained from (3.30) is R(X) = X. It turns out that in this regard the Gaussian case is not particularly illuminating, although we will return to it after uncovering more details about double scaling, as it still has important lessons to teach us. Moving on, we return to the quartic potential. The large-N free energy F˜ of the theory has interesting behavior at X = 1 F˜ = −N 2 ∫ 1 0 dX(1 − X) log R(X), (4.1) where R(X) is the large-N version of the recursion coefficient Ri . We can obtain an expression for R(X) by solving (3.31) in the large-N limit. The dependence of this expression on R at different values of the index indicates that it will become a differential equation in the large-N limit. However, the derivatives will appear in a 1/N expansion. At leading order, the equation is R(X)(1 + 12g4R(X)) = X, (4.2) which has the solution R(X) = 1 24g4 (−1 + √ 1 + 48g4X). Inserting this into the free energy, we can perform an expansion in g4. At the endpoint X = 1, the leading order contribution to the free energy diverges as we send g4 → − 1 48 . The function R also takes the value R(1; g4 = − 1 48 ) = 2 ≡ Rc. The significance of this divergence can be understood by remembering that the matrix integral enumerates random tessellations of surfaces with all genuses. The leading order contribution in that enumeration corresponds to the sphere. When we tune g4 in this way, the contribution from the spherical tessellations 79 diverges, which is interpreted as creating smooth sphere-topology surfaces with a macroscopic area. For this reason the value g c 4 = − 1 48 earns the title of a critical value. Unfortunately the higher-topology surfaces do not survive the large-N limit in its current form, as they are suppressed by powers of 1/N. To make matters worse, only tuning g4 → g c 4 further emphasizes the sphere topology. In order to keep smooth, macroscopic surfaces of all genus, we must cream off the scaling pieces of important quantities. For the k th multicritical model, scale away from X = 1 via X = 1 + (x − µ)δ 2k , where δ → 0 and x ∈ (−∞, µ]. The parameter µ is referred to as the Fermi level in the many-body formalism, and can also be thought of as the renormalized Liouville cosmological constant. Since the eigenvalues are ordered, the point X = 1 corresponds to the largest eigenvalue. Therefore we scale away from that point via λ = a − Eδ2 , where E ∈ [0,∞). The recursion coefficient R(X) takes the critical value R(1) = 1 at the edge of the spectrum. Its scaling form is R(X) = Rc − u(x)δ 2 . Finally, we define N−1 = hδ̵ 2k+1 . Since correlation functions, like the ones that contribute to the free energy F˜, were expanded in powers of N−1 prior to double scaling, h̵ will be the new topological expansion parameter in the double scaling limit. By double scaling the outcome of (3.30) one arrives at a differential equation for the function u. For the quartic potential, the equation of motion is − h̵2 3 u ′′ + u 2 + x = 0, where primes denote derivatives with respect to x. The equation of motion in the Gaussian case ends up being u(x)+x = 0. These two equations have the same general structure, namely a “differential polynomial in u plus x equals 0.” In a matrix model with potential V = ∑ k i=1 g2iM2i , each of the coupling constants g4, . . . , g2k can be tuned to a critical value. It turns out that for any value of k, the differential polynomial is none other than the k th Gelfand-Dikii polynomial Rk, and the equation of motion is Rk+x = 0. A double 80 scaled model with this equation of motion is called the k thmulticritical model. This is the first indication that double scaled matrix models have a deep connection with the KdV hierarchy. We can also define the double scaling limit of the orthogonal polynomials via 1 √ hi e −NV (λ)/2 pi(λ) double scale ÐÐÐÐÐÐ→ ψ(x, E). (4.3) Other important objects related to the wavefunction have double scaled versions as well. First, in the large-N limit the operator Λˆ can be written in terms of two shift operators Λˆ = √ R(X + ϵ) exp{ϵ ∂ ∂X } + √ R(X) exp{−ϵ ∂ ∂X }, (4.4) where ϵ = 1/N. In the double scaling limit this becomes Λˆ = 2 − δ 2 ( − h̵2 ∂ 2 + u(x)) ≡ 2 − Hδ 2 , (4.5) where we have defined the Schrodinger Hamiltonian H = −h̵2∂ 2+u(x), and ∂ ≡ ∂/∂x. On the other hand, we also have λ = a − Eδ2 . Therefore, in a particular normalization where a = 2, the relation Λˆψ = λψ implies the double scaled wavefunction ψ(x, E) satisfies the Schrodinger equation with u as the potential −h̵2 ∂ 2ψ(x, E) + u(x)ψ(x, E) = Eψ(x, E). (4.6) The Schrodinger equation provides an elegant way of determining the double scaled oscillator wavefunctions. Recall that two important statistical quantities, the kernel K and the eigenvalue density ρ˜, can be computed in the finite-N regime using the wavefunctions. In the double scaling limit, K(E, E′ ) = ∫ µ −∞ ψ(x, E)ψ(x, E′ )dx, & ρ(E) = ∫ µ −∞ ∣ψ(x, E)∣2 dx. (4.7) 81 By using the Wentzel–Kramers–Brillouin (WKB) approximation for ψ, given to one loop by ψ(x, E) = 1 √ πh̵(E − u0(x))1/4 cos ( 1 h̵ ∫ x √ E − u0(x ′)dx′ + π 4 ) , (4.8) we arrive at the formula for the leading eigenvalue density ρ0(E) = 1 2π ∫ 0 −xc dx √ E − u0(x) , (4.9) which contributes to the full density at O(h̵−1 ). See Appendix A for more information about the WKB approximation and its relationship to matrix models. As we have seen it is possible to determine the leading eigenvalue density in the Gaussian theories using saddle point analysis1 . Upon performing double scaling with the relation λ = λ (β) c − δ 2E, with λ (2) c = 2 and λ (1) c = √ 2, the Wigner semi-circle densities in (3.18) become ρ(E) = √ E πh̵ , ρ(E) = 2 3/4 √ E πh̵ , (4.10) for β = 2, 1 respectively2 . In fact, it is possible to obtain a closed form expression for the full nonperturbative eigenvalue densities in both the GUE and GOE. In the double scaling limit we have u(x) = −x, and hence the Schrodinger equation is h̵2 ∂ 2ψ + (E + x)ψ = 0, (4.11) 1 In fact the same analysis can be repeated for different values of k with some extra effort. 2 Some compensating factors of N need to be included to extract the scaling part of the density. These factors of N combine with the left over power of δ to form the renormalized parameter h̵. 82 which can be manipulated into the Airy differential equation, yielding the wavefunctions ψ(x, E) = h̵− 2 3Ai( − h̵− 2 3 (E + x)). (4.12) By utilizing an integral identity for the Airy function, the full double scaled eigenvalue density is ρ(E) = h̵− 2 3 (Ai′ (E) 2 − EAi(E) 2 ), (4.13) where E = −h̵− 2 3 E and the prime denotes a derivative with respect to E. The density for the double scaled GOE will be discussed in Chapter 7. Recall that we also have the operator Lˆ that implements the derivative on the Hilbert space L 2 . Prior to double scaling, Lˆ and Λˆ satisfy the commutation relation [L, ˆ Λˆ] = −1. We require this to continue to be true in the double scaling limit. Denote the double scaling limit of Lˆ in the k th model by Pk. In order to make a guess at the form of Pk, we must finally begin to understand the connection between the matrix model and the KdV hierarchy. The Witten-Kontsevich theorem states that the square root of the matrix integral Z is a τ -function of the KdV hierarchy. Moreover, if Z = e −F , then the function v ∼ −∂ 2F satisfies the KdV flow equations. The function F is precisely the free energy that has led us to the double scaling limit. Moreover, recall the large-N expression for F in terms of R(X) in (4.1). If we insert the scaling ansatz for X and R, we get F(µ) = 1 h̵2 ∫ µ −∞ (x − µ)u(x)dx. (4.14) This relationship can be inverted to give u(x) = −h̵2 ∂ 2F(x) ∂x2 . (4.15) 83 This confirms that the function u, which we obtained as the scaling part of the large-N recursion coefficient R, is the same as the function v in the Witten-Kontsevich theorem. Therefore u will satisfy the KdV flows. An outcome of this revelation is that the Schrodinger Hamiltonian H is identified as the Lax operator Q of the KdV hierarchy. Then for free, we know that there must be a conjugate operator (Hk− 1 2 ) + which generates the flow of Q. The motivation to identify Pk with (Hk− 1 2 ) + is two-fold3 : first, it’s the only easily identified differential operator apart from H in the double scaled matrix model; second, we have already noted that the equation of motion for u is Rk + x = 0. Recall that [(H k− 1 2 ) + , H] = R ′ k , (4.16) and we have determined that [Pk, H] = −1. If we identify Pk = (Hk− 1 2 ) + , then the two commutation relations imply that R ′ k + 1 = 0, or Rk + x = 0 if we discard an integration constant. Hence, if we identify Pk with (Hk− 1 2 ) + , the matrix model’s canonical commutation relation is equivalent to the equation of motion for u. Despite the incorporation of the KdV hierarchy into the matrix model story, we are missing the equivalent of the KdV times. One can construct more complicated models in the double scaling limit by taking linear combinations of the Gelfand-Dikii polynomials4 ∑ k tkRk + x = 0. (4.17) Such models are referred to as massive interpolations, and can be thought of as renormalization group (RG) flows between the different multicritical models [21]. For historical reasons, equations of the form (4.17) are called string equations, or as we saw in Chapter 2, Novikov equations. 3Of course we have used the benefit of hindsight to judiciously choose the name Pk, just as we did in the KdV hierarchy. 4 Strictly speaking this is true for models where the double scaled spectrum is one connected part of R, the single-cut matrix models. A slightly different approach is needed to describe the double-cut models, for instance. 84 The coupling constants tk are so-named because they can be thought of as specific values of the KdV times. This leads us to the idea of a “generic” matrix model defined by (4.17) with all of the coupling constants left unfixed. The string equation is formally an infinite-order differential equation, but as we will demonstrate later in detail, it is still possible to perturbatively solve for u in terms of x and the coupling constants tk in an asymptotic series in h̵. As a preview of what is to come in the following chapters, recall that the leading order solution u0 can be used to compute the density of states. Actually, the exact solution does not need to be known, only that it satisfies (4.17) to O(h̵0 ) and what its value at x = µ is. In the cases where u0(µ) = 0, ρ0(E) = 1 2π ∞ ∑ k=1 √ πΓ(k + 1) Γ(k + 1 2 ) tkE k− 1 2 . (4.18) The numbers tk that define the string equation of the model have an interpretation in the string theory language as being the coupling constants for gravitationally dressed CFT primaries Ok, or closed string operators [20]. This idea is captured by the path integral mnemonic in (2.36). Recall that in the CFT language, the primaries were indexed by two integers whose allowed values were constrained by the numbers p and q that define the model. It turns out that β = 2 Wigner-Dyson models are dual to minimal string-type models with p = 2. Therefore there is only one allowed value of the index constrained by p. Hence the primaries are indexed by one number, and we choose to name them σk. Then, keeping the path integral mnemonic in mind, we have ⟨σk⟩ = ∂tk F [20, 45]. From the point of view of the underlying KdV structure, the operators σk are dual to ∂/∂tk inside correlation functions. Hence we can also make an association between the connected correlation functions of σk and the KdV vector fields ξk [21] h̵2 ∂ 2 ∂µ2 ⟨σk1⋯σkn ⟩µ = ξkn+1⋯ξk1+1 ⋅ u(µ), (4.19) 85 where the µ subscript on the correlation function denotes that it is evaluated at x = µ. All correlation functions will be connected unless noted otherwise. The factor of h̵2 comes from the relationship between u and F in (4.15). Double scaled matrix models with a string equation of the form (4.17) describe closed string physics in non-supersymmetric models. One way of interpreting this statement is that the surfaces described by the model are closed and hence look like closed string worldsheets. However, it is possible to introduce asymptotic boundaries to these theories using macroscopic loop operators5 . In the finite-N regime, these operators are represented by insertions of thermal partition functions tr e −lM into the matrix integral. An example of such a scenario is depicted in Figure 4.1. In a holographic interpretation, this is like saying there is a dual quantum system with this partition function living on the boundary [9]. Figure 4.1: A pre-double scaling diagram with an insertion of tr e −lM. In the double scaling limit, a macroscopic loop is represented by the operator e −βH , where H is the Hamiltonian of the auxiliary quantum mechanics. Its expectation value ⟨e −βH⟩ has the interpretation of a gravity path integral on a surface with an asymptotic boundary of renormalized length β [21]. General 5The use of “asymptotic” here will be justified at the end of this chapter. The qualifier is included to differentiate the boundary conditions from brane-related boundaries. 86 connected correlation functions of the macroscopic loop operator have a topological expansion similar to Z ⟨e −β1H⋯e −βnH⟩ = ∞ ∑ g=0 h̵2g−2+nZg,n(β1, . . . , βn), (4.20) where Zg,n is interpreted as the path integral over surfaces with genus g and n asymptotic boundaries. The operator e −βH has an expansion in terms of the closed string operators σk that will be used repeatedly later on e −βH = h̵ 2 √ πβ ∞ ∑ k=1 (−1) kβ k k! σk−1. (4.21) The mass dimension of u is defined to be [u] = 1, with h̵ dimensionless, and the variable x has dimension [x] = − 1 2 . The mass dimension of β must be [β] = −1 to make βH dimensionless. Meanwhile, the k th Gelfand-Dikii, polynomial has dimension [Rk] = k, and the coupling constants satisfy [tk] + [Rk] = [x], or [tk] = −k − 1 2 , in order to make the string equation dimensionless. For the combination tkσk to be dimensionless, we must have [σk] = k+ 1 2 . This implies that (k − 1 2 ) [β]+[σk−1] = 0, as necessary because e −βH is dimensionless. This justifies the powers of β relative to the labelling of the σk in the expansion of the macroscopic loop operator. In the double scaling limit it is convenient to introduce a complex uniformizing coordinate z to replace the scaling part of the eigenvalue λ. The two are related by z 2 = −E. This is most commonly seen when considering the correlation functions of the matrix resolvent R and the eigenvalue density ρ. One then considers a slightly altered form of the functions Rg,n in (3.19) Wg,n(z1, . . . , zn) = (−1) n z1⋯znRg,n(−z 2 1 , . . . , −z 2 n ). 87 The functions Wg,n satisfy a double scaled version of the recursion relation (3.20), which here are referred to as topological recursion relations [8] Wg,n(z1, . . . , zn) = Res z→0 ⎧⎪⎪ ⎨ ⎪⎪⎩ 1 z 2 1 − z 2 1 4y(z) [Wg−1,n+1(z,−z, J) + ′ ∑ I∪I ′=J g1+g2=g Wg1,∣I∣+1(z, I)Wg2,∣I ′ ∣+1(−z, I)]⎫⎪⎪ ⎬ ⎪⎪⎭ , (4.22) where y(z) is the double scaled spectral curve of the matrix model, J = {z2, . . . , zn}, and ∑ ′ denotes a sum over stable configurations6 . The spectral curve is proportional to the leading eigenvalue density ρ0 (4.18), evaluated using z = √ −E. 4.2 Double Scaled Altland-Zirnbauer Theories The double scaling procedure for the (α, β) = (1 + 2Γ, 2) Altland-Zirnbauer models goes through similarly to how it does in the β = 2 Wigner-Dyson models, and all of the important objects defined for those models — for example the free energy, eigenvalue density, wavefunctions, etc. — are important here. In particular, recall from Section 3.3.3 that one can still introduce orthogonal polynomials that obey a recursion relation under multiplication by λ with the same structure as the β = 2 Wigner-Dyson models. Therefore this recursion relation will also double scale to a Schrodinger equation. The double scaled version of the recursion coefficients, still referred to here as u, will satisfy a differential equation, albeit a different one than before. Multicritical theories can still obtained for this class of matrix models by tuning the coupling constants in the matrix potential to critical values during the double scaling procedure. It is easiest to begin with the case Γ = 0. For the k th multicritical model, the string equation is [26] uR 2 k − h̵2 2 RkR ′′ k + h̵2 4 (R ′ k ) 2 = 0, (4.23) 6These are configurations where I, I′ ≠ ∅ and h, h′ ≠ 0. 88 where Rk ≡ Rk + x. This is the most general equation of motion for the function u that is both consistent with the KdV flows and which reproduces perturbative closed string physics – the regime in which that perturbation theory is recovered will be discussed in the following chapter. One outcome of still having the KdV flows is that the operator content will still consist of the σk introduced in the preceding section. Thus there will still exist composite observables like the macroscopic loop operator, as well as the others that we will discuss later. The string equation for the double scaled (α, β) = (1, 2) Altland-Zirnbauer model has two generalizations. To start, we note that just as with the Wigner-Dyson models, one can consider the massive interpolation between models uR 2 − h̵2 2 RR′′ + h̵2 4 (R ′ ) 2 = 0, (4.24) where now the more general R ≡ ∑k tkRk + x is defined. The first generalization of (4.24) is obtained by noticing that the function u together with the coupling constants tk have a scaling symmetry [25] ∞ ∑ k=1 (k + 1 2 ) ∂u ∂tk + 1 2 x ∂u ∂x + u = 0. (4.25) The mass dimensions can be found in the discussion following Figure 4.21. The scaling relation (4.25) is just a Callan-Symanzik equation for the model. Assuming that u satisfies the KdV flow equations (2.10), using the recursion relation for the Gelfand-Dikii polynomials, and integrating once, (4.25) becomes uR 2 − h̵2 2 RR′′ + h̵2 4 (R ′ ) 2 = h̵2Γ 2 , which it turns out is the correct string equation for the double scaled (α, β) = (1+2Γ, 2) Altland-Zirnbauer model [26]. There is an apparent tension between the definition of R provided above and what one would actually obtain by following the procedure described to turn the scaling relation into the string equation. 89 The choice to include the explicit factor of (k+ 1 2 ) is conceptually necessary to call (4.25) a scaling symmetry. In the formalism used throughout this paper, unless noted otherwise, we choose to absorb any extra factors into the coupling constants tk. The second generalization comes from slightly altering the model starting at the level of the matrix integral. Recall that the (α, β) = (1, 2) and (α, β) = (1 + 2Γ, 2) models involved non-negative hermitian matrices, which is to say that the eigenvalues satisfied λ ≥ 0. One can instead consider an ensemble of matrices whose eigenvalues satisfy E ≥ s in the double scaling limit for some s ∈ R. Doing so in a (α, β) = (1 + 2Γ, 2) model yields the double scaled string equation [27] (u − s)R 2 − h̵2 2 RR′′ + h̵2 4 (R ′ ) 2 = h̵2Γ 2 . (4.26) The full differential equation (4.26) will henceforth be referred to as the Dalley-Johnson-Morris (DJM) equation. Note that from the point of view of the differential equation Γ need not be an integer, despite the fact that from the matrix model perspective it does not make sense to consider a matrix with non-integer size. This generality remains true in the physical interpretation of the equation of motion: there will be some cases where Γ should naturally be a non-negative integer, cases where the exact value of Γ is not important, and even cases where Γ = − 1 2 . The DJM equation (4.26) exhibits a somewhat miraculous universality. It was originally derived as the string equation describing double scaled multicritical complex matrix models via the process described here [24, 26]. It was subsequently argued that the DJM equation provides a consistent non-perturbative formulation of the multicritical bosonic closed string theory, since as we will see in Chapter 5 one regime of its perturbation theory reproduces results from double scaled β = 2 Wigner-Dyson models [25, 27]. It was later discovered that (4.26) also has the capacity to describe type 0A and 0B superstrings [65, 66], with 90 further explorations in a modern context in [67, 68]. Different models are defined by the different types of perturbative solutions that (4.26) allows. This is discussed in depth below. There is yet another context through which we encounter the DJM equation. In [69] Kostov showed that a β = 2 Wigner-Dyson model describing both open and closed string sectors simultaneously in the double scaling limit is obtained from just the closed string matrix model by including the deformation of the matrix potential Vopen(M) = γ N tr log(1 − m2M2 ), (4.27) which can also be thought of as the insertion of a determinant operator in the matrix integral. Here γ is the ratio of the open string coupling to closed string coupling, and m represents the endpoint mass of an open string. The details concerning the double scaling limit of the pure gravity model are contained in [69], as well as the beginnings of the generalization to arbitrary multicritical models. The story was made more complete in [24, 51, 70, 71], where the connections to the KdV hierarchy in the double scaling limit were established in generality. The following is a review of [51]. Recall that a general non-supersymmetric string equation is written7 R˜ ≡ ∞ ∑ k=0 tkR˜ k = 0. (4.28) The string equation describing the open-closed string system is similar to the closed string equation, but involves The Gelfand-Dikii resolvent Rˆ introduced in Section 2.3.1. The new string equation is R˜ + 2h̵ΓRˆ(x, s) = 0, (4.29) 7We are reverting to the normalizations used in [47] because they are used in [51]. This is the last time the old normalizations will be used. 91 which we will refer to as the Kostov-Johnson (KJ) equation. In the above, h̵ is the closed string coupling constant, Γ is the scaling part of γ, and ρ is the scaling part of m2 . Recall that the resolvent Rˆ solves the Gelfand-Dikii differential equation (2.22), which we reproduce here for convenience 4(u + s)Rˆ2 − 2RˆRˆ′′ + (Rˆ′ ) 2 = 1, (4.30) where the spectral parameter is −s. By solving (4.29) for Rˆ and substituting that into the Gelfand-Dikii equation, one obtains the (open-closed) string equation (u + s)R˜2 − 1 2 R˜R˜′′ + 1 4 (R˜′ ) 2 = h̵2Γ 2 , (4.31) which we can plainly see is the DJM equation. Thus in addition to the models described previously that share this string equation, it also describes open string physics. The differential equation (4.31) can be obtained from the closed string equation (4.17) by a shift in the coupling constants tk. Define the shifted variables tk → tk + 2h̵Γ(−s) −(k+ 1 2 ) , x → x − h̵Γs − 1 2 . (4.32) By substituting these into the closed string equation (4.17) and invoking the expansion of the resolvent Rˆ, one gets Kostov’s string equation (4.29). We pause here to clarify some terminology, since the same objects are used to describe different physics. The work of [51, 69, 71] demonstrated a duality between purely closed strings and an interacting openclosed string system. In the “KdV frame” both matrix models are organized by the KdV flows and a string equation, each of which is dependent on a set of KdV times, or coupling constants. Before establishing any equivalence between the two models, the closed string matrix model is described by the string equation 92 (4.17), while the open-closed system is described by the string equation (4.31). The open-closed system can be described by the closed string equation (4.17) once one makes the change of coordinates t (closed) k = t (open) k + 2h̵Γ(−s) −(k+ 1 2 ) , x (closed) = x (open) − h̵Γ(−s) − 1 2 . 4.2.1 Non-Perturbative Solutions Before wrapping up this discussion of double scaled Hermitian matrix models and their string equations, it is worth discussing the numerical procedure by which one obtains a full solution to the DJM equation (perturbative solutions will be the focus of the next chapter). Full solutions will be smooth and therefore more amenable to the numerical procedures used to solve the Schrodinger equation, which we will utilize in Chapter 7 to extract results for unoriented gravity. This discussion will also serve to introduce perturbation theory for the string equations. A full exposition of this nonperturbative framework can be found in [11]. The first step toward obtaining a full solution to (4.26) in a particular model defined by the coupling constants tk is to identify the (asymptotic) boundary conditions. The boundary conditions arise from solving the differential equation to leading order in either an expansion in h̵ or 1/∣x∣. The conditions are different for x → −∞ and x → +∞. For the former, the boundary condition is ∑k tku k 0 + x = 0, where the sum is over all nonzero couplings. For the latter, the boundary condition is u0 = 0. However, it is often easier to solve the DJM equation for h̵ = 1 rather than h̵ ≪ 1, and so we will actually supply the boundary condition u → h̵2 4Γ2−1 4x2 as x → +∞. As we will see in the next chapter, this contribution to the solution is actually universal, being completely independent of the coupling constants. 93 -400 -300 -200 -100 0 100 200 x -5 0 5 10 15 20 u k = 2 Nonperturbative DJM Solution -10 -8 -6 -4 -2 0 2 4 6 8 10 x -0.5 0 0.5 1 1.5 2 2.5 3 3.5 u Interior Well Nonperturbative Classical Figure 4.2: The left panel shows the full solution on the desired range of x values. The right panel is a zoomed in portion around the origin. The nonperturbative solution develops a well for Γ = 0, which disappears as Γ is increased. Some of the nonlinearity of the DJM equation can be removed by differentiating it once and factoring out an overall factor of R, which results in u ′R + 2uR ′ − h̵2 2 R ′′′ = 0. (4.33) The boundary conditions above will still be used in this equation. Notice though, that the parameter Γ no longer explicitly shows up in the differential equation, so it will only appear through the x → +∞ boundary condition. There are two models that will be utilized in Chapter 7. The first is the k = 2 model with t2 = 1. The boundary condition for x → −∞ is u0 → √ −x. The equation (4.33) is solved in MATLAB using the differential equation solver bvp5c on a spatial grid from x = −400 to x = 200. It is convenient to start with Γ = 1 and iterate through decreasing Γ until one obtains a solution for Γ = 0 which is accurate to a specified tolerance of 10−9 . The solution is displayed in fig. (4.2). The second model is the (2, 3) minimal string with t1 = 1 and t2 = 4π 2 9 (see Table (4.1) below for more details about the coupling constants). The boundary condition for x → −∞ is u0 = (−t1+ √ t 2 1 − 4t2x 2)/2t2. 94 -400 -300 -200 -100 0 100 200 x -2 0 2 4 6 8 10 u (2; 3) Minimal String Nonperturbative DJM Solution -10 -8 -6 -4 -2 0 2 4 6 8 10 x -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 u Interior Well Nonperturbative Classical Figure 4.3: The left panel shows the full solution on the desired range of x values. The right panel is a zoomed in portion around the origin. The nonperturbative solution develops a well for Γ = 0, which disappears as Γ is increased. The solution is obtained in exactly the same manner as before, and is displayed in fig. (4.3). It is qualitatively very similar to the purely k = 2 model, with some slight differences introduced by the facts that t2 ≠ 1 and that the k = 1 model is included. 4.3 Notable Example Models We end the review of double scaled matrix models with some important examples that we will see again later. Each example will be a matrix model description of a preexisting theory. The general procedure for finding the coupling constants that define the matrix model is to compare the density of states of the preexisting theory to the general formula for the eigenvalue density in the matrix model in (4.18). A definition of a gravity (plus matter) theory in terms of multicritical double scaled matrix models is often called a minimal model decomposition. The example models are listed in table 4.1. The parameter Γ in the supersymmetric theories has several context-dependent interpretations, which will be discussed more later. The parameter Q = b + b −1 in the two Virasoro Minimal Strings is the Liouville background charge and Qˆ = b −1 − b is a dual Liouville charge. 95 Table 4.1: A list of important example matrix models and their coupling constants. Type of Model Model Name Coupling Constants β = 2 JT Gravity [10, 72] π 2k−2 k!(k − 1)! β = 2 (2, 2p − 1) Minimal String [73] π 2k−2 k!(k − 1)! 4 k−1 (p + k − 2)! (p − k)!(2p − 1) 2k−2 β = 2 Virasoro Minimal String [74, 75] 2 √ 2 π 2k+1 (k!) 2 (Q 2k − Qˆ2k ) (α, β) = (1 + 2Γ, 2) N = 1 (0A) JT Supergravity [60, 67] π 2k (k!) 2 (α, β) = (1 + 2Γ, 2) N = 1 Virasoro Minimal String [76] 2 √ 2 π 2k+1 (k!) 2 (Q 2k + Qˆ2k ) (α, β) = (1 + 2Γ, 2) N = 2 JT Supergravity [77, 78] 2 kπ 2k−2 (2π √ E0) k(k − 1)! Jk(2π √ E0) The coupling constants for the (2, 2p − 1) minimal string are zero k > p, and hence is an interpolation between the first k multicritical models. Consider evaluating the spectral curve for these coupling constants using (4.18). It will be a rank-(2p − 1) polynomial in the complex coordinate z with p − 1 independent roots8 . It turns out that the roots of y(z) correspond directly to the principal ZZ branes of the dual minimal string [35]. Jackiw-Teitelboim (JT) gravity is a particularly important model because it both informs and motivates many of the conclusions drawn about general double scaled matrix models that will be made later. Due to its distinguished role in the story presented here, it is worth spending some time covering basic facts 8Most of the roots will consist of p − 1 pairs (z, z ∗ ), and one of them will be z = 0. Ignoring the complex conjugates and 0 gives p − 1 roots. 96 about it. The field content of (pure) JT gravity is the two-dimensional metric γ and the dilaton Φ. For a Euclidean manifold M (in general with boundary ∂M) the full action is I[g, Φ] = −S0χ[M] + IJT [g, Φ], χ[M] = 1 4π ∫M √ γR + 1 2π ∮ ∂M √ hK, IJT [g, Φ] = − 1 16πGN ∫M √ γΦ(R + 2) − 1 8πGN ∮ ∂M √ hΦ(K − 1). (4.34) Here χ is the Euler characteristic and K is the extrinsic curvature of the boundary. In the bulk, Φ acts as a Lagrange multiplier whose equation of motion sets R = −2. Classical solutions to JT gravity are thus patches of AdS2 in the bulk. The equation of motion obtained from varying the metric in the bulk is ∇µ∇νΦ − γµν∇ 2Φ + γµνΦ = 0. (4.35) The metric is determined independently of the dilaton. Since the bulk geometry is AdS2, if the boundary ∂M is the asymptotic conformal boundary, it needs to be regularized. This can be partially dealt with through boundary conditions. Take the bulk metric to be ds2 = dT2+dZ2 Z2 . A suitable set of boundary conditions for the metric and dilaton are γ∣ bdry = 1 ϵ 2 , Φ∣ bdry ≡ Φb = a 2ϵ . (4.36) The renormalized length of the boundary is commonly denoted by β. Parametrize the boundary curve by T = F(τ ) and Z = Z(τ ). In order to satisfy the metric boundary condition, we must have Z(τ ) = ϵF′ (τ ). The bulk portion of IJT vanishes on-shell, leaving only the boundary term IJT = −C ∫ dτ{F, τ}, C = a 16πGN , (4.37) 97 where {⋅, ⋅} denotes the Schwarzian derivative. The function F is often referred to as the Schwarzian mode, and completely dominates the on-shell dynamics of the theory in the ϵ → 0 limit. The presence of the Euler characteristic allows us to perform a topological expansion of the path integral, in analogy with what is done in string theory. Consider n sets of the boundary conditions in (4.36), with boundary lengths β1, . . . , βn. Then the full JT path integral Z JT(β1, . . . , βn) is given by Z JT(β1, . . . , βn) = ∞ ∑ g=0 h̵2g−2+nZ JT g (β1, . . . , βn), (4.38) where h̵ ≡ e −S0 . The term “full path integral” refers to integrals over both γ and Φ, as well as all possible topologies connecting the boundaries. The quantity Z(β1, . . . , βn) denotes the path integral over γ and Φ on n-boundary surfaces with a fixed number of handles g. The path integral contribution Z(β1, . . . , βn) can be computed using a cutting-and-gluing procedure. A surface Mg,n with genus g with n asymptotic boundaries can be decomposed into a bulk piece Σg,n and n cylinders. The surface Σg,n has g handles and n geodesic boundaries, and each cylinder has one geodesic boundary and one asymptotic boundary. The cylinders are often referred to as trumpets. An example of this procedure is shown in Figure 4.4. The contribution to the path integral from Σg,n becomes the Weil-Petersson volume Vg,n(b1, . . . , bn) introduced in Section 2.6, where bi are the geodesic lengths of the boundaries. They arise in this case from integrating over the moduli of Σg,n. The remaining degree of freedom is the Schwarzian mode (or rather n copies of it), which are confined to live near each asymptotic boundary on the trumpets. The Schwarzian path integral on a trumpet with geodesic boundary length b and asymptotic boundary length β is Ztr = 1 √ 4πβ e − b 2 4β . (4.39) 98 Figure 4.4: Gluing procedure for a surface with g = 2 and one asymptotic boundary. The Schwarzian mode is shown in green near the asymptotic boundary on the trumpet. In [9] Saad, Shenker, and Stanford showed that JT gravity has a dual description in terms of a double scaled β = 2 Wigner-Dyson model with leading eigenvalue density ρ0(E) = sinh(2π √ E) 2π 2h̵ . (4.40) They also argued that the full JT path integral with n asymptotic boundaries is computed in the matrix model by the correlation function of macroscopic loop operators, ⟨e −β1H⋯e −βnH⟩ = Z JT(β1, . . . , βn). Consequently, the perturbative contributions Z JT g are computed by perturbative contributions to the matrix model correlator. This is to say that in JT gravity, macroscopic loop operators create asymptotic boundaries. This provides the motivation to continue relating the two in future chapters. Additionally, it is necessary to differentiate the two kinds of boundaries that appear on the trumpet. One is the asymptotic — in this case AdS — renormalized boundary and the other is a geodesic. As we will show in Chapter 6, trumpet objects can be defined in more general models via the correlation function of a macroscopic loop operator and a geodesic loop operator. 99 The spectral curve of JT gravity is y(z) = sin(2πz) 4π . This in turn implies that the resolvent functions Wg,n in this theory are related to the Weil-Petersson volumes Vg,n via Laplace transform. Indeed, inverse Laplace transforming the recursion relation (4.22) with this spectral curve reproduces Mirzakhani’s recursion relation for Vg,n [7, 8]. With this in mind, we note that the resolvent recursion relation (4.22) exists in the broader context of generic matrix models. Therefore it stands to reason that the inverse Laplace transform of Wg,n ought to be a generalization of the corresponding Weil-Petersson volume [36, 79]. We will apply the geodesic loop operator formalism to this question in Chapter 6. Generalizations of JT gravity (including orientability, defects, supersymmetry, etc.) have been studied from both a field theory [80–82] and random matrix perspective [60, 67, 68]. The coupling constants defining a particular JT supergravity can be found in Table 4.1. We will encounter unoriented JT gravity in Chapter 7. 100 Chapter 5 String Equation Perturbation Theory In this chapter we take some time to develop the perturbation theory of the string equations (4.17) and (4.26). This chapter has three purposes: to display the general techniques for solving these differential equations, discuss the interpretations of the solutions, and to collect results that will be useful later. The results presented here are original [36], and represent a comprehensive analysis of the methods used to obtain perturbative solutions of these string equations, organized in a way to make them useful for the perturbative computation of correlation functions. There is some overlap with [83], where the authors also consider perturbative solutions to the Novikov equations describing the closed string sectors of minimal strings and JT gravity. In particular, we begin with a discussion of the closed string sector of β = 2 Wigner-Dyson models, which are by far the most well-known differential equations in the general string equation program. However, the equation (4.17) is often solved perturbatively with only a finite number of coupling constants turned on. Therefore in our discussion we will focus on a new systematic approach to obtaining perturbative solutions to the generic string equation. Some specific example models from Table 4.1 will be studied as well. We then turn to the open string sector of β = 2 Wigner-Dyson and (α, β) = (1 + 2Γ, 2) AltlandZirnbauer models by studying the KJ equation (4.29) (or equivalently the DJM equation). This provides 101 the first in-depth look at how multi-faceted this string equation is. We will be able to use the general methodology developed to solve (4.17), however some adjustments will of course need to be made. Using the open string sector as a bridge, we then study the closed string solutions in 0A theories described by (α, β) = (1 + 2Γ, 2) Altland-Zirnbauer models. Some more specific examples will be considered. Finally, we use the knowledge gained from the open and closed string sectors of the 0A theories to develop perturbation theory for the DJM equation as it applies to N = 2 supersymmetric theories. The DJM equation naturally incorporates the open string parameters corresponding to the endpoint mass and coupling constant. It turns out though they have context-dependent interpretations. One of the key differences is that the parameters exist in closed sectors of the supersymmetric theories as well. In closed string sector of 0A theories, the parameter Γ counts background RR flux on the worldsheet instead of being a coupling constant. Another key difference is the interpretation of the spatial variable x. In non-supersymmetric systems, both the open and closed physics is described in the region x < 0. On the other hand, in the 0A and N = 2 systems the open string physics is still in the x < 0 region, but the closed string physics is captured by the x > 0 part of the solution. This is related to the fact that the Ramond ground state could only be supergravitationally dressed for one particular sign of the cosmological constant in N = 1 minimal superstring theories [35], which we saw in Section 1.6.1. The comparisons are illustrated in the following table. Table 5.1: Locations of open and closed string physics Supersymmetry Open String Closed String N = 0 x < 0 x < 0 N = 1 x < 0 x > 0 N = 2 x < 0 x > 0 102 As mentioned at the end of the preceding chapter, we will transition to a different normalization for the Gelfand-Dikii polynomials than the one used in [47]. In the new normalization, the first several Rk are given by R0 = 1, R1 = u, R2 = u 2 − h̵2 3 u ′′, R3 = u 3 − h̵2 2 (u ′ ) 2 − h̵2 uu′′ + h̵4 10 u (4) . (5.1) The normalization used here fixes Rk = u k + ⋯. Notice also that we have incorporated explicit factors of h̵ into the definition of Rk by rescaling d dx → h̵ d dx . The Gelfand-Dikii polynomials satisfy the recursion relation Rk+1 = 2k + 2 2k + 1 [uRk − h̵2 4 R ′′ k − 1 2 ∫ x dxu′ (x)Rk] . (5.2) They also have an organization in powers of h̵ (see [68], for example) Rk = r (0) k + h̵2 r (2) k + ⋯, r (0) k = u k , r (2) k = − k(k − 1) 12 [(k − 2)(u ′ ) 2 + 2uu′′]u k−3 . (5.3) Before proceeding with calculations, we take some time to review and preview the importance of the function u. Recall that knowing the matrix model free energy F amounts to having solved the theory: it is used to evaluate the partition function Z, and is a generating function for correlation functions of the closed string operators F — in fact it will continue to be a generating function in the open string sector of the theories, as we will see later. The function u is directly related to F, and so knowing u yields the same information. Moreover, the expansion of matrix model quantities in powers of h̵ is interpreted as a topological expansion. Therefore knowing perturbative contributions to F or u gives specific information about surfaces with different topologies. Moreover, certain sectors of perturbation theory in the open string models will correspond to having different numbers of D-branes present. The work presented here is drawn from [37], as well as some new results in Section 5.3. 103 5.1 Novikov Equations The equation of motion for a general double scaled β = 2 Wigner-Dyson model is ∞ ∑ k=1 tkRk + x = 0. (5.4) A solution to this differential equation can be obtained in an asymptotic series in small h̵/∣x∣ u(x) = ∞ ∑ g=0 h̵2g ug(x). (5.5) By plugging this ansatz into (5.4), we find that the leading order contribution u0 satisfies f(u0) + x = 0, f(u0) ≡ ∞ ∑ k=1 tku k 0 . (5.6) Equation (5.6) is commonly referred to as the disk level string equation. When only the p th coupling constant tp is non-zero, the solution is u0 = (−x/tp) 1/p . Typically in these models the Fermi level is located at µ = 0. A feature of the individual multicritical models is that u0(µ) = 0, which generalizes to massive interpolations: the disk level equation at x = 0 is f(u0(0)) = 0, which is always solved by u0(0) = 0 since f has no constant term. One could also see this by using the Lagrange inversion theorem. The derivatives of u0 with respect to x can be expressed in terms of the solution u0 and the function f. The first several derivatives with their values at the Fermi surface x = µ are u ′ 0 [u0] = − 1 ˙f , u′ 0 (µ) = − 1 t1 u ′′ 0 [u0] = − ¨f ˙f 3 , u′′ 0 (µ) = − 2t2 t 3 1 , u ′′′ 0 [u0] = 1 ˙f 3 ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ ... f ˙f − 3 ( ¨f ˙f ) 2⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , u′′′ 0 (µ) = 6t1t3 − 12t 2 2 t 5 1 . (5.7) 104 A dot denotes a partial derivative with respect to u0 in these expressions. The subsequent perturbative contributions to u are determined by the h̵-organization of the GelfandDikii polynomials (5.3) and the disk level string equation. The general structure is simply ∑ k tkr (g) k = 0, (5.8) where r (g) k denotes the full contribution to Rk at order h̵2g when the expansion of u is taken into account. The g = 1 equation is ˙fu1 − 1 12 [ ... f (u ′ 0 ) 2 + 2 ¨fu′′ 0 ] = 0. (5.9) The solution and its value at x = µ, expressed in terms of the function f and the coupling constants, are u1 = 1 12(f ′) 2 ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ ... f ˙f − 2 ( ¨f ˙f ) 2⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , u1(µ) = 3t1t3 − 4t 2 2 6t 4 1 . (5.10) It will prove convenient to consider the relationship between the perturbative corrections ug and the derivatives of the leading order solution u0. For example, the second derivative u ′′ 0 is related to the first derivative u ′ 0 by u ′′ 0 = u ′ 0 d du0 u ′ 0 , (5.11) via a straightforward application of the chain rule. At g = 1 one finds u1 = − u ′ 0 12 d 2 du2 0 u ′ 0 . (5.12) It is likely that there are similar relationships expressing each contribution ug in terms of derivatives of lower order contributions to u. A proof of this would probably also involve the recursion relation of the 105 Gelfand-Dikii polynomials and their h̵-expansion. If such a total derivative relation exists at each g, it would simplify certain correlation functions in the closed string sector. At the next order in h̵, the string equation is u2 ˙f + 1 2 (u 2 1 − 1 3 u ′′ 1 ) ¨f + 1 6 ( 1 10 u (4) 0 − u ′′ 0u1 − u ′ 0u ′ 1) ... f + 1 2 ( 1 20 (u ′′ 0 ) 2 + 1 15 u ′ 0u ′′′ 0 − 1 6 (u ′ 0 ) 2 u2) f (4) + 11 366 (u ′ 0 ) 2 u ′′ 0 f (5) + 1 288 (u ′ 0 ) 4 f (6) = 0. (5.13) The solution, expressed purely in terms of f, is u2 = 1 1440 ˙f 9 ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ − 980 ¨f 5 − 420f (4) ˙f 2 ¨f 2 + 1760 ... f ˙f ¨f 3 + ˙f 3 (102 ... f f(4) − 5f (6) ˙f) + ˙f 2 (64f (5) ˙f − 545 ... f 2 ) ¨f ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ . (5.14) When we evaluated it at x = µ, it is u2(µ) = −3920t 5 2 + 10560t1t3t 3 2 − 5040t 2 1 t4t 2 2 + 15t 2 1 (128t1t5 − 327t 2 3 ) t2 + 18t 3 1 180t 9 1 + (102t3t4 − 25t1t6) 180t 9 1 . (5.15) It is possible to determine which coupling constants appear when the function u (n) g is evaluated at x = µ. Let k ′ g,n denote the largest index of the tk that shows up in u (n) g (µ). It can be shown by induction that k ′ g,n = 3g + n. For g = 0, the function u (n) 0 will contain one appearance of f (n) [u0], and f (n) [0] ∝ tn. For g ≥ 1, the result follows from analysis of the h̵-expansion of the Gelfand-Dikii polynomials. The base case g = 1, n = 0 has an appearance of f (3) . Inducting on g follows, and any nonzero value of n is included trivially. 106 5.1.1 Examples Here we provide a collection of the functions ug and their derivatives evaluated for interesting examples: JT gravity, the minimal string, and the Virasoro minimal string. Their coupling constants can be found in Table 4.1. These examples will be reconsidered when we look at the perturbative expansion of correlation functions later, but nevertheless this subsection is particularly result-heavy. The first model we consider is JT gravity. The leading order contribution u0 satisfies √ u0 2π I1(2π √ u0) + x = 0. (5.16) The first derivative u ′ 0 is given in terms of u0 by u ′ 0 [u0] = − 1 0F˜ 1(1, π2u0) , (5.17) where the tilde denotes that the hypergeometric function is regularized and we have used 0F˜ 1 (a + 1 2 , x 2 16) = ( x 4 ) 1 2 −a Ia− 1 2 ( x 2 ) . (5.18) Further, the second and third derivatives are u ′′ 0 [u0] = − π 2 0F˜ 1(2, π2u0) 0F˜ 1(1, π2u0) 3 , u ′′′ 0 [u0] = π 4 0F˜ 1(1, π2u0) 0F˜ 1(3, π2u0) − 3 0F˜ 1(2, π2u0) 2 0F˜ 1(1, π2u0) 4 . (5.19) Hence the first correction to the potential is given by u1[u0] = π 4 12 0F˜ 1(1, π2u0) 0F˜ 1(3, π2u0) − 2 0F˜ 1(2, π2u0) 2 0F˜ 1(1, π2u0) 4 . (5.20) 107 Its first derivative is given by u ′ 1 [u0] = π 6 12 70F˜ 1(1, π2u0)0F˜ 1(2, π2u0)0F˜ 1(3, π2u0) − 80F˜ 1(2, π2u0) 3 0F˜ 1(1, π2u0) 6 − π 6 12 0F˜ 1(1, π2u0) 2 0F˜ 1(4, π2u0) 0F˜ 1(1, π2u0) 6 . (5.21) Second, we consider the (2, 2p − 1) minimal string. The leading order contribution u0 satisfies u0 2F1 (1 − p, p, 2,− 4π 2u0 (2p − 1) 2 ) + x = 0. (5.22) The first and second derivatives are u ′ 0 [u0] = − 1 2F1 (1 − p, p, 1,− 4π2u0 (2p−1) 2 ) , u ′′ 0 [u0] = − 4p(p − 1)π 2 2F1 (2 − p, p + 1, 2,− 4π 2u0 (2p−1) 2 ) (2p − 1) 2 2F1 (1 − p, p, 1,− 4π2u0 (2p−1) 2 ) . (5.23) The first correction to u is given by u1[u0] = − 8p 2 (p − 1) 2π 4 2F1 (2 − p, p + 1, 2,− 4π 2u0 (2p−1) 2 ) 2 3(2p − 1) 4 2F1 (1 − p, p; 1;− 4π2u0 (2p−1) 2 ) 4 + 2p(p − 1)(p 2 − p − 2) 2F1 (1 − p, p, 1,− 4π 2u0 (2p−1) 2 ) 2F1 (3 − p, p + 2, 3,− 4π 2u0 (2p−1) 2 ) 3(2p − 1) 4 2F1 (1 − p, p; 1;− 4π2u0 (2p−1) 2 ) 4 . (5.24) Finally, consider the Virasoro minimal string, and recall that Q = b + b −1 , Qˆ = b −1 − b, and that b is the parameter from the Liouville action. The leading order contribution u0 satisfies 2 √ 2 ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ I0(2πQ√ u0) − I0(2πQˆ √ u0) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ + x = 0. (5.25) 108 The first and second derivatives are u ′ 0 [u0] = − b 2 2 √ 2π 3 1 (b 2 + 1) 2 0F˜ 1 (2, Q2π 2u0) − (b 2 − 1) 2 0F˜ 1 (2,Qˆ2π 2u0) , u ′′ 0 [u0] = − b 2 8π 4 (b 2 + 1) 4 0F˜ 1 (3, Q2π 2u) − (b 2 − 1) 4 0F˜ 1 (3,Qˆ2π 2u) ( (b 2 + 1) 2 0F˜ 1 (2, Q2π 2u0) − (b 2 − 1) 2 0F˜ 1 (2,Qˆ2π 2u0) ) 3 . (5.26) 5.2 The DJM Equation For the remainder this work, when solving the DJM equation we will redefine the constant s as z 2 = −s. When plotting or applying the solutions to statistical analysis, we will replace s with E to match the naming convention in the double scaling procedure. This is the standard uniformizing coordinate transformation familiar from studying the spectral curve in matrix models, and is not a coincidence since standard interpretations of FZZT branes in matrix models link the parameter z to the matrix eigenvalues. The DJM equation written in terms of z is (u + z 2 )R 2 − h̵2 2 RR′′ + h̵2 4 (R ′ ) 2 = h̵2Γ 2 . (5.27) Unlike the string equation (5.4) in double scaled β = 2 Wigner-Dyson models, the DJM equation admits real solutions on all of R, which we saw briefly in Section 4.2.1. As discussed above, open string perturbation theory is always recovered in the limit x → −∞. An interesting feature of supersymmetric theories described in this fashion is that they naturally incorporate D-branes by having this perturbative regime built into their equations of motion, which is in contrast to the non-supersymmetric theories where extra work had to be done to add branes. Nevertheless, the x → −∞ solutions to the DJM equation provide a sort of universal description of the open string sectors of the class of matrix models describing (2, #) minimal (super)strings. 109 5.2.1 Open String Sector The Gelfand-Dikii resolvent Rˆ has the expansion Rˆ(x, z) = 1 h̵ ∞ ∑ k=0 (−1) k z 2k+1 (2k)! 2 2k+1(k!) 2 Rk, (5.28) in terms of the differently normalized Gelfand-Dikii polynomials. The shifted coupling constants (4.32) that map the Novikov equation for the non-supersymmetric closed string sector to the open string sector become tk → tk + 2h̵Γ (−1) k (2k)! 2 2k+1(k!) 2 z −(2k+1) , x → x + 2h̵Γ(2z) −1 . (5.29) Define the shift coefficients ζk = (−1) k (2k)! 2 2k+1(k!) 2 , (5.30) for k ≥ 0. The full string equation of the open-closed model can be written ∞ ∑ k=1 (tk + 2h̵Γζkz −(2k+1) )Rk + x + h̵Γz −1 = 0. (5.31) Denote the expansion of u by u = ∞ ∑ g,h=0 h̵2g+hΓ h ug,h, (5.32) with u0,0 ≡ u0. At leading order we still have the disk level string equation f(u0) + x = 0. The equation for u0,1 can be written ˙fu0,1 + 2φ + z −1 = 0, (5.33) 110 where φ = ∑ ∞ k=1 ζkz −(2k+1)u k 0 . The function φ can be written in closed form 2φ + 1 z = 1 √ u0 + z 2 , (5.34) and thus u0,1 = − 1 ˙f √ u0 + z 2 . (5.35) The function φ + (2z) −1 is merely the leading solution to the Gelfand-Dikii equation for the resolvent Rˆ. The functions f and φ are ubiquitous in perturbation theory of the open-closed string equation, much like f is in solving the closed string equation. A comparison of the leading order solution u0 and u0,1 for different values of z 2 = −E for the (2, 3) minimal string is shown in Figure 5.1. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 Figure 5.1: The leading order (left) and first open string sector correction (right) to the potential u for the (2, 3) minimal string. As the energy grows, the nonzero vertical asymptote in u0,1 gets pushed toward x → −∞. The general structure of the string equation for higher order corrections to u0 is ∞ ∑ k=1 (tkr (g,h) k + 2ζkz −(2k+1) r (g,h−1) k ) = 0, (5.36) 111 where r (g,h) k is the full contribution to Rk at order h̵2g+hΓ h . For g = 0, h = 2 we have ˙fu0,2 + 1 2 ¨fu2 0,1 + φu˙ 0,1 = 0. (5.37) The solution u0,2 is u0,2 = 1 2 ˙f d du0 [ ˙fu2 0,1 ] = − ˙f + 2(u0 + z 2 ) ¨f 4(u0 + z 2) 2 ˙f 2 . (5.38) For g = 0, h = 3 the string equation is ˙fu0,3 + ¨fu0,1u0,2 + 1 6 ... f u3 0,1 + φu˙ 0,2 + 1 2 φu¨ 2 0,1 = 0, (5.39) with the solution u0,3 = − 3 ˙f 2 + 9 ˙f(1 + 2(u0 + z 2 ) ¨f) + 8(u0 + z 2 ) 2 (3 ¨f 2 − ... f ) 48 ˙f 4(u0 + z 2) 7/2 (5.40) For g = 1, h = 1 ˙fu1,1 − 1 12 ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ f (4) u0,1(u ′ 0 ) 2 + 2 ... f (u0,1u ′ 0 ) ′ − 2 ¨f(6u0,1u1,0 − u ′′ 0,1 ) + φ¨(u ′ 0 ) 2 + 2 ˙φu′′ 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ + φu˙ 1,0 = 0. (5.41) Notice that the g = 1 equation involves both h = 0 and h = 1 functions. The formalism so far is adequate to describe any number of open strings boundaries as long as they all have the same endpoint mass, and in order for the masses to be different we need a slight generalization of the string equation. The difference in the two scenarios (i.e. multiple boundaries on the same brane versus multiple boundaries spread out across several branes) is depicted in fig. 5.2. The more general framework developed below will be able to simultaneously handle both of these types of interactions. 112 Figure 5.2: The left panel shows a contribution at O(Γ 2 ) that involves only one brane. The right panel shows a contribution at O(Γ 2 ) that involves two different branes. At the level of the Hermitian matrix model the appropriate change to make is to introduce a potential for each mass Vopen(M) = γ N ∑ i tr log (1 − m2 i M2 ) . (5.42) Therefore after double scaling the string equation will be R + 2h̵Γ∑ i Rˆ(x; zi) = 0, (5.43) which will be related to a dual closed string equation by the shifted coupling constants tk → tk + 2h̵Γζk ∑ i z −2k−1 i . (5.44) The solution u will now have the slightly more complicated expansion u(x; z1, . . . ) = ∞ ∑ g=0 ug,0(x) + h̵Γ∑ i u0,1(x; zi) + h̵2Γ 2 ∑ i,j u0,2(x; zi , zj) + ⋯, (5.45) 113 where the contribution at O(Γ h ) will depend on h of the z variables at a time. The functions ug,h are not exactly the same as before, but will be related. We will suppress the x-dependence for convenience. At O(Γ) the string equation is ˙f ∑ i u0,1(zi) + 2∑ i φ(u0, zi) = 0, (5.46) which is solved by setting u0,1(zi) to be the same as (5.35). Define the total contribution at this order as u0,I ≡ ∑i u0,1(zi). At the next order in Γ ∑ i,j [ ˙fu0,2(zi , zj) + 1 2 ¨fu0,1(zi)u0,1(zj) + 2 ˙φ(zi)u0,1(zj)] = 0, (5.47) which has the solution u0,2(zi , zj) = − 1 ˙f [ 1 2 ¨fu0,1(zi)u0,1(zj) + 2 ˙φ(zi)u0,1(zj)] . (5.48) This is the natural generalization of the result for one string mass, and although each of these is not symmetric under i ↔ j, the total contribution to u is. A succinct way to express the whole contribution at Γ 2 , which is more reminiscent of the one-mass solution, is u0,II = − 1 ˙f [ 1 2 ¨fu2 0,I + 2Φ˙ u0,I ] , (5.49) where Φ = ∑i φ(zi). We still have the identity u0,II = 1 2 ˙f d du0 ( ˙fu2 0,I). (5.50) 114 It is unclear a priori how to do the same analysis for the open string sector of supersymmetric theories starting from the DJM equation. Such a generalization may necessitate a function like the Gelfand-Dikii resolvent, but which depends on an arbitrary number of the spectral parameters and solves a more complicated differential equation than the Gelfand-Dikii equation. However we will develop an open-closed duality in Chapter 6 that will allow us to bypass this complication. 5.2.2 0A Closed String Sector The results presented above for the non-supersymmetric models rely entirely upon perturbation theory in the x → −∞ regime, where the function u is determined via the initial condition R = 0. It was shown in [65, 66] that the closed string perturbation theory of the type 0 models is captured by the x → +∞ part of the solution to the DJM equation. In both systems the open string sector is described in the x → −∞ region, but the closed string contribution to the function u is determined differently. Put more succinctly, the open string sector is described by the formulae derived from the R = 0 formalism for the bosonic theories as well as the type 0 theories. The initial condition for perturbation theory in the positive-x region is simply u0 = −z 2 [11]. Solutions are typically found for z = 0, especially in the closed string sector. The results collected below all have smooth transitions from z ≠ 0 to z = 0, and so we take the point of view that it is worthwhile to present the more general family of solutions. When we consider perturbation theory in the closed string sector in the next section we will set z = 0 when considering supersymmetric theories. The nontrivial solutions for the first couple values of h with g = 0 are u0,2 = 1 (f(−z 2) + x)(f(−z 2) + 2z 2 ˙f(−z 2) + x) u0,4 = − ˙f(−z 2 )[32xz2 (f(−z 2 ) + x) ˙f(−z 2 ) + 16x(f(−z 2 ) + x) 2 ] 8x 2(f(−z 2) + x) 3 (2z ˙f(−z 2) + f(−z 2) + x) 3 (5.51) 115 The contributions at odd powers of h vanish, which is not a feature unique to g = 0. At g = 1 we have u1,0 = − 1 4(f(−z 2) + x)(f(−z 2) + 2z 2 ˙f(−z 2) + x) , u1,2 = ˙f(−z 2 )[z 2 ˙f(−z 2 ) + f(−z 2 ) + x] x 2(f(−z 2) + x) 3 (2z 2 ˙f(−z 2) + f(−z 2) + x) 3 × [2z 2 (f(−z 2 ) + 3x) ˙f(−z 2 ) + (f(−z 2 ) + 6x)f(−z 2 ) + 5x 2 ] x 2(f(−z 2) + x) 3 (2z 2 ˙f(−z 2) + f(−z 2) + x) 3 , (5.52) and at g = 2 u2,0 = ˙f(−z 2 )[ − 4z 4 (f(−z 2 ) + 3x) ˙f(−z 2 ) 2 − 2z 2 (3f(−z 2 ) + 10x)(f(−z 2 ) + x) ˙f(−z 2 )] 8x 2(f(−z 2) + x) 3 (2z 2 ˙f(−z 2) + f(−z 2) + x) 3 − ˙f(−z 2 )[(f(−z 2 ) + x) 2 (2f(−z 2 ) + 9x)] 8x 2(f(−z 2) + x) 3 (2z 2 ˙f(−z 2) + f(−z 2) + x) 3 . (5.53) These perturbative contributions are typically summed to be displayed based on overall factors of h̵ as u = u0 + h̵2u1 + h̵4u2 + ⋯, where u1 = u1,0 + Γ 2u0,2 and u2 = u2,0 + Γ 2u1,2 + Γ 4u0,4. There are three features of note in the perturbative solution u. First, the result depends only on even powers of Γ. This has the interpretation that Γ counts R-R flux and is consistent with the dependence of the RR field in the 0A string theory on the Liouville field [65, 84]. Second, notice that when z = 0, the function f(0) = 0, so the O(h̵2 ) result becomes independent of the coupling constants and the O(h̵4 ) contribution depends only on t1. Clearly the dependence of the highest rank coupling constant on the order of perturbation theory of u(µ) is different as compared to the non-supersymmetric case. Finally, the presence of the function f in each perturbative contribution to u means that there is no longer a universal term at O(h̵) for z ≠ 0. With the parameter z turned off, each 0A model is characterized at leading order 116 by the universal Bessel model. For a generic value of z, the FZZT brane associated to it probes theorydependent information at each order in perturbation theory past leading order. 0 1 2 3 4 5 6 -3 -2 -1 0 1 2 3 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 9 Figure 5.3: The combined leading and first subleading contributions to u in the k = 1 0A multicritical model for different values of E, for Γ = 0 (left) and Γ = 1 (right), all with h̵ = 0.1. We can see as E is increased the function is translated up and to the right. The potential switches concavity at Γ = 1 2 . Only the positive branch of each solution is displayed. The simplest multicritical 0A model is the (2, 4) super-minimal model coupled super Liouville theory and corresponds to having only t1 nonzero, which we set to 1. The solution up to O(h̵4 ) is u(x; z) = −z 2 + h̵2 (Γ 2 − 1 4 ) x 2 − z 4 − h̵4 (4Γ2 − 1) ((4Γ2 − 9) x 2 + (1 − 4Γ2 ) z 4 ) 8x (x 2 − z 4) 3 + ⋯. (5.54) To visualize this potential, note that −z 2 = E is meant to be positive. Plots for Γ = 0, 1 are displayed for various values of E in Figure 5.3. 5.3 N = 2 Theories There have been several recent advancements in relating JT supergravity with extended supersymmetry to matrix models, and hence advancements in the study and use of matrix models themselves. With the increase in the amount of supersymmetry comes the added complexity of dealing with R-charge multiplets and BPS states. It was shown in [77] that each multiplet is described by a matrix model in the 117 (α, β) = (1 + 2Γ, 2) Altland-Zirnbauer class, where here Γ is interpreted as the number of BPS states, which is dependent on the R-charge of the supermultiplet and the R-charge of the supercharge. The authors of [77] use this to deduce the matrix model describing N = 2 JT gravity. A decomposition of this theory in terms of multicritical models, akin to the ones done for JT gravity and 0A JT supergravity, was constructed in [78]. It is apparent from this minimal model decomposition that, although the N = 2 multicritical models are still described by a variation of the DJM equation, the boundary conditions differ from the 0A case. Further, the coupling constants t˜k and Fermi surface µ˜ are necessarily functions of E0, the smallest energy of the non-BPS states in the multiplet. For the duration of the discussion of N = 2 models we will use a tilde to implicitly denote when an object has E0 dependence. 5.3.1 Closed String Sector An arbitrary N = 2 model is defined by coupling constants t˜k, which can be written t˜k = tkJ (E0), where J (0) = 1. The function J is model dependent and can be fixed by the BPS density of states. The theory will be controlled by the DJM equation, but with some modification, and closed string perturbation theory will once again be contained in the x → +∞ regime. The desired solution for u is seeded by the initial condition u0(x) = µ˜ 2E0 ( ˜f(−z 2) + x)( ˜f(−z 2) + 2z 2 ˜˙ f(−z 2) + x) − z 2 , (5.55) which is obtained by taking the simultaneous limit h̵ → 0 and Γ → ∞, such that h̵Γ = µ˜ √ E0 remains constant and finite, of the 0A solution, keeping only the leading contributions1 [78]. In principle, this limit allows one to obtain the N = 2 solution directly from the 0A one. The desired leading order solution u0 is a viable leading order solution to the DJM equation with z = 0 and h̵2Γ 2 = µ˜ 2E0 because perturbation theory is meant to be an asymptotic series in both h̵ and 1/∣x∣. However, this poses an issue if we wish to apply the techniques developed in this chapter, which have 1The combination h̵Γ is necessarily small, which is why higher powers of h̵Γ are not included in the limit to obtain the new u0. 118 relied solely on an expansion in h̵ — in the systems studied earlier, the initial conditions allowed us to ignore the values of x, though as we saw each progressive perturbative contribution had increasingly negative powers of x attached. A natural question to ask is whether or not we can adapt the DJM equation so that it still produces the desired N = 2 results using our techniques. We will make two proposals, each with their own benefit. The first will involve uniformly rescaling the coupling constants by h̵2 . This method will not involve any new parameters, but interpreting how the solution is organized relative to the 0A solution will be difficult. The second method will involve a scaling transformation on the coupling constants that will depend on each of their specific mass dimensions. This will involve introducing a new parameter ϵ, but the organization of the solution will be more straightforward. First, consider the slight changes R = h̵2 ∑ k t˜kRk + x, h̵2Γ 2 ≡ α 2 , (5.56) with the DJM equation otherwise unchanged. This can be obtained from the 0A equation by rescaling the coupling constants by h̵2 in addition to the E0-dependent factor. If we expand u = u0 + ⋯ as before, then the leading order solution to this modified equation is our desired result, and perturbation theory proceeds as before for higher powers of h̵, with u = ∑ ∞ n=0 h̵2nun. The first correction to u is simply given by u1 = − 1 4x 2 − 2α 2 ˜f ( α 2 x2 ) x 3 . (5.57) The next perturbative correction is u2 = 4α 6 ... ˜f ( α 2 x2 ) ˜˙ f ( α 2 x2 ) + 3x 2 [α 2x (8α 2 ˜f ( α 2 x2 ) + 9x) ˜˙ f ( α 2 x2 )] 6x 9 + 4α 4 ( ˜˙ f ( α 2 x2 ) + 1) ¨˜f ( α 2 x2 ) + x 3 ˜f ( α 2 x2 ) (6α 2 ˜f ( α 2 x2 ) + x) 6x 9 . (5.58) 119 The simultaneous limit that keeps α 2 constant leaves behind terms proportional to h̵2gα 2n . Define the deficit ∆ = 2g − 2n between the powers of α and the remaining power of h̵ in the 0A solution. Let c be the total power of the coupling constants in a particular term (e.g. t 2 1 and t1t2 both have c = 2). Then a term from the 0A solution should appear in the N = 2 solution at O(h̵∆+2c ). To check, consider again the simplest theory, k = 1. Leaving the E0-dependent coupling constant t˜1 arbitrary, we find u = α 2 x 2 − h̵2 ( 2t˜1α 4 x 5 + 1 4x 2 ) + h̵4 ( 7t˜2 1α 6 x 8 + 5t˜1α 2 x 5 ) − h̵6 ( 30t˜3 1α 8 x 11 + 217t˜2 1α 4 4x 8 + 9t˜1 8x 5 ) + ⋯. (5.59) Comparing this to (5.54) confirms the assertion. By rescaling the 0A coupling constants by h̵2 , it is possible that we have interfered with the topological interpretation of u. Previously, the power of h̵ indicated the Euler characteristic of the associated surface. Even after absorbing the powers of h̵ coming from RR flux insertions into powers of α, the remaining factors of h̵ in the 0A solution still ostensibly correspond to genus-counting. However, the powers of h̵ in the N = 2 solution are also affected by the rescaled coupling constants. In order to maintain the same topological interpretation we would have to undo the rescaling, which corresponds to summing over terms with the same value of ∆ and dropping the portion related to c. The second proposal is motivated by the scaling dimensions of the quantities that make up the string equation and takes inspiration from the previous method, where we saw that the specific presence of the coupling constants can be used to organize the solutions. Therefore we might wonder if there is a way to track when certain coupling constants, or certain powers of them, appear. Make the change R = ∑ k ϵ k t˜kRk + x. (5.60) 120 This is equivalent to shifting the coupling constants tk → ϵ k tk, including t0 = x, which makes this a transformation based on length dimension instead of mass. The expansion of u is given by u = ∞ ∑ g,n=0 h̵2g ϵ n ug,n, (5.61) with u0 ≡ u0,0 as before. Recall also that [u]mass = 1, and that the DJM equation is the outcome of the Callan-Symanzik equation representing the scaling symmetry of the theory. If we were to further rescale u → ϵ −1u, then the DJM equation would be invariant. The fact that we are not doing this makes the transformation non-trivial. Perturbation theory proceeds in powers of h̵ and ϵ. At h̵0 , R = x + f(ϵu0) + ϵ 2 u0,1 ˙f(ϵu0) + ϵ 4 2 ( ¨f(ϵu0)u 2 0,1 + 2 ˙f(ϵu0)u0,2) + ⋯, (5.62) which yields the solutions u0 = α 2 x 2 , u0,1 = − 2α 4 t˜1 x 5 , u0,2 = α 6 (7t˜2 1 − 2t˜2x) x 8 . (5.63) At the next order in h̵, we use the h̵-expansion of the Gelfand-Dikii polynomials as well as the same expansion in ϵ to get u1,0 = − 1 4x 2 , u1,1 = 5α 2 t˜1 x 5 , u1,2 = − 7α 4 (31t˜2 1 − 10t˜2x) 4x 8 . (5.64) 121 Clearly these solutions contain many of the same terms as the previous approach, and thus also parts of the 0A solution. In this case we can also derive a distinct relationship between the perturbative N = 2 solutions and the 0A ones. By direct comparison, it turns out that ug,h 0A ≅ u∆,h+g−1 N =2 , (5.65) where ∆ = g − h 2 . That we recover the 0A perturbative solution is not surprising, since we are still fundamentally solving the DJM equation. However, the non-trivial scaling transformation has lead to a different organization of perturbation theory, with perhaps the most drastic change being the leading order solution u0. 5.3.2 Open String Sector The simultaneous limit involving h̵Γ should also be applied to the open string sector, since it is also controlled by the same differential equation as the closed string sector. However, based on the argument used when matching the leading order eigenvalue density, the behavior of u0 in the x < 0 region should be unchanged by this. Therefore the string equation must not change in this region, and perturbation theory will be organized in terms of h̵ and α. On the other hand, it is interesting to consider what would happen if the closed string scaling transformation were applied in the open string sector. The brane cosmological constant is dimensionful, with [z]mass = 1, and will need to be scaled as well. It is convenient here to use ϵ 2 as the parameter, sending tk → e 2k tk and z → z/ϵ. The string equation is obtained by plugging these transformations into (5.31) along with the replacements h̵Γ = α and tk → t˜k, to get ∞ ∑ k=1 ϵ 2k (t˜k + 2ϵαζkz −2k−1 )Rk + x + ϵαz−1 = 0. (5.66) 122 Denoting the N = 2 solution by u, if one continues to not rescale u, then this string equation yields wrong perturbative results. Instead, we must also perform the transformation u → u/ϵ 2 . Since [Rk]mass = k, we get ∞ ∑ k=1 (t˜k + 2ϵαζkz −2k−1 )Rk + x + ϵαz−1 = 0, (5.67) which has the same structure as (5.31), but with h̵Γ → ϵα. Therefore the mapping between the 0A solutions and the N = 2 solutions is trivial: ug,h ≅ ug,h. (5.68) This is a direct consequence of the scaling symmetry possessed by the DJM equation that we purposefully avoided in the closed string sector. 5.3.3 Interpreting The Scaling The natural identification of ϵ in the open string sector is as a brane-counting parameter, since the contribution to u at O(ϵ h ) is the same (up to changes introduced by having α as a parameter in the solutions) as the contribution to u at O(Γ h ) in the 0A theory. There thus a sense in which the open string sector is invariant under the simultaneous limit h̵Γ → α. When we compute correlation functions in the next chapter, we will employ this identification. A crucial difference between the open and closed sectors is how the function u is scaled. In the closed sector, u is invariant under the transformation even though it is dimensionful. This means that u represents some type of fixed point, for example if ϵ was being introduced in the context of an RG flow. The significance of this difference is not presently understood. 123 Chapter 6 Open And Closed String Correlation Functions The notion of open-closed duality in two-dimensional string theory has existed since at least the early 90s [51, 69], but was vastly overshadowed by the end of the decade by another open-closed duality, the AdS/CFT correspondence [42]. As we saw in Chapters 4 and 5 there is a simple connection between matrix models dual to closed string theories and matrix models dual to open string theories which is facilitated by the KJ and DJM equations. Since the KJ equation (4.29) can be obtained from the closed string equation (5.4) by a z-dependent shift of the coupling constants tk, we have an apparent duality transformation between the two sectors of the string theory [51]. The inclusion of open strings in the matrix model implies (or necessitates) the additional inclusion of branes. Recall that there are two important types of D-brane in minimal string theory, ZZ and FZZT branes [35, 85–87], in terms of which we are afforded an alternative interpretation of the matrix model [88]. The index i = 1, . . . , N on the matrix labels a configuration of N ZZ branes, and the matrix element Mij represents a stringy degree of freedom stretching between the i th and j th branes. The matrix model naturally lives on ZZ branes because of the identification between the eigenvalues λ and the associated Liouville direction φ via φ ∼ log λ. Since ZZ branes represent Dirichlet boundary conditions on φ, they should be associated directly with the eigenvalues λ. In fact, as we saw in Section 4.3, the principal ZZ 124 branes of a minimal string are in one-to-one correspondence with the nontrivial independent zeros of the spectral curve. On the other hand, some number of FZZT branes can be inserted into the theory via determinant operators det(M + µi), where µi corresponds to the cosmological constant on each brane. Therefore in this picture the FZZT branes act as probes of the ZZ brane spacetime. By integrating in and out auxiliary degrees of freedom, one can equivalently interpret the matrix model with determinant operator insertions as describing strings stretching between the FZZT and ZZ branes, as well as strings stretching just between the FZZT branes. Put another way, a β = 2 Wigner-Dyson model naturally incorporates ZZ branes; in the finite-N regime there are N of them present. Recall that the (α, β) = (1 + 2Γ, 2) Altland-Zirnbauer models can be thought of as building an N × N positive Hermitian matrix out of a (N + Γ) × N rectangular matrix. In this sense, the new matrix model has “extra” ZZ branes, which in the closed string sector can be thought of as RR flux. On the other hand, an (α, β) = (1 + 2Γ, 2) Altland-Zirnbauer model can be obtained from a β = 2 Wigner-Dyson model via the inclusion of determinant operators, i.e. FZZT branes. In this chapter we will study correlation functions of two operators that represent boundary insertions on the worldsheet. As noted in the discussion of JT gravity in Section 4.3, the two most basic types of boundary in that theory are the asymptotic AdS boundary and bulk geodesic boundaries. Macroscopic loop operators [21] are interpreted in the context of JT gravity as creating asymptotic boundaries [9, 72]. An operator representing the insertion of geodesic loops has existed in the literature (see [51, 70] for example) but was not appreciated in the modern context fully until [36, 79]. The point of view advanced in [36] is that the geodesic boundary operator naturally exists in the closed string sector of a matrix model. Open-closed duality then implies that such observables computed in the closed string sector involving geodesic loops are equivalently computed by the Γ ≠ 0 solutions in the open string sector. The work presented here is drawn from [36], as well as new results in Sections 6.2.4, 6.2.5, and 6.3. 125 6.1 Macroscopic Loop Perturbation Theory We saw in the preceding chapter that the h̵-expansion of u is utilized to solve the non-linear equation of motion efficiently. Using the relationship between u and F (4.15) we are able to explicitly compute the genus expansion of the closed string free energy F = ∞ ∑ g=0 h̵2g−2Fg. (6.1) The free energy is the generating function of connected diagrams, and so all observables in the theory will have an expansion in powers of h̵, with the powers being linked to the Euler characteristic of the corresponding surface. In the purely closed string context we will usually limit ourselves to considering surfaces with one type of boundary, where the Euler characteristic is χ(g, n) = 2g − 2 + n. One such class of boundary-having observables is the correlation functions of macroscopic loop operators e −βH, which are dual to path integrals on surfaces with asymptotic boundaries. This is in analogy with the explicit construction in [9] for JT gravity that we discussed at the end of Chapter 4. For this reason we call Zn(β1, . . . , βn) ≡ ⟨e −β1H⋯e −βnH⟩ a path intergral in all matrix models. The topological expansion of this path integral is still denoted Zn(β1, . . . , βn) = ∞ ∑ g=0 h̵2g−2+nZg,n(β1, . . . , βn), (6.2) The approach to perturbation theory taken here will involve the expansion of e −βH in terms of the point-like operators σk given in (4.21). The intimate relationship that these operators have with both the KdV organization of the theory and the underlying closed string physics means that all formulae derived here will apply equally well to supersymmetric and non-supersymmetric theories. The prescription for performing computations in specific models will be to evaluate the functions u (n) g (µ) in the model’s closed 126 string sector. In the perturbative expansion of the one-point function we will also need to evaluate certain integrals on different supports depending on whether or not the theory is supersymmetric. 6.1.1 General Formulae Correlation functions of the macroscopic loop operators are computed, by virtue of the relationship (4.19) between the point-like operators σk and the vector fields ξk, by repeated action of the KdV vector fields on the function u [21] ⟨e −β1H⋯e −βnH⟩ = 2 n−2h̵n−2 π n/2 √ β1⋯βn ∞ ∑ ki=0 (−1) k1+⋯+kn β k1 1 ⋯β kn n (k1)!⋯(kn)! × ∫ µ dx∫ x dx′ ξk1⋯ξkn ⋅ u(x ′ ). (6.3) Therefore each perturbative contribution to Zn will be Zg,n(β1, . . . , βn) = 2 n−2 π n/2 √ β1⋯βn ∞ ∑ ki=0 (−1) k1+⋯,+kn β k1 1 ⋯β kn n (k1)!⋯(kn)! × ∫ µ dx∫ x dx′ [ξk1⋯ξkn ⋅ u(x ′ )] g . (6.4) The notation [⋅]g denotes the full contribution at order h̵2g , after taking into account the h̵-expansion of the Gelfand-Dikii polynomials and the topological expansion of u. In this language the expectation value of a single macroscopic loop operator is ⟨e −βH⟩ = h̵ 2 √ πβ ∞ ∑ k=0 (−1) kβ k k! ∫ µ −∞ dx Rk. (6.5) The full leading order contribution to Rk is u k 0 , and hence Z0,1(β) = 1 2 √ πβ ∞ ∑ k=0 (−1) kβ k k! ∫ µ −∞ dx u0(x) k , 127 a geometric representation of which is displayed in Figure 6.1. This integral can be computed in two steps. Figure 6.1: The disk geometry as it is usually presented in nAdS2 with an asymptotic boundary of renormalized length β, but without the wiggling Schwarzian mode. Even though a matrix model with arbitrary tk may not have an explicit geometric interpretation, it is useful to keep this picture in mind in the topological expansion. First, notice that the bounds of the integral straddle x = 0, where there is a change in how u0 is determined. After splitting the integral, the portion from −∞ to 0 can be computed in terms of the coupling constants by changing the integration variable to u0, using the disk level string equation to calculate the Jacobian. The result in this region is Z (−) 0,1 (β) = 1 2 √ πβ3/2 ∞ ∑ k=1 k!tkβ −k+1 . (6.6) This contribution is the same in every model we study here. However, the portion from 0 to µ depends on which model we are studying. In the non-supersymmetric case where µ = 0 this portion actually doesn’t exist. In the 0A models where u0 = 0, the only non-vanishing term in the sum is at k = 0, giving Z (+) 0,1 = hµ̵ /2 √ πβ. Since the KdV time t0 is identified with our variable x, it is natural to also make the connection t0 ↔ µ. Doing so, the full expression fo Z0,1 in 0A theories can be obtained by changing the lower bound of summation to k = 0 in the expression for Z (−) 0,1 in (6.6). This motivate a general prescription 128 for matrix model calculations. In addition to integrating over the full Fermi sea in g = 0, n = 1 objects, we must include k = 0 in sums. In the N = 2 models, where u0 is nonzero in this region, it is convenient to once again pull the infinite sum inside the integral to get Z (+) 0,1 (β) = 1 2 ( µe˜ −E0β √ πβ − µ˜ 2E0Erfc ( √ E0β)) . (6.7) In totality, we have Z0,1(β) = ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪ ⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ Z (−) (β) Non-supersymmetric, Z (−) (β) + µ 2 √ πβ 0A, Z (−) (β) + 1 2 ( µe˜ −E0β √ πβ − µ˜ 2E0Erfc ( √ E0β)) N = 2. (6.8) It is often said that the supersymmetric models incorporate a second topological point at k = 0, which we can see borne out by the additional term here in the macroscopic loop expectation value. The first perturbative correction Z1,1(β) can also be computed exactly in terms of the coupling constants. The full contribution to Rk at order h̵2 is Rk = ⋯ + h̵2 (kuk−1 0 u1 − k(k − 1) 12 u k−3 0 [(k − 2)(u ′ 0 ) 2 + 2u0u ′′ 0 ]) + ⋯. (6.9) The correction is then Z1,1(β) = − 1 2 √ πβ ∫ µ −∞ dx e−βu0(x) [βu1(x) + β 2 12 (2u ′′ 0 (x) − β(u ′ 0 (x))2 )] . (6.10) Since this expression is meant to be perturbative, the integral should only be done in the region where closed string perturbation theory is defined for the model (see Table 5.1). In the non-supersymmetric 129 models, by invoking the relationships (5.11) and (5.12) and integrating by parts, one is left with purely surface terms Z1,1(β) = − √ β 24√ π (u ′ 0 (µ)β + u ′′ 0 (0) u ′ 0 (0) ) , or written in terms of the coupling constants Z1,1(β) = √ β 24t1 √ π (β + 2t2 t1 ) . (6.11) In the supersymmetric models the integration is restricted to be from 0 to µ. However, due to the different choices for u0 in 0A and N = 2 theories we will consider this calculation in more detail when we look at examples in Section 6.1.2.2. The leading order contribution to the two-point function, Z0,2, relies on the fact that ∫ x dx′ [ξk1 ⋅ ξk2 ⋅ u(x ′ )] 0 = k1k2u ′ 0 (x)u0(x) k1+k2−2 . (6.12) After performing the sum and computing the remaining integral the result is Z0,2(β1, β2) = √ β1β2 π(β1 + β2) e −(β1+β2)u0(µ) . (6.13) In non-supersymmetric and 0A theories, we usually have u0(µ) = 0, and the so the exponential factor is trivial. However, it will be present in N = 2 models. The matrix model calculation confirms the well-known fact from topological recursion that the path integral ‘double trumpet’ geometry is universal (independent of the coupling constants). 130 The g = 1 contribution to ⟨σkσl⟩ is ∬ [ξk ⋅ R ′ l ] 1 = −kl[u1u k+l−2 0 + (u ′ 0 ) 2u k+l−3 0 12 (−5 + 4(k + l) − k 2 − kl − l 2 ) + u ′′ 0u k+l−3 0 6 (2 − (k + l))]∣ x=µ . (6.14) In theories with u0(µ) = 0 this produces formally divergent results. Throwing away the diverging pieces yields Z1,2(β1, β2) = √ β1β2 π (u1 − ( β 2 1 + β1β2 + β 2 2 12 ) (u ′ 0 ) 2 + ( βT 6 ) u ′′ 0) e −βT u0 R R R R R R R R R R Rx=µ , (6.15) where βT = β1 + β2. It is possible that these terms should be kept in N = 2 theories; determining this would require comparison to path integral calculations on the gravity side. The g = 0 contribution to the three-point function uses [ξk1 ⋅ ξk2 ⋅ ξk3 ⋅ u] 0 = k1k2k3 d 2 dx2 u ′ 0u k1+k2+k3−3 0 . (6.16) Performing the three sums yields Z0,3(β1, β2, β3) = − 2 √ β1β2β3 π 3/2 u ′ 0 (µ)e −βT u0(µ) , (6.17) where here βT = β1 + β2 + β3. The g = 1 contribution to ⟨σkσlσn⟩ is ∬ [ξk ⋅ ξl ⋅ R ′ l ] 1 = kln d dx [u1u k+l+n−3 0 − ( k 2 + l 2 + n 2 + kl + kn + ln 12 ) (u ′ 0 ) 2 u k+l+n−5 0 − ( k + l + n 6 ) u ′′ 0u k+l+n−4 0 ] + kln[ (kln 12 ) (u ′ 0 ) 3 u k+l+n−6 0 ]∣ x=µ . (6.18) 131 This contribution also has divergent terms that must be thrown away, giving Z1,3(β1, β2, β3) = 2 √ β1β2β3 π 3/2 e −βT u0 ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ − u ′ 1 + βT u1u ′ 0 − ⎛ ⎝ β 3 1 + β 3 2 + β 3 3 12 + β1β2β3 + β1β2(β1 + β2) + β1β3(β1 + β3) + β2β3(β2 + β3) 6 ⎞ ⎠ (u ′ 0 ) 3 + ⎛ ⎝ 2(β 2 1 + β 2 2 + β 2 3 ) + 3(β1β2 + β1β3 + β2β3) 6 ⎞ ⎠ u ′ 0u ′′ 0 − βT 6 u ′′′ 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ R R R R R R R R R R Rx=µ . (6.19) A general formula exists for the g = 0 part of the higher order correlation functions starting with n = 3 [23, 28, 89] Z0,n(β1,⋯, βn) = − √ β1⋯βn 2π n 2 [( ∂ ∂x) n−3 (u ′ 0 (x)e −βT u0(x) )] x→µ , (6.20) which is easily confirmed using the matrix model technology here. For n ≥ 3 one finds that [ξk1⋯ξkn ⋅ u] 0 = k1⋯kn d n−1 dxn−1 u ′ 0u k1+⋯kn−n 0 , (6.21) which can be shown by repeated application of u k (∂/∂u) and taking the h̵0 -part. Integrating this twice and performing the sum reproduces (6.20). 6.1.2 Examples The models listed in Table (4.1) each constitute interesting matrix models defined by interpolations between multicritical models. In the non-supersymmetric cases this endows them with the attractive feature of producing finite results. As we will see, the supersymmetric models are better behaved overall. We reproduce their coupling constants and basic information here for convenience. 132 6.1.2.1 Non-Supersymmetric Theories The individual multicritical models display an unusual aversion to macroscopic loop perturbation theory for k ≥ 2. In fact the only finite results are for Z0,1 and Z0,2. For the p th model, the former is evaluated by setting tk = δkp in (6.6), while the latter is universal. Most other quantities diverge due to the combination of having u0 = (−x) 1/p and the Fermi surface at µ = 0. The failure of the multicritical models to yield finite results can be traced to the fact that the string equation is defined by the monomial f(u0) = u p 0 , which for p ≥ 2 satisfies ˙f(0) = 0. The only model in this family lacking this behavior is the Gaussian model p = 1, and in order for the matrix model to provide nontrivial finite results we have to consider an interpolation including the topological Gaussian point. The first interpolated examples we consider are the (2, 2p − 1) minimal strings [90]. Their coupling constants are tk = π 2k−2 k!(k − 1)! 4 k−1 (p + k − 2)! (p − k)!(2p − 1) 2k−2 . (6.22) A nontrivial check of the formalism above is that the g = 1 correction to Z1 is computed to be Z1,1(β) = √ β 24√ π [β + (1 − 1 (2p − 1) 2 ) π 2 ] , (6.23) which matches the result obtained in [90]. The first perturbative correction to the macroscopic loop twopoint function is readily computed to be Z1,2(β1, β2) = √ β1β2 π ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 2π 4 p(p − 1)(3p(p − 1) + 2) 3(2p − 1) 4 + ( β 2 1 + β1β2 + β 2 2 12 ) + 4π 2 p(p − 1) (2p − 1) 2 ( βT 6 ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ . (6.24) 133 Recall that JT gravity can be thought of as the p → ∞ limit of the minimal string, or equivalently as the infinite interpolation between multicritical models defined by the coupling constants tk = π 2k−2 k!(k − 1)! . (6.25) The disk path integral Z0,1 is confirmed to be [9] Z0,1(β) = 1 2 √ πβ3/2 e π 2 β . (6.26) The first perturbative correction to this is also confirmed to be Z1,1(β) = √ β 24√ π [β + π 2 ]. (6.27) In [9], the authors do not directly compute Z1,2 in the matrix model, but instead derive the general procedure to compute it and other perturbative corrections using Weil-Petersson volumes. The macroscopic loop formalism correctly gives Z1,2(β1, β2) = h̵ √ β1β2 24π [3π 4 + 2 (β 2 1 + β1β2 + β 2 2 ) + 4π 2 βT ]. (6.28) Another interesting model defined by an infinite interpolation is the Virasoro Minimal String [74, 75]. The couplings constants of this model are tk = 2 √ 2 π 2k+1 (k!) 2 (Q 2k − Qˆ2k ) , (6.29) where Q = b + b −1 , Qˆ = b −1 − b. (6.30) 134 The parameter b and the two combinations Q,Qˆ are familiar from Liouville theory, which similarly to the non-supersymmetric multicritical matrix models has a direct connection to the theory. The disk path integral is given by Z0,1(β) = √2π β [e c−1 6β − e − cˆ−1 6β ] , (6.31) where c = 1 + 6Q2 and cˆ = 1 − 6Qˆ2 are central charges. This is the result one would get from Laplace transforming the universal Cardy density of states for two-dimensional CFTs1 . The first correction to the macroscopic loop one-point function, typically called the torus path integral in the CFT context, is given by Z1,1(β) = √ β˜ 96√ π [β˜ + c − 13 24 ] , (6.32) where we have introduced a rescaled length β˜ = β/2π 2 . This includes the familiar (c − 13)/24 from torus calculations. 6.1.2.2 Supersymmetric Theories An interesting feature of the closed string sector of the DJM equation is that, while we typically set z = 0, is is entirely permissible to have Γ ≠ 0 and to interpret it in terms of RR flux. As we saw at the level of the function u, there is a dramatic difference between Γ = 0 and Γ ≥ 1. This difference continues to manifest itself in the perturbative corrections to macroscopic loop correlators. For example, in 0A theories with the initial condition u0 = 0, the term Z1,1 is given by Z1,1(β) = − √ β 2 √ π ∫ µ 0 dx Γ 2 − 1 4 x 2 = 1 2µ √ π (Γ 2 − 1 4 ) √ β, (6.33) where we’ve dropped a divergence at x = 0 and left the Fermi surface unfixed, but nonzero. The bounds of integration are different, starting at x = 0 and ending at x = µ, because the closed string sector of the 1 Indeed this is in a way by construction, since this density of states is used to determine the coupling constants [75]. 135 theory is located in the x > 0 region and the matrix model naturally depends only on sub-Fermi level information [68]. There are two features of Z1,1 worth mentioning. First, since the total contribution to u at O(h̵2 ) is universal in 0A models with z = 0, the g = 1 correction to the one-point function is also universal. This means that the each multicritical model has a well-defined Z1,1. Second, Z1,1 changes sign as Γ is increased from 0 to 1, being identically 0 for Γ = 1 2 . The special value Γ = 1 2 is the appropriate choice for 0A JT supergravity [60, 67]. For n > 2, the leading order contributions Z0,n = 0 irrespective of the value of Γ because u0 = const., which is true even if z is turned on. For z = 0, the g = 1 corrections Z1,n all have the form √ β1⋯βn (Γ 2 − 1 4 ), which can easily be predicted using the fact that u1 is the highest order correction to u that can appear. One 0A theory of note is N = 1 JT supergravity, defined by the coupling constants [67] tk = π 2k (k!) 2 . (6.34) Using the appropriately modified version of (6.6), that is starting the sum at k = 0, we get the disk path integral Z0,1 = 1 2 √ πβ e π 2 β . (6.35) A more recent interesting 0A theory is the N = 1 generalization of the Virasoro Minimal String considered above. It’s coupling constants are [76] tk = 2 √ 2 π 2k+1 (k!) 2 (Q 2k + Qˆ2k ) , (6.36) giving the disk path integral Z0,1(β) = √2π β [e c−1 6β + e − cˆ−1 6β ] . (6.37) 136 Note that this differs from the ordinary Virasoro Minimal String only in the relative sign between the exponentials. To conclude the supersymmetric examples, we consider the N = 2 theory. Due to the unique nature of u0, the g = 0 contributions to both the macroscopic loop two- and three-point functions have nontrivial exponential factors Z0,2(β1, β2) = √ β1β2 πβT e −βT E0 , Z0,3(β1, β2, β3) = 4E0 √ β1β2β3 µπ˜ 3/2 e −βT E0 . (6.38) In N = 2 JT supergravity the Fermi surface is given by µ˜(E0) = sin(2π √ E0) 4π2 √ E0 , and at leading order in small E0 is µ˜ ≈ 1/2π [78]. Inserting this into the expression for Z0,3 gives precise agreement with [77], up to an overall factor of 4. Most importantly, the factor of 2π that comes from the U(1) R-symmetry and the exponential dependence on E0 and β are accounted for. Evidently the matrix model agrees with the path integral calculation only in the small E0 limit. Since µ˜ and E0 are related to the density of BPS states, E0 is constrained to have a relatively small value. It is possible that the matrix model definition captures physics away from the strictly small E0 limit. To compute path integrals at higher genus we need to choose one of the methods of defining perturbation theory for the N = 2 string equation discussed in section 5.3. We will choose the epsilon prescription. There are infinitely many terms contributing at O(h̵2 ), labelled by their corresponding power of ϵ. Taking ϵ small for convenience, the leading order contribution at g = 1 will be computed by inserting u0 = α 2 /x 2 and u1,0 = −1/4x 2 into (6.10). The result is Z1,1(β) = √ β 24√ πµ˜ (2βE0 − 3)e −E0β + O(ϵ). (6.39) 137 6.1.3 KdV Flows An important fact about the non-supersymmetric examples considered here has to do with which coupling constants the perturbative corrections to Zn depend on. What we have seen is that the quantity Zg,n generically depends on the coupling constants t1 through t3g−2+n. Notice that the maximum possible index is very nearly the virtual dimension of the moduli space Mg,n of genus-g Riemann surfaces with n punctures. In practice, the maximum value of the index is determined by the minimum of 3g − 2 + n and the largest index of the nonzero coupling constants. The models that always have 3g − 2 + n as the maximum index in Zg,n must have all coupling constants turned on. This feature can be proven generally. Due to the origin of the contributions Zg,n in the topological expansion, each one contributes at order h̵χ(g,n) , where χ(g, n) = 2g − 2 + n is the Euler characteristic. On the other hand, as we have seen these partition functions depend on the h̵-expansion of u and its derivatives evaluated at x = µ. Since the function u (n ′ ) 2g ′ contributes at h̵2g ′+n ′ , the partition function Zg,n can depend on combinations of the functions u (n ′ ) 2g ′ as long as 2g ′+n ′ ≤ χ(g, n). Let kg,n denote the largest index of the coupling constants appearing in Zg,n, and let k ′ g ′ ,n′ denote the same quantity for u (n ′ ) 2g ′ (µ). The index kg,n will be the maximum value of k ′ g ′ ,n′ subject to the constraint 2g ′ + n ′ ≤ 2g − 2 + n. The maximum occurs when g ′ = g and n ′ = n − 2 and is kg,n = max g ′ ,n′ k ′ g ′ ,n′ = 3g − 2 + n, (6.40) as desired. A possible interpretation for why kg,n is one more than the dimension of the corresponding moduli space is the fact that t1 is a special coupling constant. It is the variable dual to the topological k = 1 model and therefore can be discarded as a “degree of freedom” when compared to the models with higher values of k. Evidently, since perturbation theory forces most quantities to depend on t −1 1 , the 138 topological model must be included to provide a sort of stability, because otherwise setting it to 0 introduces divergences. Excluding t1 from the count decreases kg,n by one, matching the moduli space dimension. Macroscopic loop perturbation theory is able to access information about an underlying moduli space because of the deep and rich connection between the KdV hierarchy and topological gravity. The point-like KdV operators σk can be thought of as being dual, in a loose sense, to the tautological classes ψk on Mg,n, whose intersection numbers are generated by the matrix model free energy F. There is a strict limit on how many of the ψk can be considered in an intersection number determined precisely by the dimension dg,n = 3g − 3 + n. 6.2 Geodesic Loop Perturbation Theory In the preceding section we studied the perturbative expansion of macroscopic loop operator correlation functions by computing them entirely in the closed string sector, since the function u was evaluated in the closed string regime of its string equation. In this section we incorporate the open string solutions obtained in Section 5.2. This will allow us to accomplish two things: we will study surfaces with both asymptotic and geodesic boundaries, and we will establish a duality between operators in the closed string sector and operations done on the free energy in the open string sector. Since the function u can be determined in two different regimes, corresponding to open and closed strings, we can define two different free energies, each determined using (4.15). In the context of the DJM equation, open-closed duality has historically consisted either of the statement that there is a transformation which maps open string perturbation theory to closed string perturbation theory in non-supersymmetric models [51], or that the x → ±∞ regimes of perturbation theory of the DJM equation describe the different sectors [66]. We will exhibit a different — although fundamentally still related — kind of duality here that applies for non-supersymmetric and supersymmetric theories alike, which relates the correlation functions of a certain operator in the closed string sector to derivatives of the open string free energy. 139 The general topic of boundaries in two-dimensional quantum gravity, string theory, and matrix models has been studied for a long time, often with different terminology used to discuss the same things. For example, there is a strong connection between FZZT branes and macroscopic loops, related to one another using identities for the determinant and Taylor series. For this reason the two terms have historically been linked in a way that could be confusing, especially in the context of the results presented here. As will become evident in this section, there are two types of boundaries that one can naturally consider: ones created by the insertion of a macroscopic loop operator into a closed string model, and ones created by the insertion of an FZZT brane. The holographic dictionary established for dilaton gravity in [9] relates macroscopic loop correlators in the matrix model to gravity path integrals on hyperbolic surfaces with asymptotic boundaries. For this reason, it is convenient to picture the macroscopic loops of the previous section in this fashion, as depicted in fig. 6.1. The new type of boundary that is introduced by the brane has a natural geometric interpretation as well. To continue the comparison to [9], the authors utilize a gluing procedure that involves hyperbolic surfaces with geodesic boundaries. The worldline of an open string with a nonzero endpoint mass inside the worldvolume of a brane will naturally be a geodesic with a finite length. For this reason, the boundaries introduced in the matrix model by the brane will be considered to be geodesic boundaries. This identification will be justified by computing the trumpet partition function, which in JT gravity is the path integral on the surface shown in Figure 6.3, as well as the matrix model resolvents Wg,n. The trumpet partition function is a feature that is universal to the types of models considered here, including the supersymmetric ones. We will continue to use the letter h to count powers of Γ, which as we will see corresponds to the number of geodesic boundaries. The free energy in the open string sector has the topological expansion F = ∑ g,h=0 h̵2g−2+hΓ hFg,h, (6.41) 140 where each Fg,h is found by integrating ug,h. It is once again possible to consider correlation functions of the point-like operators. When it is not clear from context we will write ⟨⋅⟩ open g,h to indicate that the open string free energy is involved, with a similar notation for the closed string sector. The statement of the open-closed duality involves a special operator which we will call ωz, defined by ωz = 4h̵Γ ∞ ∑ k=1 ζkz −2k σk−1. (6.42) The duality is expressed by ⟨ωz1⋯ωzhO⟩ closed g = 2 h! ∂ ∂z1 ⋯ ∂ ∂zh ⟨O⟩ open g,h ∣ conn. , (6.43) where O is any operator consisting of the σk’s. The interpretation of this is that the operator ωz represents the insertion of a geodesic boundary with open string endpoint mass −z 2 . The right hand side of (6.43) is to be evaluated using the solution to the multi-brane string equation, but because that solution takes into account the possibility of the worldsheet ending on the same brane multiple times, we only want to look at the case where it lands on each brane once. Even though a multi-brane generalization of the DJM equation to describe multiple open strings in supersymmetric theories does not currently exist, the desired open string quantities will be computed using the left hand side of (6.43), in the closed string sector. Actually, a way to generate the solutions that should exist — although they aren’t obtained from a differential equation — is to apply the duality relation to the identity operator to calculate the open string free energy, and then invert the relationship between F and u. 6.2.1 Boundary Operator Thinking of the coupling constants tk and the open string endpoint mass −z 2 as parameters of the theory, since derivatives with respect to tk correspond to inserting the operators σk into correlation functions, 141 it is natural to expect that a derivative with respect to z should be represented by an operator insertion as well. This was shown to be true in the open string matrix model for the case of an individual multicritical model in [51]. One can show by direct matrix model computation that [24] ∂u ∂z = 4zh̵ΓRˆ′ . (6.44) By expanding Rˆ′ in terms of the Gelfand-Dikii polynomials and using the KdV flow equations it is clear that the definition of ωz in (6.42) is the appropriate operator dual to the z-derivative. By construction then, the expectation value of ω (in the closed string sector) will be intimately related to the Gelfand-Dikii resolvent. In particular ⟨ωz⟩ = 4Γz h̵ ∫ µ −∞ R dx. ˆ (6.45) Therefore knowing the Gelfand-Dikii resolvent, which is the solution to the Gelfand-Dikii differential equation (2.22), is enough information to determine the expectation value of ωz at any order in perturbation theory. But, this integral of the Gelfand-Dikii resolvent is nothing but the complexified spectral density of the closed string theory, the leading order contribution to which is often referred to as the spectral curve. The spectral density contains important statistical information about the model, making its connection to a brane insertion especially interesting. We will explore this more in Section 6.2.5. As we have seen, it is common to work in terms of the complexified variable z when computing the correlation functions of the matrix resolvent (M−λ) −1 in the double scaling limit. Recall that it is natural to redefine the components of the topological expansion of one of these correlation functions in terms of the objects Wg,n(z1, . . . , zn). Given the strong connection between the matrix resolvent and the determinant, 142 it is no surprise that there is still a close connection between FZZT brane quantities and the functions W. In fact, using open-closed duality the connection can be written as Wg,n(z1, . . . , zn) = 2⟨ωz1⋯ωzn ⟩g = 2 n! ∂ nFg,n ∂z1⋯∂zn (6.46) This claim will be explored more thoroughly in the next section. It is also well-known that the (real) spectral density is given by the discontinuity of the resolvent R across the real axis. A web of connections is beginning to unfold. The story continues with the fact that the resolvent functions Wg,n are the Laplace transforms of the generalized Weil-Petersson volumes [8]. Hence the functions Wg,n contain information about the intersection numbers on a generalized moduli space, the generating function for which — at least in the case where the matrix model is dual to JT gravity — is the open string free energy F [50, 51]. The connections are summarized below in fig. 6.2. Figure 6.2: The web of connections forming in the string equation formalism, which includes important matrix model objects and the Weil-Petersson volumes. 143 6.2.2 Trumpet Partition Function Studying macroscopic loops is still a fruitful endeavor in the open-closed theory. A modern motivation for studying them is the trumpet partition function defined in the JT gravity context in [9] Ztr(β, b) = 1 √ 4πβ e − b 2 4β , (6.47) which is the Schwarzian path integral on a (hyperbolic) surface with a geodesic boundary of length b and an asymptotic boundary of length β. The same quantity was computed in [60], where they found it has the same functional dependence on b and β. In both instances Ztr was computed specifically using the (super) Schwarzian action, and so given the specificity one might expect that the corresponding object computed in the matrix model would at least be dependent on the coupling constants tk. In [79] it was argued that FZZT branes have a natural place in the landscape of general topological gravity. Moreover they argued that the trumpet partition function was related to an FZZT brane insertion, and that it should be independent of the coupling constants tk. Their argument relates the trumpet partition function to the Liouville wavefunction, which has no explicit knowledge of the coupling constants used to define the matrix model. In the context of the present work, this independence will arise from a specific dependence on the function f that defines the leading order open string equation. In order to begin justifying the duality, we will compute the trumpet partition function two ways. The trumpet, shown in fig. 6.3, is an asymptotic disk with a geodesic boundary insertion in the bulk. Hence it should show up at order Γ in the macroscopic loop expectation value ⟨e −βH⟩ open 0,1 in the open string theory. The expectation value of each σk is computed as before in terms of the KdV vector fields, but now we expand the Gelfand-Dikii polynomial Rk to order Γ. Therefore the g = 0, h = 1 contribution to Z1 is given by ⟨e −βH⟩0,1 = − 1 2 √ β π ∫ µ −∞ dx e−βu0 u0,1. (6.48) 144 Figure 6.3: The trumpet geometry with one asymptotic boundary of renormalized length β, and one geodesic boundary (shown in red) of length b. Plugging in the solution for u0,1 in terms of its Taylor series in u0 and changing the integration variable to u0 using x = −f(u0), the bounds of integration go from u0(µ) = E0 to ∞. The result is ⟨e −βH⟩0,1 = 1 √ 4πβ ∞ ∑ k=0 ζkΓ(k + 1, βE0)z −2k−1 β −k , (6.49) where Γ(a, b) is the incomplete gamma function. Recall that the mass of the endpoint of the open string serves as a cosmological constant for the open string boundaries. This fact is captured in the FZZT language by the fact that the determinant operator is really related to the Laplace transform of a fixed-length insertion. The open string endpoint mass is therefore Laplace conjugate to geodesic boundary length. In order to retrieve the trumpet partition function we must take the inverse Laplace transform of ⟨e −βH⟩0,1. This gives L −1 z [⟨e −βH⟩0,1](b) = 1 √ 4πβ ∞ ∑ k=0 Γ(k + 1, βE0) (k!) 2 (− b 2 4β ) k , (6.50) 145 By expanding the gamma function in a power series and resuming, we get L −1 z [⟨e −βH⟩0,1](b) = Ztr(b, β) − βE0 √ 4πβ ∞ ∑ k=0 (−βE0) k k!(k + 1) 1F2 (k + 1; 1, k + 2;− E0b 2 4 ) . (6.51) For nonzero E0, there is an infinite tower of corrections to the trumpet partition function, whereas for E0 = 0 the corrections disappear leaving the desired result. Comparing this to the N = 2 where in general E0 ≠ 0, this is a departure from the result obtained in [77], where they found that the N = 2 trumpet was simply e −βE0Ztr. In the following section we will establish a different gluing procedure that utilizes just Ztr, with a volume that takes into account the exponential E0-dependence. Although the result for that volume will also differ from [77], the path integral calculations will match up. Crucially, this expression is nevertheless still independent of the coupling constants, making it universal for the class of models considered here. Next consider the correlator ⟨ωze −βH⟩, computed in the closed string sector. We can utilize the previously stated g = 0 two-point function of the σk’s from (6.12). There will be one remaining integral with respect to x, which we can change into an integral over u0 by using the factor of u ′ 0 present in the integrand. The bounds of the integral are once again from E0 to ∞. Integrating once with respect to z we get L −1 z [⟨ωze −βH⟩0](b) = 2Γ √ 4πβ ∞ ∑ k=0 Γ(k + 1, βE0) (k!) 2 (− b 2 4β ) 2 , (6.52) which is equal to the result derived in the open string sector, up to the desired factor of 2Γ. Apart from computing the trumpet partition function purely using the matrix model, we have also established the duality in this particular case. 146 As a bonus, and in furtherance of justifying the proposed duality, consider the multi-brane correlator ⟨e −βH⟩0,2 in the open string sector. Expanding the Gelfand-Dikii polynomial Rk to order h̵2Γ 2 and performing the sum, ⟨e −βH⟩0,2 = − √ β 4 √ π ∫ ∞ E0 du0e −βu0 ˙f(u0) (βu2 0,I − 2u0,II) . (6.53) Using the multi-brane generalization of the identity (5.38), integrating by parts, and keeping the term with no repeated factors of z1 or z2 yields the surface term ⟨e −βH⟩0,2 = − √ β 2 √ π u ′ 0 (µ) √ E0 + z 2 1 √ E0 + z 2 2 e −βE0 . (6.54) In the closed string sector, we get the result ⟨ωz1ωz2 e −βH⟩0 = − Γ 2 2 √ β π z1z2u ′ 0 (µ) (E0 + z 2 1 ) 3/2(E0 + z 2 2 ) 3/2 e −βE0 (6.55) which establishes the duality for this case as well. 6.2.3 FZZT Brane Partition Function It is worth pausing briefly to compare the open-closed duality methodology to the more standard way of dealing with FZZT branes in the closed string sector of matrix models. Recall that FZZT branes are represented by determinant operators FZZT brane ←→ det(λ − M). (6.56) 147 By using the identity det(λ + M) = exp{tr log(λ + M)}2 , it is possible to rewrite the FZZT brane operator in terms of macroscopic loop insertions tr log(λ + M) ≅ −∫ ∞ ε dβ β e −λβ tr e −βM, (6.57) where ε → 0 and we have dropped a log ε divergence. It must also be understood that we will need to make the substitution λ → −λ later. Therefore det(λ + M) = ∞ ∑ n=0 (−1) n n! n ∏ i=1 ∫ ∞ 0 dβi βi e −λβi tr e −βiM. (6.58) Recall that the operator tr e −βiM represents a macroscopic loop prior to double scaling. The expectation value of a single brane operator is denoted by ⟨Ψ(E)⟩ = ψ(µ, E) in the double scaling limit: ψ(µ, E) = ∞ ∑ n=0 (−1) n n! ⟨ n ∏ i=1 ∫ ∞ 0 dβi βi e βiE e −βiH⟩ , (6.59) where now each expectation value includes connected and disconnected parts. The leading contribution comes from the leading parts of the totally disconnected terms in the correlation functions. That is, ψ(µ, E) ≈ ∞ ∑ n=0 (−1) n n! n ∏ i=1 ∫ ∞ 0 dβi βi e βiE ⟨e −βiH⟩ 0 . (6.60) Using the result derived in section 6.1, the brane expectation value is ψ(µ, E) ≈ ∞ ∑ n=0 (−1) n n! n ∏ i=1 1 2h̵ √ π ∫ µ −∞ dx∫ ∞ 0 dβi β 3/2 i e −βi(−E+u0) . (6.61) 2This is precisely the reason why the matrix model describing open strings has a logarithmic potential. 148 Noting that ∫ ∞ 0 dβi β 3/2 i e −βi(−E+u0) = −2 √ π √ −E + u0, (6.62) we find ψ(µ, E) ≈ exp { i h̵ ∫ µ √ E − u0} . (6.63) This is the expected result, since before double scaling the brane expectation value computes the Nth orthogonal polynomial pN (λ) associated to the matrix potential. As discussed, these orthogonal polynomials double scale to wavefunctions solving the Schrodinger equation with potential u. The leading order WKB approximation to the wavefunction is given by ψ(x, E) ≈ exp {± i h̵ ∫ x √ E − u0} , (6.64) and the index N is located at the Fermi surface x = µ. By incorporating higher order corrections to Rk, we can compute higher orders of the WKB expansion of the wavefunction. Although the preceding result was derived via the h̵-expansion, it should be noted that the outcome represents leading-order physics. The next term in the perturbative expansion of ψ(µ, E), which is really the first true perturbative contribution, comes from breaking the terms with an even number of macroscopic loops into connected two-point functions and keeping only the leading contributions there. Each of those terms contributes a factor of h̵0 . The calculation requires the integral I = 1 π ∫ ∞ 0 dβ1dβ2 β1β2 √ β1β2 β1 + β2 e −(β1+β2)(−E+u0(µ)) ≅ − 1 2 log(−E + u0(µ)), (6.65) 149 where we’ve dropped a divergent piece. Each term with an even number 2n of macroscopic loops will contribute I n with a degeneracy of (2n)! 2nn! . Thus we find ∞ ∑ n=0 1 (2n)! (2n)! 2 nn! I n = e −iπ/4 (E − u0(µ))−1/4 . (6.66) At this order the brane wavefunction is ψ(µ, E) ≈ 1 (E − u0(µ))1/4 exp { i h̵ ∫ µ √ E − u0 − iπ 4 } , (6.67) which is consistent with what one calculates in the Airy model, including the phase. The natural object to describe a single FZZT brane insertion in the open-closed duality language is the free energy of the open string sector. Using the relationship between F and u, we find F0,1 = ∫ µ √ u0 + z 2. (6.68) Recalling that z 2 = −E, the leading order contribution to the brane partition function would then be e F0,1/h̵ = ψ(µ, E) at leading order. We have to divide by h̵ to account for the fact that F0,1 is defined without any explicit h̵ present. The first perturbative correction to the partition function comes from exponentiating F0,2/2, where the free energy is evaluated using the single-brane solution. We only need to consider F0,2 at this order because we must have h ≠ 0 in order to be describing the brane. The factor of 2 is inserted to account for the symmetry to swap the endpoints of the string. Using (5.38) we have F0,2 = − 1 2 log(E0 + z 2 ). (6.69) The exponential and extra factor of 2 convert this into the desired portion of the WKB wavefunction. The calculations done the more traditional way become increasingly tedious at higher orders, and it can be 150 somewhat hard to parse when disconnected versus connected correlators need to be used. In the openclosed duality language, one simply computes the open string free energy and exponentiates. The next contribution would come from F1,1. 6.2.4 The Matrix Kernel The double scaled matrix kernel K(E, E′ ) primarily exists to calculate statistical properties of the random matrix model, and as such provides a powerful tool for probing the spectrum of the theory. It has been utilized recently to access the discrete nature of the black hole spectrum in JT gravity using a proposed non-perturbative completion (see for example [59]). The kernel itself is typically calculated using the double scaled wavefunctions ψ via K(E, E′ ) = ∫ µ −∞ dx ψ(x, E)ψ(x, E′ ). (6.70) As discussed in the previous subsection, the wavefunction ψ(µ, E) is the partition function of an FZZT brane with cosmological constant E. This implies that the kernel is a “composite D-brane probe that is well-suited to detecting the ‘location’ of individual energies" [59]. Since the one-point function of ωz computes the full eigenvalue density, we may expect that its correlation functions compute the correlations between multiple eigenvalues. The two-point function of ω is ⟨ωz1ωz2 ⟩ = 16h̵2Γ 2 ∞ ∑ k1,k2=0 ζk1 ζk2 z −2k1 1 z −2k2 2 ∬ µ ξk1 ⋅ R ′ k2 . (6.71) Performing the sums, we can write ⟨ωz1ωz2 ⟩ = 16h̵2Γ 2 z1z2 ∬ µ ˆξ(z1) ⋅ Rˆ′ (z2), (6.72) 151 where we have introduced ˆξ(z) = ∑ ∞ k=0 Rˆ(k+1) (z) ∂ ∂u(k) . We note that the partial derivatives with respect to u (k) are understood to only act on functions in the closed string sector. Three interesting observations arise from this. First, much like we have the identification σk ↔ ∂ ∂tk ↔ ξk, now we also have ωz ↔ ∂ ∂z ↔ ˆξ(z), (6.73) which is evidenced by rewriting (6.44) as ∂u(z) ∂z = 4h̵Γˆξ(z) ⋅ u. Second, using open-closed duality we have the double-flow of the two-brane open string sector solution ∂ 2u(z1, z2) ∂z1∂z2 = 16h̵2Γ 2 ˆξ(z1) ⋅ Rˆ′ (z2) = 16h̵2Γ 2 ˆξ(z1) ⋅ ˆξ(z2) ⋅ u, (6.74) which generalizes (6.44). This is a specific example of the comment made at the beginning of this section about constructing the supersymmetric multi-brane solutions without having a string equation. Third, the connection between the matrix resolvent and the kernel can be exploited to arrive at Wg,2(z1, z2) ∼ 4z1z2h̵2K(−z 2 1 ,−z 2 2 ) 2 [91]. Therefore ∬ µ ∂ 2u(z1, z2) ∂z1∂z2 (dx) 2 = 4h̵2K(−z 2 1 ,−z 2 2 ) 2 = 16∬ µ ˆξ(z1) ⋅ Rˆ′ (z2)(dx) 2 . (6.75) The left equality relates the kernel to the open string sector solution, while the right one relates the kernel to the closed string sector solution. These equalities provide a sharper image of the relationship between the kernel and its statistical importance, and the presence of branes. 6.2.5 Eigenbranes And Microstates In [92] the authors observed that if one inserts two FZZT branes with identical cosmological constants into the finite-N matrix integral and performs the double scaling procedure, the eigenvalue density will 152 Frozen eigenvalue Disk ℏ = .1 Γ = 1 0 2 4 5 6 8 10 0 2 4 6 8 10 E ρ(E) Density of States With a Frozen Eigenvalue: GUE Figure 6.4: The two-brane density of states ρ0(E) h̵ + h̵Γ 2ρ0,2(E; 5) with h̵ = 0.1 and Γ 2 = .1 is represented by the solid purple line. The disk density ρ0(E) = √ E/π is represented by the dashed green line. An opening has appeared at E = 5, representing the fact that the eigenvalue has been “frozen” there. The width of the opening is directly related to the size of h̵Γ. The density actually asymptotes to −∞ both as E → 0 and E → 5. Divergences like this are common in the perturbative expansion of the density. contain a sort of cutout around the value of the cosmological constant. In particular, consider an ensemble of (N + 1) × (N + 1) Hermitian matrices. After diagonalizing, label the eigenvalues (λ0, . . . , λN−1, x). Then the (N + 1) × (N + 1) Vandermonde determinant can be factored into ∆N+1(λ0, . . . , λN−1, x) = det(x − M)∆N (λ0, . . . , λN−1), (6.76) where M is the random matrix. Therefore the full matrix integral can be written ZN+1 ∝ ∫ dx ⟨Ψ(x) 2 ⟩N , Ψ(x) = 1 √ hN det(x − M)e −NV (x)/2 . (6.77) The observable Ψ double scales to the wavefunction ψ(µ, E) evaluated at the Fermi surface, which we have seen is the FZZT brane partition function. 153 Frozen eigenvalue Disk ℏ = .1 Γ = 1 0 2 4 5 6 8 10 0 100 200 300 400 500 E ρ(E) Density of States With a Frozen Eigenvalue: (2,3) Minimal String Figure 6.5: The two-brane density of states ρ0(E) h̵ + h̵Γ 2ρ0,2(E; 5) with h̵ = 0.1 and Γ 2 = .1 is represented by the solid purple line. The disk density ρ0(E) = √ E π + 16πE 3 2 27 is represented by the dashed green line. The scale of the y-axis is different that the k = 1 model due to the presence of the k = 2 model and the fact that the extra coupling constant t2 > 1. Evidently, the two-brane open string sector of the matrix model should have an eigenvalue density with a frozen eigenvalue. This can be readily contextualized in the framework we have developed. Recall that the perturbative expansion of the density can be computed by inserting the WKB approximation of the wavefunction into eq. (4.7). See Appendix A for more details. A single frozen eigenvalue will thus be visible at O(Γ 2 ) in the expansion of ρ, with the brane cosmological constants set to be equal. In a general non-supersymmetric model the formula for the leading order two-brane correction is given in (A.15), which we reproduce here for convenience ρ0,2 = − 1 2π ∫ 0 du0 ˙f(u0) ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ ¨f(u0) (E′ − u0)(E − u0) 1 2 − 5 2(E′ − u0)(E − u0) 3 2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ . (6.78) The lower bound of integration is conventionally disregarded, as it often causes divergences in perturbative corrections like this. We evaluate this formula for the k = 1 model and for the (2, 3) minimal string. The 154 Frozen eigenvalue Disk ℏ = .1 Γ = 1 0 2 4 5 6 8 10 0 2 4 6 8 10 E ρ(E) Density of States With a Frozen Eigenvalue: chGUE Figure 6.6: The two-brane density of states in the simplest 0A model with h̵ = 0.1 and Γ 2 = .1 is represented by the solid purple line. The disk density ρ0(E) = √ E π + µ 2π √ E is represented by the dashed green line. plots are shown in figures (6.4) and (6.5). In these cases3 care must be taken with the result of the integral because of E = E ′ will be a branch point. The general behavior we expect to be caused by having a frozen eigenvalue at the level of perturbation theory is ρ(E) ∼ ρ0(E) − pδ(E − E ′ ), where p is a number. That is, there should be an infinitesimal puncture in the density at the eigenvalue E ′ . The story in 0A theories is very similar. The eigenvalue density will have extra terms coming from integrating over the region [0, µ]. The leading order part gets an extra term given by µ/2π √ E, and the two-brane correction ρ0,2 gets an extra term given in (A.17). The result for the simplest model, k = 1, is shown in Figure 6.6. In principle, with a large number of branes inserted in pairs at different values of E ′ , we could recover a perturbatively discrete looking spectrum. This is in contrast to the non-perturbatively discrete spectrum that is underlying the matrix model [59]. This allows us to construct a cartoonish geometric picture of twodimensional quantum gravity microstates using branes. The density ρ0,2 is associated to the correlation 3This will actually be true for all of the models that produce finite results. 155 function ⟨ω 2 z e −βH⟩0, which is an asymptotic disk with two holes in the bulk in the geometric interpretation. A large (finite) number of frozen eigenvalues would therefore correspond to a large number of holes in the bulk, which we can picture as approaching a “ribbon simplex,” not dissimilar to the ’t Hooft diagrams drawn pre-double scaling. Since the density is naturally constructed in terms of the brane cosmological constant, the sizes of the holes are constrained4 but allowed to fluctuate. With all eigenvalues frozen, i.e. a discrete spectrum, the geometric picture is completely destroyed. In order to pursue this cartoon further, we would need to compute ρ0,2h for increasingly large h. It is possible that, given the relationship between matrix model quantities and Weil-Petersson volumes explored in the next section, a large-h limit of the volume V0,2h or the resolvent W0,2h could be useful. 6.3 Generalized Weil-Petersson Volumes Recall that JT gravity path integrals can be expressed in terms of the Weil-Petersson volumes Vg,n. The purpose of the volume is to account for constituent surfaces in the gluing procedure that have only geodesic boundaries. In the geometric interpretation of the matrix model offered here, this is precisely the role of the correlation functions of ωz. A natural question to ask is whether other two-dimensional gravity path integrals can be computed in terms of corresponding volumes, and if so, is there a corresponding matrix model dual? In [73] minimal string theory was compared to its matrix model dual, and moduli space volumes were identified on the field theory side. In the Virasoro Minimal String, the authors and linked the theory to the moduli spaces of Riemann surfaces on the gravity side [74] . In [79] the authors start on the matrix model side and consider a generalization of Vg,n in any topological gravity background. While the process for constructing moduli space volumes for ordinary Riemann surfaces is clear, the picture becomes more complicated when supersymmetry is included. The jump to N = 1 super-Riemann 4 In the sense that large holes are penalized in the partition sum or path integral based on the value of the cosmological constant. 156 surfaces is fairly straightforward, but as the supersymmetry is extended further things become more difficult [60, 77]. As we have pointed out, a remarkable feature of the DJM equation (5.27) is that it has the capacity to describe both N = 1 (i.e. the 0A theories) [65, 66] and N = 2 superstrings [78]. This can be leveraged to make predictions not only about the physics of asymptotic boundaries in the theories but geodesic ones, and hence moduli space volumes, as well. Similar to the trumpet partition function in the preceding section, our aim in this section is to further justify the claim that correlation functions of ω compute the matrix model resolvent functions Wg,n. By computing them in the matrix model using the closed string operator formalism, we will show that it is possible to study more complicated supersymmetric theories and calculate their generalized Weil-Petersson volumes. Using the known results as a testing ground, we will be able to confidently make predictions about these quantities in N = 2 theories. By computing the resolvent functions Wg,n in terms of the closed string solution u we will be able to compute generalized Weil-Petersson volumes for arbitrary coupling constants. A similar procedure was performed in [79]. The added benefit of performing the analysis using the string equation is that it is much easier to define the volumes for supersymmetric theories. In both the non-supersymmetric and supersymmetric cases, we will define the volumes b1⋯bnVg,n(b;tk) = L −1 z [⟨ωz1⋯ωzn ⟩g](b), (6.79) where the boldface notation indicates taking the multi-variable inverse Laplace transform, and all powers of h̵ and Γ are disregarded. This allows us to make a general statement about the generalized volumes. 157 By evaluating the Laplace transforms, we can write Vg,n in terms of correlation functions of the σk in the closed string sector Vg,n(b;tk) = 2 n ∞ ∑ k1,...,kn=1 ζk1⋯ζkn (2k1)!⋯(2kn)! b 2k1−2⋯b 2kn−2 ⟨σk1−1⋯σkn−1⟩g. (6.80) Compare this to the form of the standard Weil-Petersson volumes in (2.45). Clearly the summand is structured exactly the same way, with a correlation function at genus-g and nearly identical powers of the geodesic lengths5 , but the sums themselves appear different: in (2.45) the sum is constrained by the dimension of the moduli space, whereas here the indices are summed over infinite ranges. But, recall the discussion in Section 6.1.3. At the time we were discussing the coupling constants that could appear in a non-supersymmetric macroscopic loop correlator at given g and n. Secretly though, the argument was about the σ correlation functions, meaning that we should also find here that each (non-supersymmetric) volume only depends on the first 3g − 2 + n coupling constants. This motivates the following conjecture, which is related to a comment made previously about the Gelfand-Dikii polynomials: the correlation function ⟨σk1−1⋯σkn−1⟩g must involve the Gelfand-Dikii polynomials arranged in such a way that when the sums in (6.80) are performed and the integrals are carried out, the result can be written as a sum over indices constrained in a similar way to (2.45). If this is true and the proof can be carried out, it would yield a relationship between the generalized intersection numbers and the σ correlators. The formula (6.80) for the generalized volumes allows us to predict the specific form of the volumes at g = 0 for any n ≥ 3. By utilizing the formula (6.21) for the repeated action of the KdV vector fields on u at g = 0, and performing the sums in (6.80) we predict V0,n(b;tk) = d n−3 dxn−3 {u ′ 0 n ∏ i=1 [ − J0(bi √ u0)]} R R R R R R R R R R Rx=µ , (6.81) 5The discrepancy has to do with the fact that our sums start at k = 1. 158 where J is the Bessel function of the first kind. In order for this to match expected results it would need to be multiplied by (−1) n , which means perhaps the relationship between the volumes and correlators of ω needs to be adjusted when n ≥ 3. This result can be compared to the large-n asymptotics for the standard Weil-Petersson volumes [93]. Curiously, the formulas there involve the modified Bessel function I0, but otherwise we find qualitative agreement. The benefits of using this approach are that this is exact for any value of n, and applies equally well to non-supersymmetric, 0A, and N = 2 theories (which we will confirm in the case n = 3 shortly). It will be clear from the following results that the operator ωz makes a lovely connection between objects naturally considered the KdV organization of double scaled matrix models and quantities typically computed using algebraic geometry. Additionally we will see, primarily in the one-boundary case, that the Gelfand-Dikii resolvent resumes its distinguished role in the story. That the Gelfand-Dikii resolvent, which is naturally constructed using the operator ωz, should be so closely related to the matrix model resolvent Wg,n was anticipated in [75], where similar results were found. 6.3.1 Non-Supersymmetric Theories Let us first explicitly show that ⟨ωz⟩ = 4Γzy(z) with some examples, starting at leading order. By plugging z = √ −E into the formulae for the disk density (4.9) and (4.18) we get y0(z) = 1 2 ∫ µ −∞ dx √ u0 + z 2 = 1 2 ∞ ∑ k=1 √ πΓ(k + 1) Γ(k + 1 2 ) tkz 2k−1 . (6.82) At leading order the Gelfand-Dikii resolvent is Rˆ = 1 2 (u0 + z 2 ) −1/2 , which we can easily see produces the equivalence between ⟨ωz⟩0 and y0(z). Due to the stated relationship between the matrix model resolvent and the spectral density, the first resolvent function Wg,1(z) is equivalent to yg(z). 159 At g = 1, the Gelfand-Dikii resolvent is given by Rˆ 1 = 5(u ′ 0 ) 2 64 (z 2 + u0) 7/2 − u ′′ 0 16 (z 2 + u0) 5/2 − u1 4 (z 2 + u0) 3/2 . (6.83) After changing the integration variable to u0, the g = 1 contribution to ⟨ωz⟩ is ⟨ωz⟩1 = 4h̵Γz ∫ ∞ 0 du0 1 u ′ 0 ⎡ ⎢ ⎢ ⎢ ⎣ 5(u ′ 0 ) 2 64 (z 2 + u0) 7/2 − u ′′ 0 16 (z 2 + u0) 5/2 − u1 4 (z 2 + u0) 3/2 ⎤ ⎥ ⎥ ⎥ ⎦ . (6.84) By using the special relationships between u ′ 0 , u ′′ 0 , and u1 and integrating by parts, the resulting surface terms give ⟨ωz⟩1 = h̵3Γ 24 2(u ′′ 0 /u ′ 0 )z 2 − 3u ′ 0 z 4 R R R R R R R R R R Rx=µ . (6.85) with the rest cancelling. Using the JT gravity solution, the coefficients in ⟨ωz⟩1 are π and −1, respectively, which agrees with the result obtained via topological recursion. Another noteworthy case that can be checked quickly is the Airy model, where u0 = −x. There, one can explicitly show using the WKB wavefunction (see Chapter 4) that the first perturbative correction to the spectral curve is 1/32z 5 . Evaluating (6.85) on the Airy model potential and using the relationship between ⟨ωz⟩1 and y(z) produces the desired result. Generally, we can refer to (5.7) to write the expectation value at g = 1 in terms of the coupling constants. By taking the inverse Laplace transform we obtain the generalized volume V1,1(b;tk) = 1 24t1 (b 2 + 4t2 t 2 1 ) . (6.86) 160 We have already confirmed that this will be the correct Weil-Petersson volume when the coupling constants are chosen to be those of JT gravity, by virtue of checking W1,1. But, this also gives us the opportunity to state the generalized volumes for the other example theories. In the (2, 2p − 1) minimal string V (p) 1,1 (b) = 1 24 ( 8π 2 p(p − 1) (2p − 1) 2 + b 2 ) , (6.87) and in the Virasoro Minimal String V VMS 1,1 (L) = 1 96√ 2π ( L 2 2π 2 + c − 13 6 ) , (6.88) where we have changed the length to L so as not to confuse it with the Liouville parameter b. Much like we did in the macroscopic loop calculation, this could be brought to a more standard form by introducing a rescaled length L. Consider the g = 0 contribution to the ω three-point function. Using (6.16) one finds that ⟨ωz1ωz2ωz3 ⟩0 = − u ′ 0 (µ) z1z2z3 . (6.89) When we take the inverse Laplace transform to change this into a fixed length quantity, one finds V0,3(b) = −u ′ 0 (µ) = 1 t1 . (6.90) It is no surprise that, given the poor behavior of the macroscopic loop correlators in individual multicritical models, it is not possible to define V0,3 for such a model. Once again in order to get finite results the topological point k = 1 needs to be included in an interpolation. 161 6.3.2 N = 1 Before proceeding to compute the volumes in supersymmetric theories we stress again here that the overall formalism does not change, only over what ranges certain integrals are performed and where the function u is evaluated. As we will see, minimal extra effort is required to calculate the 0A volumes, and we never have to directly interact with supermanifolds. The first notable change occurs in the spectral density. Since the Fermi surface is nonzero in 0A theories, the integration extends into the closed string sector of the theory, where we take the solution u0 = 0. Recall that this was important for the macroscopic loop expectation value as well. If we take t0 = µ, then the leading order spectral density is given by (6.82) but with the sum starting at k = 0, the second topological point in the KdV hierarchy. The integral defining ⟨ωz⟩1 must be once again modified to only go from 0 to µ. The Gelfand-Dikii resolvent at g = 1 is Rˆ 1 = − u1 4z 3 and hence ⟨ωz⟩1 = h̵Γ Γ 2 − 1 4 µz2 . (6.91) This implies that the moduli space volume is given by V1,1 = Γ 2 − 1 4 µ . (6.92) Recall that in the closed string sector the parameter Γ counts RR flux insertions. This result is consistent with the discussion in [60], in which the authors point out that the flux insertions are intrinsic to the theory and therefore constitute extra degrees of freedom or moduli. When Γ is set to 0 here, it reproduces the negative volume one would expect in a 0A theory, although our result differs from [60] by a normalization choice. 162 The Gelfand-Dikii resolvent at g = 2 is Rˆ 2(x, z2 ) = − u ′′ 1 − 3u 2 1 + 4z 2u2 8z 5 . (6.93) Performing the x integral and taking the inverse Laplace transform gives the volume V2,1(b;tk) = − (4Γ2 − 1) (4Γ2 − 9) (b 2µ + 12t1) 384µ4 (6.94) The coupling constant takes the value t1 = π 2 and the Fermi surface is µ = 1 in this particular N = 1 JT supergravity. Plugging those values in and turning the RR flux off, we once again have agreement with the expected result, up to the same normalization factor of 1 2 . We note also that any theory with t1 = 0, for example any of the individual multicritical models, will compute the same volume V 0A 2,1 (b, 0). This is another example of how the 0A multicritical models are better behaved than their non-supersymmetric counterparts. In order to calculate the volume V1,2, we no longer can rely on the Gelfand-Dikii resolvent. However we can once again adapt the result from (6.14) for the σ two-point function. We get ⟨ωz1ωz2 ⟩1 = − h̵2Γ 2 (4Γ2 − 1) 4µ2z 2 1 z 2 2 , (6.95) which produces the constant volume V1,2 = − 4Γ2 − 1 4µ2 . (6.96) Once again, this agrees with the expected result up to the normalization choice when we set Γ = 0. 163 Recall that the N = 1 generalization of the Virasoro Minimal String [76] is a 0A theory. Hence the results for V1,1 and V0,3 are valid there, since they are independent of the closed string coupling constants. The first coupling constant is t1 = 4 √ π/b, which means the g = 2, n = 1 volume is V2,1(L) = − 3 2 9 (L 2 + 16√ 2π b ) . (6.97) 6.3.3 N = 2 We begin here with the three-point function, since it does not require considering any part of u past leading order. During the derivation of the result for the theories with u0(µ) = 0, that fact was used to simplify the result. For a model where this does not happen, the result is ⟨ωz1ωz2ωz3 ⟩0 = − h̵3Γ 3 z1z2z3 [(E0 + z 2 1 )(E0 + z 2 2 )(E0 + z 2 3 )]3/2 , (6.98) The inverse Laplace transform gives the volume V0,3(b1, b2, b3) = −J0(b1 √ E0)J0(b2 √ E0)J0(b3 √ E0)u ′ 0 (µ˜). (6.99) Note that this matches the general result stated at the beginning of this section. By gluing the exponential part of the N = 2 trumpet to this volume three times, we recover the matrix model prediction for Z0,3. The simplest volume that depends on contributions to u past leading order is g = 1 and n = 1. To do this, we consider the Gelfand-Dikii resolvent at g = 1, further expanded in ϵ. At leading order in ϵ, ⟨ωz⟩1 = − z 3 4(z 2 + E0) 5/2 + O(ϵ) (6.100) 164 which gives the volume V1,1(b) = − √ π 8˜µ (J0(b √ E0) − b √ E0 3 J1(b √ E0)) + O(ϵ). (6.101) 6.4 Gluing Procedure We have alluded to the general fact that integrating trumpets and volumes together yields the macroscopic loop correlation functions, much like it does in JT gravity. That this works out in general can be shown rather easily. As an example, consider the genus-g contribution with one asymptotic boundary. The volume Vg,1 is given by Vg,1(b;tk) = ∞ ∑ k=1 (−1) k b 2k−2 4 kk!(k − 1)! ⟨σk−1⟩g. (6.102) Then integrating this against the trumpet partition function Ztr, with an extra factor of b, gives 1 √ 4πβ ∫ ∞ 0 db ∞ ∑ k=1 (−1) k b 2k−1 4 kk!(k − 1)! e − b 2 4β ⟨σk−1⟩g = 1 4 √ πβ ∞ ∑ k=1 (−β) k k! ⟨σk−1⟩g, (6.103) which is the macroscopic loop expectation value at genus g, up to a factor of 2. The cases with n boundaries follow in precisely the same manner. Note that this is a truly model independent argument, because all of the data that differentiates the models (the coupling constants) is contained in the correlation functions of the σ operators. Naively, it may seem like this is a happy coincidence then. However this relationship between the volumes, the trumpet, and the macroscopic loop operators can be predicted based on the relationship between FZZT brane insertions and macroscopic loops displayed in (6.58). Nevertheless, it is still nontrivial that the combinatorial factors cancel correctly. 165 6.5 Topological Recursion There is a common thread beneath the surface of the web in fig. (6.2). The Gelfand-Dikii polynomials, which as we have seen are central to the KdV organization of these matrix models, obey the recursion relation (5.2). The solutions to the string equation and the perturbative expansions of correlators are intimately linked to the structure the Gelfand-Dikii polynomials are endowed with by virtue of this recursion relation. Also within the KdV organization of the models, the closed string operators σk, which are related to the vector fields that generate the KdV flows, obey a sort of recursion relation in the form of Ward identities that result from the Virasoro conditions [22, 51, 71]. Finally, the Weil-Petersson volumes famously obey Mirzakhani’s recursion relation [7]. This can be thought of as a special case of the matrix resolvent recursion relation in [8], where the matrix model is taken to be dual to JT gravity. Still though, if we define the volumes Vg,n(b;tk) via the inverse Laplace transform of Wg,n(z;tk), the generalized volumes should satisfy a recursion relation as well. Given the fact that on the surface of fig. (6.2) each of these topics is related to each other, it would be surprising if one could not somehow map their respective recursion relations onto one another. In this section we will focus on the connection between the Virasoro conditions and the generalized volumes, although we will make some comments about the connection between the Gelfand-Dikii polynomials and topological recursion. The fact that the Virasoro conditions are tied to the recursion between the Weil-Petersson volumes was noticed immediately after Mirzakhani’s discovery in [50]. In that work, the authors approached the problem more from the point of view of intersection theory, and as a result dealt with matrix models more formally, in the spirit of the Kontsevich model’s role in proving the WittenKontsevich theorem. In this sense, the results of [50] are somewhat limited in scope compared to what can be accomplished using the open-closed duality described here, since there is clearly a difference between using the matrix model purely as a generating function for a specific set of intersection numbers and volumes, and defining a general family of such objects which depend on the coupling constants tk and that 166 reproduce the Weil-Petersson case as an example. Moreover, the matrix model technology displayed above allows for the computation of volumes in both 0A and N = 2 supersymmetric theories. Since the Virasoro conditions are attached to the KdV organization of the matrix model, and not specifically the string equation6 , their Ward identities and recursion relations should manifest in the supersymmetric cases as well. It is clear from open-closed string duality that the recursion relation between (4.22) amongst the resolvents Wg,n will maintain a geometric interpretation, insofar as we know that Wg,n is intimately connected to the structure of worldsheets stretching between a set of n branes. In particular, we can still imagine that the recursion relation is describing the number of ways to decompose a surface with n geodesic boundaries and genus g into constituent surfaces, either by pinching a handle of the initial surface or by cutting it open. As we will see, the close string operators σk of the matrix model naturally interact with each other in such a way. An important part of the Witten-Kontsevich theorem is that the matrix integral (3.3), with β = 2, is related to the tau function of the KP hierarchy [45, 46] 7 . In particular, the tau function is τ = √ Z = e −F/2 , where F is the free energy. By combining the KdV flow equations and the string equation, once can show that τ satisfies an infinite tower of partial differential equations called Virasoro conditions. The closed string Virasoro generators are L−1 = ∞ ∑ k=1 tk ∂ ∂tk−1 + x 2 4h̵2 , L0 = ∞ ∑ k=0 tk ∂ ∂tk + 1 16 , Ln = ∞ ∑ k=0 tk ∂ ∂tk+n + 4h̵2 n ∑ k=1 ∂ 2 ∂tk−1∂tn−k , (6.104) 6The non-supersymmetric closed string equation and the DJM equation are actually implied by two of the Virasoro conditions, but there are infinitely many other ones to derive Ward identities from. 7 In the double scaling limit, it is related to the tau function of the KdV hierarchy, which is of course our primary interest. 167 and the constraints are Ln ⋅ τ = 0 [22]. The key ingredients in deriving these equations can be generalized to models incorporating open strings and supersymmetry, and a version of the Virasoro conditions for such models was derived in [51] and expanded on in [71]. By plugging in the coordinate transformation (5.29), we obtain L−1 → ∞ ∑ k=1 tk ∂ ∂tk−1 + 2h̵Γ ∞ ∑ k=1 ζkz −(2k+1) ∂ ∂tk−1 + (x + h̵Γz −1 ) 2 4h̵2 , L0 → ∞ ∑ k=0 tk ∂ ∂tk + 2h̵Γ ∞ ∑ k=0 ζkz −(2k+1) ∂ ∂tk + 1 16 , Ln → ∞ ∑ k=0 tk ∂ ∂tk+n + 2h̵Γ ∞ ∑ k=0 ζkz −(2k+1) ∂ ∂tk+n + 4h̵2 n ∑ k=1 ∂ 2 ∂tk−1∂tn−k . (6.105) The full form of the open string sector Virasoro operators is actually L˜ n = Ln − (2n + 2) Γ 2 4 z 2n − z 2n+2 ∂ ∂z , where Ln is denotes the shifted operators directly above [24, 51]. Consider the correlation function ⟨σkI−1⟩ where kI denotes multi-index notation σkI−1 ≡ ∏ i∈I σki−1, (6.106) for some set I. The open string sector Virasoro generators annihilate a new τ -function τ = e −F/2 , where F is the Free energy of the open theory. The point-like operators σk naturally act on the free energy via derivatives with respect to tk. Notice that the Virasoro conditions can be rewritten to so that the operators act nonlinearly on F Ln ⋅ τ˜ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ − 1 2 ∞ ∑ k=0 (k + 1) tk ∂F ∂tk+n − h̵Γ ∞ ∑ k=0 ζkz −(2k+1) ∂F ∂tk+n + 4h̵2 n ∑ k=1 (− 1 2 ∂ 2F ∂tk−1∂tn−k + 1 4 ∂F ∂tk−1 ∂F ∂tn−k ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ e −F/2 = 0. (6.107) 168 By acting on such correlation functions with the Virasoro generators, we arrive at a set of Ward identities. Performing the topological expansion on the correlation functions with a Virasoro generator inserted produces the recursion relation [51] ⟨(z 2m+1ωz + ∞ ∑ k=0 (k + 1)tkσk+m−1)σkI−1⟩ g,h − ∑ i∈I (ki + 1)⟨σki+m−1σkIˆ−1⟩g,h = − m ∑ k=1 ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 4⟨σk−1σm−kσkI−1⟩g−1,h + 2 ∑ g1+g2=g Q∪R=I ⟨σk−1σkQ−1⟩g1,h⟨σm−kσkR−1⟩g2,h ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , (6.108) for any m ≥ 1. Structurally, the Ward identity (6.108) is already very close to the recursion relation for the matrix resolvents Wg,n in (4.22). Notice that on both sides the multi-indices are so far unfixed, as is the set I. We can use the multi-indices on each side to introduce factors of the operator ω in the correlators on each side. Specifically, let I = {2, . . . , n} and multiply by factors of ζki z −2ki . When we sum over kI we get ⟨(z 2m+1ωz + ∞ ∑ k=0 (k + 1)tkσk+m−1)ωzI ⟩ g,h − ∞ ∑ kI=1 ζkI z 2kI I ∑ i∈I (ki + 1)⟨σki+m−1σkIˆ−1⟩g,h = − m ∑ k=1 ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 4⟨σk−1σm−kωzI ⟩g−1,h + 2 ∑ g1+g2=g Q∪R=I ⟨σk−1ωzQ ⟩g1,h⟨σm−kωzR ⟩g2,h ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , (6.109) We still need two more factors of ω on the right hand side. Thinking of it as a two-index object Sk−1,m−k, notice that the sum over k is a sum over the mth anti-diagonal. A sum over m with a coefficient Cm, with m going from 1 to ∞, will include each element of S exactly once. Thus it can be rewritten ∞ ∑ m=1 Cm ∞ ∑ k=1 Sk−1,m−k = ∞ ∑ i,j=1 Ci+j−1Si−1,j−1 169 Written in this manner, the indices of S are decoupled, but it comes at the price of complicating the indexing of the coefficient C. Evidently we want Ci+j−1 ∝ ζiζjz −2i−2j . A natural candidate for the numerical portion of Cm comes from a polynomial P built out of the Gelfand-Dikii polynomials via [47] R˜ iR˜′ j = P˜′ i,j , (6.110) where we have used the Gelfand-Dikii normalization. Translated into the normalization used here, one can define a resolvent-esque generating function for Pi,j P(x; z1, z2) = ∞ ∑ i,j=0 ζiζjz −2(i+j+1)Pi,j (6.111) The function P does not have any definite symmetry property under the exchange z1 ↔ z2, but its symmetric and anti-symmetric parts obey explicit relations determined by the resolvent Rˆ. Since Pi,j is a single polynomial, P can be resummed to depend on just a single index m. The coefficients in the expansion of P written this way should essentially be the coefficient Cm that we desire. By introducing the sum over m with the coefficient just described, picking ∣I∣ = n − 1, and including the other necessary sums to produce all insertions of ω, the right hand side of (6.108) becomes 4⟨ωzωzωI ⟩g−1,h + 2 ∑ g1+g2=g Q∪R=I ⟨ωzωQ⟩g1,h⟨ωzωR⟩g2,h = Wg−1,n+1(z, z, I) + ∑ g1+g2=g Q∪R=I Wg1,1+∣Q∣(z, Q)Wg2,1+∣R∣(z, R), (6.112) which is the essential content of the right hand side of the recursion relation (4.22). 170 The left hand side of (6.108) needs to factorize in a very particular way after we introduce the sums. Change the explicit factor of z to z1, so that the all other explicit factors introduced in the extra sums are z. Then we desire the following outcome ∑Left hand side ∝ (z1 + z)y(z)Wg,n(z, I). (6.113) If this happens, then we can divide both sides of the recursion relation by (z 2 1 − z 2 )y(z). On the left hand side this would leave Wg,n(z, I)/(z1 − z), and the right hand side would match the argument of the residue in (4.22). But, taking the residue at z = 0 of both sides would yield Wg,n(z1, I) on the left hand side8 , yielding the correct recursion relation. 8We know that the resolvents will only have poles at z = 0, so taking the residue at z = 0 will still force the replacement z → z1. 171 Chapter 7 Unoriented Two-Dimensional Gravity In [60] Stanford and Witten discuss unoriented JT gravity and its connection to random matrix theory. They develop perturbation theory using the loop equation formalism, which focuses on the resolvent operator of the random matrix. By applying techniques developed in [9] they conclude that the presence of even a single cross-cap can cause divergences in otherwise well-behaved objects. See [94] for recent progress relating to the loop equation formulation of unoriented JT gravity. Following the work in [37], an effort is made in this chapter to define perturbation theory in general β = 1 Wigner-Dyson models from a different perspective, in the hopes that a more matrix model-focused approach may resolve the apparent issues with the loop equation formalism. This alternate point of view is met with mixed results, as it turns out to be difficult to extract results from the formalism even for the simplest non-trivial minimal model. Our focus on double scaling thus far has been on models with Hermitian matrices, primarily because of the lack of results in the string equation formalism for real symmetric matrices. As we have seen, perturbation theory in Hermitian matrix models, for both correlation functions and statistical quantities, focuses on the topological expansion of the function u. It is clear based on the presence of orthogonal polynomials in the solution of β = 1 models that the function u should continue to play a role in their double scaling limits. Therefore there should still be some organization in terms of the KdV hierarchy, and hence some of the operator content should involve the σk. However the full integrable structure of 172 double scaled β = 1 models is currently undetermined, so operator correlators are still out of reach. On the other hand, we will see that we have enough information to determine an important statistical quantity, the double scaled eigenvalue density. 7.1 Orthogonal Polynomial Approach Recall from section 3.3.2 that we computed a general correlation function GN in the finite-N regime of the GOE by writing it as the Pfaffian of a matrix J, whose matrix elements were defined in terms of orthogonal polynomials pi . We then used the invariance of the determinant to rewrite J by taking linear combinations of the rows and columns, which resulted in the formula (3.47) for the density. We can employ a similar approach for more general matrix potential V . Begin by setting J2α,2γ(u, v) = fαγ(u, v). Next, define gαγ(u, v) = − N 2 +k−1 ∑ σ=0 cσγJ2α,2σ+1(u, v) = ∫ ∞ −∞ dx∫ ∞ −∞ dy u(x)v(y)(ψ2α(x)ψ ′ 2γ (y) − ψ2α(y)ψ ′ 2γ (x)). (7.1) Notice that the values of the dummy index σ go outside of the “N × N block” in index space in which the matrix model technically lives. The impact of this on the eigenvalue density will be discussed shortly. Finally, define µαγ(u, v) = N 2 +k−1 ∑ µ,ν=0 cµαJ2µ+1,2ν+1cνγ = ∫ ∞ −∞ dx∫ ∞ −∞ dy u(x)v(y)(ψ ′ 2α(x)ψ ′ 2γ (y) − ψ ′ 2α(y)ψ ′ 2γ (x)). (7.2) 173 After some rearranging to ensure antisymmetry, we find the Pfaffian can be written Pf(J) → det1/2 ⎛ ⎜ ⎜ ⎜ ⎝ fαγ gαγ −gγα µαγ ⎞ ⎟ ⎟ ⎟ ⎠ , (7.3) and hence the correlator GN can be expressed in terms of this new determinant. If u, v are both even functions, the diagonal blocks of this new matrix are zero and the Pfaffian reduces to det(g). The result for the GOE (3.47) nearly holds for more general multicritical models with k > 1. Denoting the density for the kth multicritical model by ρ˜ (k) , the result is actually ρ˜ (k) = ρ˜ (1) + surface terms, (7.4) where the oscillator functions used in ρ˜are k-dependent as well. The additional terms, called surface terms here, are localized around α = N/2 and arise from the fact that we included terms outside of the Fermi sea to obtain ψ ′ 2α when rewriting the matrix J. The number of extra terms is set by k in the upper limit of summation in (3.56). The surface terms, denoted by K(λ), have the form K(λ) = N 2 −1 ∑ α=0 Nα N 2 −1+k ∑ σ= N 2 cσαψ2σ+1(λ), Nα = ∫ ∞ −∞ ψ2α(λ ′ )dλ′ . (7.5) In the simplest case k = 2, K(λ) = N 2 −1 ∑ α=0 Nα(c N 2 ,α ψN+1(λ) + c N 2 +1,α ψN+3(λ)). (7.6) It is not possible to determine any of the quantities in this formula in closed form for finite N. 174 7.2 Previous Double Scaling Results As we have noted, the double scaling limits of numerous matrix models with Gaussian potentials, including the β = 1, 2, 4 theories, can be determined by using known facts about the Hermite polynomials. See [81] for a review. In particular, the double scaled eigenvalue density for the GOE can be expressed in terms of the Airy function ρGOE(E) = ρ (2) (E) + 1 2 Ai(−E) (1 − ∫ E −∞ Ai(−E ′ )dE′ ) , (7.7) where ρ (2) is the eigenvalue density for the double scaled GUE given in (4.13) The double scaling limits of the operators a, b introduced in (3.3.2) were computed in [63]. In the normalization used here, they are a → Tk = h̵k−1 ∂ k−1 + k−2 ∑ i=0 h̵i gi∂ i , b → Sk = h̵k ∂ k + k−1 ∑ i=0 h̵i hi∂ i . (7.8) The double scaled version of (3.59) is S † k Tk = Pk. The authors of [63] outline an algorithm to determine the coefficients in Tk, Sk which we will describe below. Using this, they compute the free energy of a pure unoriented gravity theory (corresponding to the single multicritical model k = 2). 7.3 New Results in Matrix Models 7.3.1 Double Scaled Density We begin by double scaling the modified version of the finite-N GOE density given in (7.4). We do so by applying the same scaling ansatze that were used in section 4.1 for the Hermitian case. Using the 175 scaling relation for the eigenvalues as a coordinate transformation implies that integrals over λ have the scaling part ∫ dλ → −δ 2 ∫ dE, (7.9) where the bounds of integration are updated on a case-by-case basis. By changing the large-N integration over α/N to i/N we introduce a factor of 1 2 . The double scaled density is then ρ(E) = 1 2 ∫ 0 −∞ [ψ(x, E) 2 − Pkψ(x, E)∫ ∞ E ψ(x, E′ )dE′ ] dx + Fk[ψ](0, E). (7.10) The double scaling limit of the surface terms from (7.4) are denoted by Fk[ψ]. Since before double scaling there are O(1) surface terms around α = N/2, Fk[ψ] should involve derivatives of ψ evaluated at the Fermi surface x = 0. For larger k, there are more surface terms before double scaling and hence higher order derivatives in the double scaling limit. In order to double scale the β = 1 density written in terms of the skew oscillator wavefunctions (3.54) we need to discuss the double scaling limit of the operator εˆ. Consider the application of εˆ to functions with definite parity. The results are even ∶ εˆ⋅ f(λ) = ∫ λ 0 f(λ ′ )dλ′ , odd ∶ εˆ⋅ f(λ) = ∫ λc λ f(λ ′ )dλ′ , (7.11) where we’ve assumed that f only has support on [−λc, λc]. By the scaling ansatz for the λ integral, even ∶ εˆ⋅ f(λ) double scale ÐÐÐÐÐÐ→ δ 2 ∫ ∞ E f(E ′ )dE′ , odd ∶ εˆ⋅ f(λ) double scale ÐÐÐÐÐÐ→ −δ 2 ∫ E 0 f(E ′ )dE′ . (7.12) 176 The formula for the finite-N density in (3.54) involves the skew wavefunctions, which are related to the oscillator wavefunctions via the transformations a and b as depicted in (3.60). The operators a, b double scale to differential operators Tk, Sk, which are order−(k − 1) and k respectively. The double scaled limit of the relationship between ζ and ψ is expressed as1 even ∶ ζ2α(λ) = [a ⋅ ψ]2α(λ) double scale ÐÐÐÐÐÐ→ δ −1Tˆ kψ(x, E) odd ∶ ζ2α+1(λ) = [b ⋅ ψ]2α+1(λ) double scale ÐÐÐÐÐÐ→ δ −1Sˆ kψ(x, E) (7.13) Putting together these pieces, we find the expression for the double scaled β = 1 density ρ(E) = 1 2h̵ ∫ 0 −∞ dx ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ Tkψ(x, E) (∫ E 0 Skψ(x, E′ )dE′ ) + Skψ(x, E) (∫ ∞ E Tkψ(x, E′ )dE′ ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ . (7.14) This expression is exact and does not ignore any surface terms. The normalization is chosen to extract the finite piece after double scaling. The full expression for the density gives us the opportunity to roughly determine what the surface terms were in the orthogonal polynomial approach. Consider the first term in the above formula. If we were to integrate by parts with respect to x in order to shift Tk inside the E ′ integral, we would pick up a contribution at x = 0 and get the term ψ(x, E)∫ E 0 T † k Skψ(x, E′ )dE′ , (7.15) 1The powers of δ appearing in these relationships may not actually both be −1. What is important is that the sum of the two powers is −2, as it is for the operator Pk. 177 where the adjoint is taken in L 2 (R). Now, the specific combination T † k Sk is proportional to Pˆ k, which is canonically conjugate to the Hamiltonian H for which ψ(x, E) is an eigenfunction. Thus Pk has a representation Pk ∼ ∂E, and ψ(x, E)∫ E 0 T † k Skψ(x, E′ )dE′ ∝ ψ(x, E)(ψ(x, E) − ψ(x, 0)), (7.16) which produces the ψ 2 term familiar from the study of β = 2 theories. Consider the second term in the density. Once again we integrate by parts to shift the location of Tˆ k. This picks up more terms at x = 0 and gives T † k Skψ(x, E)∫ ∞ E ψ(x, E′ )dE′ ∝ Pkψ(x, E)∫ ∞ E ψ(x, E′ )dE′ . (7.17) Being more careful about signs, we conclude ρ(E) = 1 2 ∫ 0 −∞ dx ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ ψ(x, E) 2 − Pkψ(x, E)∫ ∞ E ψ(x, E′ )dE′ ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ + surface terms, (7.18) where the surface terms are generated by moving the operator Tˆ k around. In the case k = 2, F2[ψ](0, E) = − 1 4hE̵ (P2ψ(0, E)ψ(0, 0) − ψ(0, E)P2ψ(0, 0)) + 1 2 √ 2h̵ [ψ(0, E)∫ E 0 S2ψ(0, E′ )dE′ + S2ψ(0, E)∫ ∞ E ψ(0, E′ )dE′ ] . (7.19) One thing that is obscured by this analysis is what happens in the GOE. For that theory, where k = 1, the operator T1 is a constant, meaning it costs nothing to move it around inside the integral. Thus there are no surface terms in that case ρGOE(E) = 1 2 ∫ 0 −∞ dx ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ ψ(x, E) 2 − P1ψ(x, E)∫ ∞ E ψ(x, E′ )dE′ ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ . (7.20) 178 (a) Plot of (7.20). (b) Plot of (7.7). (c) Both the plots. Figure 7.1: Our result from (7.20) exactly overlaps with the standard result (7.7), as can be seen from (c). Two plots are indistinguishable. We can confirm numerically that (7.20) matches the previously known result given in (7.7); the plots are shown in Figure 7.1. 7.3.2 New String Equations It is evident from (7.14) that we have to determine the operators Tk, Sk, and there is a well defined algorithm to determine them outlined in [63]. For the k th minimal model we can write the following factorization equation2 Pk = − k−1 ∏ i=1 (h∂̵ − 1 2 gi) h∂̵ k−1 ∏ j=1 (h∂̵ + 1 2 gk−j) . (7.21) The operator Pk is (2k − 1)-order differential operator where the coefficients are functions of u and its derivatives. Notice how this differs from writing Pk = S † k Tk with Tk, Sk previously defined in eqn. (7.8). 2Technically we will be working with the operator −iPk, if Pk = (H k− 1 2 ) + . This factor of i can be restored by modifying the factorization in terms of Tk and Sk, but the functions gi would remain unchanged, and so would all of our other results. 179 There, the two individual operators were defined in terms of separate families of functions, which we called gi , hi . By imposing that the factorization be anti-self-adjoint and setting some integration constants to 0 (see [63]) one can reduce the number of functions so that Tk, Sk depend on the same family of functions. The new factorization in eqn. (7.21) amounts to setting Sk = k−1 ∏ j=1 (h̵2 ∂ 2 + h̵ 2 gk−j∂ + h̵ 2 g ′ k−j) , Tk = k−1 ∏ j=1 (h∂̵ + 1 2 gj) . (7.22) Expanding both sides of (7.21) and equating powers of ∂ we get (2k − 1) coupled differential equations for (k − 1) gi functions. These equations will not all be independent, and we can find (k − 1) independent differential equations. While we do not have exact forms for each equation obtained using the algorithm outlined above for arbitrary k, it is straightforward to show that the equation determined by ∂ 2k−3 is always [63] (k − 1 2 ) u = k−1 ∑ j=1 [−hjg ̵ ′ j + 1 4 g 2 j ] . (7.23) This can be adapted to accommodate a general interpolation. The argument is as follows. The highest order derivative term in Pk will be obtained from (−h̵2 ∂ 2 ) k−1 × ih∂̵ = i(−1) k−1 h̵2k−1 ∂ 2k−1 . (7.24) The next non-vanishing term will be proportional to ∂ 2k−3 . We can think of there being two contributions to this. The first is (−h̵2 ∂ 2 ) k−1 × h̵−1 q−1∂ −1 , (7.25) 180 where all of the derivatives are commuted past q−1 to leave the desired power of ∂. The coefficient from this term is (−1) k ih̵2k−3 2 u. The second contribution comes from the next highest order cross-term in Hk−1 , which is obtained by replacing one factor of −h̵2∂ 2 with u, and commuting the derivatives to the right u(−h̵2 ∂ 2 ) k−2 × ih∂. ̵ (7.26) There are k − 1 ways to do this, giving the coefficient (−1) k (k − 1)ih̵2k−3u. Therefore the operator Pk is −iPk = (−1) k−1 h̵2k−1 ∂ 2k−1 + (−1) k (k − 1 2 ) uh̵2k−3 ∂ 2k−3 + ⋯. (7.27) This justifies the coefficient on the left hand side of (7.23). Based on the discussion above, we may be interested in linear combinations of Pi for different values of i. For instance, consider the superposition P = k ∑ j=1 tjPj , (7.28) for some k ≥ 2. The highest order derivative in Pj is strictly increasing with j, so the maximum power of ∂ appearing in P will be ∂ 2k−1 , and the only term in the superposition that can contribute to that term is Pk. The next power of ∂ appearing in P is ∂ 2k−3 , whose coefficient will receive contributions from Pk and Pk−1. Thus −iP = (−1) k−1 tkh̵2k−1 ∂ 2k−1 + (−1) k ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ (k − 1 2 ) tku + tk−1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ h̵2k−3 ∂ 2k−3 + ⋯. (7.29) Note that no other terms in the superposition can contribute at these orders. 181 There are two types of term that show up in the coefficient of ∂ 2k−3 on the right hand side of (7.21). The first comes from commuting all derivatives past functions. Since none of the functions will be differentiated, the mixed terms involving gigj with i ≠ j will cancel by symmetry. This leaves h̵2k−3 ( k−1 ∑ i=1 g 2 i ) ∂ 2k−3 . (7.30) The other type of term comes from letting one of the derivatives in the factorization act on one of the gi . The possible outcomes can be visualized by breaking the factorization up into the operators T and S as in (7.22), corresponding to lumping the terms of the form ∏(∂−g) into T and the ones of the form ∂ ∏(∂+g) into S. The function gj can be differentiated once k − j times in T † and j + k times in S, since the terms in S can also be differentiated by the terms in T † . Taking into account the minus signs and factors of 2, this gives −h̵2k−3 ⎛ ⎝ k−1 ∑ j=1 jg′ j ⎞ ⎠ ∂ 2k−3 . (7.31) Hence the factorization is (−1) k−1 itk k−1 ∏ i=1 (h∂̵ − 1 2 gi) h∂̵ k−1 ∏ j=1 (h∂̵ + 1 2 gk−j) = (−1) k−1 itkh̵2k−1 ∂ 2k−1 + (−1) k−1 itk ⎛ ⎝ k−1 ∑ j=1 −jg′ j + 1 4 g 2 j ⎞ ⎠ h̵2k−3 ∂ 2k−3 + ⋯. (7.32) Therefore the proper way to modify (7.23) is (k − 1 2 ) tku + tk−1 = tk k−1 ∑ j=1 [−hjg ̵ ′ j + 1 4 g 2 j ] . (7.33) It turns out that the equation determined by ∂ 2k−4 is simply the derivative of the above equation. This implies that g0 ∼ √ u0, even for more complicated massive interpolations 182 7.4 Unoriented Gravity And Unoriented Minimal Strings The simplest minimal model, k = 1, is dual to topological gravity. Although it is not a genuine theory of surfaces it is interesting to study in its own right, and provides the opportunity to do many analytical calculations. The first nontrivial minimal model, k = 2, is dual to pure gravity. The simplest non-trivial minimal string is the (2, 3) model. Since the goal of this work is to construct a minimal model decomposition of unoriented JT gravity, the (2, 3) unoriented minimal string provides an important testing ground for the methods developed here. 7.4.1 The GOE We can use the fact that the wavefunctions ψ(x, E) and the eigenvalue density ρ are both known exactly in terms of the Airy function to test some of the framework above. Note that the GOE does not require an extra function g to compute the density. Recall that the Schrodinger equation for the k = 1 model is3 h̵2 ∂ 2ψ + (x + E)ψ = 0. (7.34) Taking the standard WKB form of the wavefunction, we find the two possibilities ψ(x, E) = 1 √ 2πh̵ (x + E) −1/4 [1 − ( i5h̵ 48(x + E) 3/2 + 5h̵2 64(x + E) 3 )] e ±i 2 3h̵ (x+E) 3/2 + ⋯. (7.35) 3 Strictly speaking this is true for the GUE. The Schrodinger equation for the GOE is related to this by a rescaling of the energy. See the appendix for discussion. 183 As noted before, the differential equation can be solved exactly in terms of the Airy function, whose Taylor series expansion in small-h̵ is a linear combination of the two perturbative solutions obtained above h̵−2/3Ai (−h̵−2/3 (x + E)) ≈ 1 √ πh̵ (x + E) −1/4 e − 5h̵2 64 (x+E) −3 × cos [ 2 3h̵ (x + E) 3/2 − 5h̵ 48 (x + E) −3/2 − π 4 ] . (7.36) The WKB expansion is inherently an expansion in small h̵. However, the non-perturbative oscillating piece has to be dealth with since it does not simplify under the assumption that h̵ ≪ 1. We circumvent that by using the complex version of the wavefunction, as described in Appendix A, and using the following trick. The formula for the density is ambiguous as it does not allow for the possibility that the wavefunction ψ is complex. The formula’s derivation from the matrix model does not fix this ambiguity, so it may be done ad hoc. The simplest option is to replace one wavefunction in each term with its conjugate. Taking the permutations of which wavefunctions get replaced ensures the outcome is real. Upon doing this, the non-perturbative pieces generally cancel out. For the Gaussian theory, T ∝ 1, S ∝ ∂ ∂x. (7.37) Using the GOE wavefunctions ψ(x, 2 √ 2E) one finds ρ(E) = 2 3/4 √ E πh̵ + 7h̵ 2 3/4256πE5/2 + ⋯. (7.38) We note three things about the outcome. First, the disk level density has the normalization one would expect from the saddle point analysis (see appendix). Second, the O(h̵1 ) term has a different coefficient than the β = 2 density, but the same functional dependence on E. The two are not related to each other by the simple rescaling of E – this is not an issue because we do not expect the theories to be related 184 in a simple way past the leading order in perturbation theory. Third, the O(h̵0 ) term has a coefficient of 0 according to the calculation. If this were truly a (perturbative) topological expansion in a theory of unoriented surfaces, then the surface with one boundary and one crosscap (i.e. the Mobius strip) would appear at this order. There is a non-vanishing, non-perturbative contribution to the O(h̵0 ) integral, but it actually contributes at O(h̵1 ). As a corollary of this, there are no terms in the perturbative series at O(h̵2n ). 7.4.2 The k = 1 To k = 2 Interpolation Take an interpolation between k = 1 and k = 2 with coupling constants t1 and t2. The momentum operator for this interpolation is given by P = t2h̵3 ∂ 3 + ( 3t2 2 u + t1) h∂̵ + 3t2 4 hu̵ ′ . (7.39) There is only one independent function g in the factorization of P, so S2 = h̵2 ∂ 2 + 1 2 gh∂̵ + 1 2 hg̵ ′ , T2 = t2 (h∂̵ + 1 2 g) . (7.40) The independent equation of motion for g is 3t2 2 u + t1 = t2 (hg̵ ′ + 1 4 g 2 ) . (7.41) 185 Each function will have a perturbative expansion in h̵ 4 u = u0 + h̵2 u2 + h̵4 u4 + ⋯ g = g0 + hg̵ 1 + h̵2 g2 + ⋯. (7.42) The first orders in the expansion of g are given by g0 = ± √4t1 t2 + 6u0, g1 = 3t2u ′ 0 2t1 + 3t2u0 , g2 = 3 √ t2 (8t 2 1u2 + 8t1t2u ′′ 0 + 24t1t2u0u2 + 12t 2 2u0u ′′ 0 − 15t 2 2u ′2 0 + 18t 2 2u 2 0u2) 2 √ 2(2t1 + 3t2u0) 5/2 . (7.43) It is easy to show that as t1 → 0, the perturbative expansion smoothly transitions to the single k = 2 case. In the following two subsections we will use these results to study the k = 2 theory, which is dual to pure unoriented gravity, and the (2, 3) unoriented minimal string. 7.4.3 Pure Unoriented Gravity Consider just the k = 2 minimal model. The leading contribution to the density of states for the k th minimal model is proportional to E k− 1 2 , so we expect to see E 3/2 behavior. The density admits a topological expansion in powers of h̵, ρ(E) = ∞ ∑ g,c=0 h̵2g+c−1 ρg,c(E), (7.44) where g is the number of handles of the surface and c is the number of crosscaps. By convention we define ρ0,0 ≡ ρ0. By keeping the leading order contributions from T2, S2 we find the disk density 4The subscripts here are used to denote the order of the perturbative expansion of the single function g, whereas before the subscript enumerated a family of different functions in the factorization of P. 186 ρ0(E) = ∫ 0 −∞ dx [ 1 4 ∂(g0ψ)∫ ∞ E (g0ψ)dE′ − 1 4 (g0ψ)∫ E 0 ∂(g0ψ)dE′ ] . (7.45) Isolating the first subleading contributions gives the crosscap density ρ0,1(E) = ∫ 0 −∞ dx ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 1 4 ∂(g1ψ)∫ ∞ E (g0ψ)dE′ + 1 4 ∂(g0ψ)∫ ∞ E (g1ψ)dE′ − 1 4 (g1ψ)∫ E 0 ∂(g0ψ)dE′ − 1 4 (g0ψ)∫ E 0 ∂(g1ψ)dE′ + 1 2 ∂(g0ψ)∫ ∞ E (∂ψ)dE′ + 1 2 ∂ 2ψ ∫ ∞ E (g0ψ)dE′ − 1 2 (∂ψ)∫ E 0 ∂(g0ψ)dE′ − 1 2 (g0ψ)∫ E 0 (∂ 2ψ)dE′ ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ . (7.46) The functions g0 and g1 are obtained from (7.43) by setting t = 0. Both (7.45) and (7.46) contain more information than is desired, unless we compute the wavefunctions perturbatively using WKB analysis. Recall that the leading order contribution to u, found by solving R2 + x = 0, is u0 = √ −x. The WKB analysis yields ψ(x, E) = 1 √ πh̵ (E − √ −x) −1/4 cos [ 4 15h̵ √ E − √ −x (2E 2 + E √ −x + 3x)] . (7.47) Unfortunately it is difficult to maintain neither analytical nor numerical control over the calculation of the density using the WKB wavefunctions. In particular, the combination of the complicated E-dependence of the WKB wavefunction and the presence of the E integrals in the formula for the density renders analytical computation very difficult. Ordinarily in a Hermitian matrix model, one would use an averaging argument to get rid of the oscillatory part of the WKB wavefunction to compute perturbative contributions to the density. However in this case the E-integrals interact with this oscillatory functions and impact the counting of powers of h̵ in a nontrivial way, as is evident from performing the analogous calculation in the GOE. 187 As a last resort we appeal to the power of the non-perturbative numerical framework described in [11], which we briefly reviewed in Section 4.2.1. Using the function shown in Figure 4.2 as the potential, the Schrodinger equation is discretized and solved using the methods of [95]. In this work, a spatial grid size of ∆x = .03 is used. The presence of the walls needed to obtain numerical solutions constrains the maximum trusted energy. The average spacing between energy levels is ∆E ≈ .02. The numerically determined wavefunctions are inserted into eq. (7.45) to produce the disk level density in fig. (7.2). The curve is oscillating because the wavefunctions are non-perturbative. In a more Figure 7.2: ρ0 vs E curve. We can see that ρ0 perfectly wraps around E 3 2 in this energy window. complete non-perturbative framework the oscillations reveal the discrete nature of the quantum system [59]. However, here they are meaningless because the formalism is only meant to be trusted at the level of perturbation theory. If we plot ρ0 up to higher energies we see a deviation of the curve from E 3 2 as shown in figure (7.3). The deviation is there because we have an integral from E to ∞ in (7.45), and this upper limit was replaced by the highest energy eigenvalue we had in our numerical solution. In this plot, that cutoff was roughly Emax = 14. This argument can be solidified by checking the plot of the function with different cutoffs, i.e. upper limits, as shown in figure (7.4). 188 Figure 7.3: ρ0 vs E curve. We can see the deviation at higher energies. (a) ρ0 with different cutoffs. (b) Zoomed in Figure 7.4: For lower energies all the curves with different cutoffs merge together. The curve deviates earlier as the cutoff is decreased. If we zoom in, we can see that for a particular energy the curve with a lower cutoff is more off from the E 3 2 curve. We can similarly plot ρ0,1 by using the numerically determined wavefunctions in eq. (7.46). To compare the values of ρ0 and ρ0,1, we plot them together in figure (7.5). We can see that ρ0 is more dominant, and ρ1 is just like a damped oscillation around the E axis, as it should be for a perturbative correction. The sum ρ0 + ρ0,1 still oscillates with the approximate E 3/2 behavior of the leading contribution. The amplitude of ρ0,1 is decreasing as E → ∞. The topological expansion of the density in (7.44) is equivalently an expansion in large E, so at large energy we expect to recover the disk contribution. 189 (a) ρ0 and ρ0,1 separately (b) ρ0 + ρ0,1 Figure 7.5: The first unoriented contribution to the eigenvalue density is small compared to the disk level contribution, even with h̵ = 1. The combined contribution to the total density still has fictitious oscillations. 7.4.4 The (2, 3) Unoriented Minimal String The (2, 2p − 1) oriented minimal string is described by a matrix model with the coupling constants [90] tk = π 2k−2 k!(k − 1)! 4 k−1 (p + k − 2)! (p − k)!(2p − 1) 2k−2 . (7.48) In particular, for the choice p = 2 the nonzero couplings are t1 = 1 and t2 = 4π 2 /9. The leading order contribution to u is found to be u0 = 3 ( √ 9 − 16π 2x − 3) 8π 2 . (7.49) It is important to consider this model for two reasons. First, it represents a proof-of-concept for the interpolation technology described here, that we apply to unoriented JT gravity in the following section, because the leading eigenvalue density can be computed analytically. Second, it is well known that JT gravity is in some sense a p → ∞ limit of the (2, 2p − 1) minimal string (see [90], for example). Thus studying the first nontrivial interpolation in this family is a step toward the full theory. It would be harder than the pure unoriented gravity case to maintain control over the WKB wavefunction in an analytical calculation, so we once again resort to the non-perturbative framework for numerics. The potential u is shown in Figure 4.3, and the Schrodinger equation is discretized and solved once again. 190 Using the numerically determined wavefunctions produces the disk level density shown in fig. (7.6). The expected analytical result, given by ρ0(E) = √ E π + 16πE3/2 27 , (7.50) is shown for comparison. Figure 7.6: ρ0 vs E curve. We can see that ρ0 perfectly wraps around the expected result in this energy window. The displayed energy window is smaller in this case than the preceding subsection due to the numerical procedure. The cutoff on the energy in this data is Emax ≈ 8. The first perturbative correction can also be calculated using (7.46). The results are displayed in fig. (7.7). (a) ρ0 and ρ0,1 separately (b) ρ0 + ρ0,1 Figure 7.7: The first unoriented contribution to the eigenvalue density in the interpolated matrix model. 191 7.5 Unoriented JT Gravity As is clear from the formalism developed above, the function u is seeded from the β = 2 theory into the β = 1 theory to determine the functions gi . Further, the communication between the two theories is facilitated by the factorization of the differential operator P, the Lax pair of the Schrodinger Hamiltonian. The operator P is a linear combination of P = ∑k tkPk for an interpolation with coupling constants tk. Since the coupling constants are determined by the leading density of states, which is the same in the oriented and unoriented theories (up to normalization), the full operator for unoriented JT gravity will be P = ∞ ∑ k=1 π 2k−2 k!(k − 1)! Pk, (7.51) where we have inserted the explicit values of the JT gravity coupling constants. We can, in principle, define operators T ,S such that P = S †T as before, and which map the wavefunctions into the skew wavefunctions. Schematically, the factorization looks like P = − ∞ ∏ i=1 (h∂̵ − 1 2 gi) h∂̵ 1 ∏ j=∞ (h∂̵ + 1 2 gj) . (7.52) The flipped bounds on the right-most product indicate that the functions gi appear in reverse order in that part P = −(h∂̵ − 1 2 g1) (h∂̵ − 1 2 g2)⋯h∂̵ ⋯(h∂̵ + 1 2 g2) (h∂̵ + 1 2 g1) . (7.53) In the same manner as before, we find enough differential equations by comparing coefficients of ∂ to determine the gi . 192 Given the extension of the operator P to the infinite interpolation that defines JT gravity and its formal factorization in terms of operators T and S, we predict that (7.14) remains true for unoriented JT gravity, giving the density ρ(E) = 1 2h̵ ∫ 0 −∞ dx ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ T ψ(x, E) (∫ E 0 Sψ(x, E′ )dE′ ) − Sψ(x, E) (∫ ∞ E T ψ(x, E′ )dE′ ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , (7.54) to all orders in perturbation theory. This formula is both self-consistent and consistent with the infinite interpolation used to define JT gravity in terms of multicritical matrix models. This theory and orientable JT gravity share the issue of having formally infinite derivatives in the definition. The string equation for u is infinite order, the operator P is infinite order, and so too are T and S. The solution is to notice that tk gets smaller and smaller for higher k and limk→∞ tk = 0. In [96] it was noted that a truncation can be made to get reliable numerical results depending on the maximum energy one is interested in probing while computing the eigenvalue density. Practically, one studies the finite interpolation up to some kmax by finding the function u satisfying5 kmax ∑ k=1 tkRk + x = 0, where tk are the same coupling constants for JT gravity. Then the Schrodinger equation for u can be solved for the wavefunctions. We believe that, in principle at least, the same can be done in unoriented JT gravity. The algorithm for computing the density is as follows. First, perform the truncated analysis to find u and the wavefunctions ψ. Then determine the interpolated operator P ≈ ∑ kmax k=1 Pk in terms of u, and use that to find the kmax − 1 functions gi . Then build the truncated operators T and S out of the gi , and finally use the formula for the density (7.54). 5This is only true perturbatively. There are difficulties associated to restricting oneself to perturbation theory, as discussed earlier in the k = 2 example. The truncation performed in [96] was for the non-perturbative formulation of JT gravity. 193 Chapter 8 Summary And Conclusions Strangers passing in the street By chance two separate glances meet And I am you and what I see is me — “Echoes” by Pink Floyd Dualities are an invaluable and powerful resource in physics and mathematics. It is often the case that when two concepts or problems are found to wholly embody and mirror one another, we are able to learn more about each individual constituent of the relationship than we would have otherwise. In this dissertation we have studied several nested dualities relating double scaled random matrix models, twodimensional quantum gravity, and string theory. The overarching relationship which has organized this entire work is the duality between matrix models and gravity, which we reviewed in Chapters 3 and 4. One approach to describing this duality is the string equation formalism, centered on a set of functions u and gi satisfying a family of differential equations, new results for which were presented in Chapters 5 and 7. A further duality between the open and closed sectors of the string theories and matrix models was explored in Chapter 6. In Appendix A we provided some supporting information about the WKB approximation, which finds its use as an important tool used for probing statistical information in matrix 194 models. In Appendix B we presented some new results about a central object in the narrative of this work, the Gelfand-Dikii resolvent. The contents of [36] are split between Chapters 5 and 6, in which we have expanded upon twodimensional open-closed string duality by exploring it in the context of FZZT branes and topological recursion, inspired by the recent developments in JT gravity and its supersymmetric generalizations. Using the general matrix models describing non-supersymmetric closed strings [16–20], open strings [51, 69–71], 0A open and closed strings [65, 66], and N = 2 open and closed strings [77, 78], we have presented an explicit equivalence between operator insertions in the closed string sector and derivatives acting on the free energy in the open string sector. This equivalence allowed us to compute perturbative contributions to correlation functions which have the interpretation of gravity path integrals in various matter backgrounds, on surfaces with both asymptotic and geodesic boundaries. The surfaces with geodesic boundaries are related to string worldsheets ending on FZZT branes. The results presented here highlight the immense power of the string equation formalism, in particular the large number of applications of the DJM equation. In Chapter 5 we collected new explicit results for the perturbative solutions of various string equations, which generalize previously existing results to include parameters like the open string endpoint mass −z 2 and open-closed string coupling constant Γ. These perturbative solutions were utilized to compute the previously mentioned correlation functions in Chapter 6. Various examples were considered, including JT gravity and supergravity, as well as general N = 2 matrix models. Matrix models describing systems with extended supersymmetry are relatively new, and any progress made in studying them is especially interesting. In particular, we have demonstrated that the open-closed string matrix model has the power to compute interesting objects, like the trumpet partition function, and to probe special brane configurations, like eigenbranes, in theories with varying amounts of supersymmetry. In all calculations of the trumpet, the result is independent of the coupling constants tk that define the matter background. This is in agreement with [79]. The trumpet partition function can be thought of as coming from the Schwarzian path 195 integral on a hyperbolic surface with one asymptotic boundary and one geodesic boundary (see e.g. [9]). The universality of the result from the open-closed duality perspective agrees with the general relationship between the Schwarzian theory and two-dimensional CFTs presented in [97]. We also demonstrated the equivalence between two approaches to computing the FZZT brane partition function, one focused on macroscopic loops and the other on geodesic loops. Geodesic boundaries are central to showing the equivalence between JT gravity and the corresponding matrix model. This dictionary between the two makes use of Mirzakhani’s recursion relation for the WeilPetersson volumes of the moduli spaces of hyperbolic Riemann surfaces with geodesic boundaries. The concept of a generalized Weil-Petersson volume, relating topological gravity in a general background to intersection theory, was explored in, e.g., [79]. In Chapter 6 we showed that these generalized volumes appear naturally in the context of open-closed duality, being the fixed length versions of certain branerelated correlation functions, which are naturally computed in terms of the branes’ cosmological constants. One key strength of the string equation formalism is that it facilitates straightforward computations of the generalized volumes in supersymmetric theories. In Section 6.5 we provided a heuristic argument connecting the Virasoro constraints of the open string sector of the matrix model with topological recursion. This connection was explored in [50] starting from the recursion relation for the moduli space volumes, whereas here we have taken a matrix-model-first approach. The interpretation of the volumes in terms of correlation functions of closed string operators makes explicit the role that the KdV organization of observables in the theories plays in setting up the recursion relation via the Virasoro conditions. The content of [37] is contained in Chapter 7, the purpose of which was to further explore the study of double scaled β = 1 Wigner-Dyson models initiated in works like [63, 98]. Recent advancements in two-dimensional quantum gravity have brought to the forefront statistical properties of the theories, with 196 a large part being played by the eigenvalue density. One model, JT gravity, has been of particular importance, and the last several years have seen the development of a cottage industry involving generalizations, deformations, and refinements of that theory. Stanford and Witten [60] have explored many such extensions of JT gravity from the matrix model perspective, including its definition on unoriented surfaces. Meanwhile, Johnson and others have developed a complementary way of defining JT gravity and its generalizations via a certain combination of simpler matrix models. It is in the spirit of the latter that this work has proceeded. We have shown that the modern perspective on double scaled matrix models, e.g. a focus on the eigenvalue density, is compatible with the results obtained in the 90s for β = 1 models, by unifying the more popular approach in the physics community at the time – orthogonal polynomials – with the more popular approach in the integrable systems at the time – skew orthogonal polynomials . While it is possible to formally define the objects necessary to compute the eigenvalue density, we found that it is difficult to extract results for specific models. We believe the primary source of difficulty is the fact that the eigenvalue density is non-local in energy space via the presence of the energy integrals. Their mere presence is enough to render the WKB approach to perturbation theory intractable. Despite appealing to the (usually) powerful numerical approach, we found that many efforts to reduce error in one place, e.g. the fidelity of the wavefunctions, resulted in increased error elsewhere, e.g. the energy integrals in ρ. Further, the numerics displayed above had h̵ = 1, which is not particularly small. There are two logistical problems associated to decreasing h̵: first, the string equation for u is actually harder to solve; second, the wavefunctions oscillate more rapidly, making the diagonalization problem in solving the Schrodinger equation more resource-intensive. Despite the numerical difficulties, the leading order contribution to the density in the k = 2 model has the desired behavior for small E – albeit with some fictitious non-perturbative oscillation. The crosscap 197 contribution to the density behaves in accordance with what one would expect of a perturbative correction with a relatively large value of h̵. To conclude, we consider possibilities for future work. It would be interesting to further explore matrix models with extended supersymmetry in the string equation formalism. The methods proposed in Chapter 5 for perturbatively solving the string equation in N = 2 theories both involve rescaling the 0A coupling constants. Compared to the better understood 0A models, the interpretation of h̵ in the N = 2 models in either approach to perturbation theory is less clear. Additional powers of h̵ complicate its topological interpretation. Adding a new parameter to the theory requires determining its meaning in order to draw conclusions from the results. The proposed interpretation is in terms of a sort of RG flow from N = 1 to N = 2, with the latter representing a fixed point. We have noted that the results obtained here for the trumpet path integral and the moduli space volumes disagree with [77], although in the end each respective gluing procedure produces the same macroscopic loop correlation functions, up to extra E0-dependent terms which are sub-leading in the small E0 limit. It is worth understanding why the two approaches agree in such a limit in the end despite the preliminary disagreements. Given the strong connection between the Schwarzian theory and 2D CFTs, it is natural to expect that N = 4 theories have matrix model descriptions in the string equation formalism, although an interpretation in terms of minimal CFTs may be not be possible. In particular, it would be interesting to determine how perturbation theory works for those string equations and how the theories’ correlation functions differ from the N = 2 ones. Matrix models with N = 1 supersymmetry require further attention as well. There is currently no string equation that describes the multi-brane open string sector in supersymmetric models. It is likely that this would require a generalization of the Gelfand-Dikii differential equation, and consequently 198 the Gelfand-Dikii resolvent, to include multiple brane cosmological constants in analogy to the nonsupersymmetric case. Some progress has been made here to construct the expected solutions, but it is possible that other contributions to the multi-brane sector could exist as well. There is reason to believe that studying FZZT brane quantities may lead to some insight into the discrete structure of non-perturbative matrix model spectra (see e.g. [59, 99] and other recent works by Johnson). In [92] the authors considered eigenbranes, which are essentially squared determinant operators that fix an eigenvalue in the double scaled spectrum. It is possible that by having an infinite number of these eigenbranes one could recover fully discrete spectrum. By interpreting this scenario in terms of branes, one might be able to extract a geometric (or lack thereof) description of non-perturbative quantum gravity microstates. The multi-brane formalism developed here is well-suited to studying this problem from a different angle. It would be interesting to use the non-perturbative framework and numerical techniques developed by Johnson to describe eigenbrane configurations in arbitrary models. Given that the minimal model construction in β = 2 theories is so intimately connected to the KdV hierarchy, there must be a similar story behind the scenes of the β = 1 construction above. Two promising avenues to pursue are Drinfeld-Sokolov systems [100], due to the fundamental dependence of the double scaled β = 1 theory on factorizing a Lax operator, and the Pfaff lattice of Adler, et al [101], due to its fundamental connection with the β = 1 matrix integral at finite N. One should expect that a non-perturbative treatment of the β = 1 Wigner-Dyson models is possible. The β = 2 Wigner-Dyson models find as a possible non-perturbative completion the Altand-Zirnbauer (α, β) = (1+2Γ, 2) models, where the parameter Γ can be interpreted as the number of background branes in the theory [10, 11, 25, 96] . The logical prediction is that the Altland-Zirnbauer (α, β) = (1+γ, 1) models, for constant γ, will provide a non-perturbative completion of the models considered here. Demonstrating the connection between the β = 2 and (α, β) = (1 + 2Γ, 2) models requires the use of the KdV integrable structure of the theory, so it is possible that a non-perturbative completion of the β = 1 models will 199 require a more careful treatment of their integrable structure as well. It would be interesting if such a non-perturbative completion of the unoriented theory could be used to study O-planes in two dimensions. Finally, it would be interesting to better understand the role of the off-diagonal Gelfand-Dikii resolvent, as well as the generalized Gelfand-Dikii polynomials Rp,n introduced in Appendix B. Double scaled Hermitian multi-matrix models already utilize families of generalizations of the Gelfand-Dikii polynomials [48, 49]. However it does not seem likely that the polynomials given in (B.14) will be members of one of those families, since they are derived in the context of other hierarchies of differential equations. An interesting possibility is that they are somehow related to the descendants of the (2, 2p−1) minimal string primaries σk, perhaps appearing in the flow equations for the coupling constants dual to those operators. 200 Appendix A The Wentzel–Kramers–Brillouin Approximation The Wentzel–Kramers–Brillouin (WKB) approximation is a general method for solving linear differential eqeuations with varying coefficients, in particular when the highest order derivative is multiplied by a small parameter. The approximation is often used when perturbatively solving the Schrodinger equation. Given a one-dimensional potential u(x), the time-independent Schrodinger equation for a particle of mass m = 1 2 is −h̵2 d 2ψ dx2 + (u − E)ψ = 0. (A.1) It is convenient to define ψ = e iϕ/h̵ , and to expand ϕ = ∑ ∞ g=0 h̵nϕn. In terms of the the function ϕ, the Schrodinger equation is −ihϕ̵ ′′ + (ϕ ′ ) 2 + u − E = 0. (A.2) The leading order equation is thus (ϕ ′ 0 ) 2 + u − E = 0, (A.3) which is solved by ϕ0(x) = ±∫ x √ E − u(x ′)dx′ . (A.4) 201 The higher order contributions are determined in the standard fashion. However, suppose that the function u has an expansion in h̵ as well. For instance, the most general expansion used herein is u(x, E) = ∑ ∞ g,h=0 h̵2g+hΓ hug,h(x, E). Then not only should we expand ϕ in powers of Γ as well, but the expansion of u should be inserted in (A.2) while solving for ϕ. This will mean that ϕ will depend on multiple energies: the eigenvalue of the Schrodinger equation and the brane cosmological constants. Denoted the expansion ϕ(x; E, E′ ) = ∑ ∞ n,h=0 h̵n+hΓ hϕn,h(x; E, E′ ), where E′ denotes the vector of brane cosmological constants. The benefit of organizing the expansion this way is, as is the case in the open string sector of the string equations, the standard “closed string” version of the wavefunction is recovered by setting Γ = 0. Sticking with convention, define ϕ0,0 ≡ ϕ0; the leading order contribution will be ϕ0(x) = ±∫ x √ E − u0(x ′)dx′ . (A.5) In order to avoid extraneous details, we will focus on the case where all brane cosmological constants have the same value. This will be sufficient for discussion of eigenbranes and microstates in section 6.2.5. Then we also have the genus-0 contributions ϕ0,1(x; E, E′ ) = − 1 2 ∫ x u0,1(x ′ , E′ ) √ E − u0(x ′) dx′ ϕ0,2(x; E, E′ ) = −∫ x u0,1(x ′ ; E ′ ) 2 + 4(E − u0(x ′ ))u0,2(x ′ ; E ′ ) 8(E − u0(x ′)) 3/2 dx′ . (A.6) The first genus-1 contribution is ϕ1,0(x; E) = c + i 4 log (E − u0(x)), (A.7) 202 where c is a constant. We will often choose c = π/4 to match the asymptotic expansion of the Airy function. The first genus-2 contribution is ϕ2,0 = −∫ x 4(E − u0(x ′ )) (4u1,0(x ′ )(u0(x ′ ) − E) + u ′′ 0 (x ′ )) + 5u ′ 0 (x ′ ) 2 32(E − u0(x ′)) 5/2 (A.8) These can be simplified by exploiting the properties of u in (5.11), (5.12), and (5.38). Two simplifications of note in non-supersymmetric models are ϕ2,0(x; E) = 1 24 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ u ′′ 0 (x) u ′ 0 (x)(E − u0(x)) 1 2 + 5u ′ 0 (x) 2 2(E − u0(x)) 3 2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , ϕ0,2(x; E, E′ ) = u0,1(x; E ′ ) 2 16u ′ 0 (x)(E − u0(x)) 1 2 . (A.9) The full wavefunction we obtain is ψ(x; E) = 1 √ 2πh̵ 1 (E − u0) 1/4 e i h̵ (ϕ0+h̵Γϕ0,1+h̵2Γ 2ϕ0,2+h̵2ϕ2,0+⋯) (A.10) By Taylor expanding the exponential in h̵, we generate an h̵-expansion of ψ. More generally, we generate an expansion in ψ and Γ: ψ(x; E) ≈ 1 √ 2πh̵ [1 + Γϕ0,1 + h̵Γ 2ϕ0,2 + hϕ̵ 2,0 + ⋯] e i h̵ ϕ0 . (A.11) 203 A.1 Computing The Eigenvalue Density The eigenvalue density is computed in terms of ψ using (4.7). By inserting the expansion of ψ, we obtain the topological expansion of the eigenvalue density ρ(E, E ′ ) = ∞ ∑ g,h=0 h̵2g−1+hΓ h ρg,h(E, E ′ ). (A.12) At leading order, ρ0(E) = 1 2πh̵ ∫ µ −xc dx √ E − u0(x) , (A.13) where xc is the zero of the denominator. The extra orders of perturbation theory that one includes depend on the context. In the purely closed string sector, each ρg,0 represents a surface with one boundary and g handles. The interpretation in the open string sector is different. For instance, in the two-brane sector with identical cosmological constants, we are only interested in contributions to ρ coming from u0,2 that appear at O(Γ 2 ). The first such term is ρ0,2 = 1 2πh̵ ∫ µ dx ϕ2,0ϕ0,2, (A.14) where the lower bound of integration is intentionally left out. This integral typically diverges at the value −xc used in the leading order density, and so only the finite portion is kept. In region (−∞, 0] we can change the integration variable to u0 using the disk string equation f(u0) + x = 0, which gives ρ0,2 = − 1 2πh̵ ∫ 0 du0 ˙f ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ ¨f (E′ − u0)(E − u0) 1 2 − 5 2(E′ − u0)(E − u0) 3 2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ (A.15) 204 Much like the macroscopic loop correlation functions and other correlators, the perturbative contributions to the density in the open string sector are not well-behaved for the individual multicritical models past k = 1. This result is sufficient in non-supersymmetric models. With minimal effort we can extend these results to 0A models. Using the leading order solution u0 = E ′ , and the perturbative corrections in (5.51) and (5.52), we find ϕ0,2 = 1 2E′f(E′) √ ∣E − E′ ∣ log ( x + f(E ′ ) − 2E ′ ˙f(E ′ ) x + f(E′) ) , ϕ2,0 = − ϕ0,2 4 , (A.16) in the region (0, µ]. Hence we find the extra contribution to the density ∫ µ 0 ϕ0,2ϕ2,0 = µ 16(E′f(E′)) 2 ∣E − E′ ∣ . (A.17) 205 Appendix B New Results Regarding The Gelfand-Dikii Resolvent In Chapter 2 we introduced the Lax operator for the KdV hierarchy, Q = −∂ 2 + u. The function u satisfies the KdV flow equations (2.10), and when it is related to the free energy of a double scaled matrix model it also satisfies a string equation. As we have seen extensively in Chapters 5 and 6, the resolvent of Q plays a major role in organizing many different facets of double scaled matrix models, from obtaining solutions to string equations, to the computation of correlation functions. However, we typically only deal with the diagonal representation of the resolvent as an integral kernel, the function we refer to as Rˆ. One use for the off-diagonal of the resolvent in the context of matrix models is in computing macroscopic loop correlation functions. In [21] it was shown that ⟨e −β1He −β2H⟩ = ∫ µ −∞ dx∫ ∞ µ dy ⟨x∣e −β1H∣y⟩ ⟨y∣e −β2H∣x⟩. (B.1) We can take advantage of the fact that the resolvent of H is related to the Laplace transform of e −βH with respect to β. It is possible to directly calculate the leading order contribution to the matrix element ⟨x∣e −βH∣y⟩ without appealing to the Gelfand-Dikii resolvent, which we will use to our advantage later. The result is (see [68] for example) ⟨x∣e −βH∣y⟩ ≈ 1 2h̵ √ πβ e −βu0− 1 β ( x−y 2h̵ ) 2 . (B.2) 206 Using the matrix model notation, where derivatives are accompanied by factors of h̵ and we call the Lax operator H, the full resolvent function is Rˆ(x, y; ζ) = ⟨x∣(H + ζ) −1 ∣y⟩. (B.3) In [47] Gelfand and Dikii provide the construction Rˆ(x, y; ζ) = ∞ ∑ l=0 l ∑ m=0 (−1) l+m 2 Bl,m[u]D mRˆ (1+ l+m 2 ) 0 (x, y; ζ), (B.4) where we have the following restrictions and definitions: 1. The sum is over pairs (l, m) such that l + m is even and m ≤ l, 2. Bl,m[u] is a polynomial in u and its derivatives satisfying the recursion relation Bl,m = −B ′′ l−2,m + uBl−2,m + 2iB′ l−1,m−1 , B0,0 = 1, (B.5) and 3. D = ih∂̵ x, and Rˆ (n) 0 = Rˆ 0 ○ Rˆ 0 ○ ⋯ ○ Rˆ 0 with Rˆ 0 denoting the resolvent at u = 0. We are going to make a further restriction on (l, m): we conjecture that the “physical” part of the offdiagonal of the resolvent is given by the terms where l and m are both even. The terms where both are odd produce non-physical results — we will discuss this more at the end. The coefficients of the differential polynomials Bl,m in the sum are computable via Fourier transform. Set l = 2p and m = 2k. The result is Rˆ(x, y; ζ) = ∞ ∑ p=0 p ∑ k=0 (−1) p+kB2p,2k[u]r2k,1+k+p(x, y; ζ), (B.6) 207 where r2k,1+k+p = 1 √ 2πh̵ ζ −p− 1 2 Γ(k + 1 2 )Γ(p + 1 2 ) Γ(1 + p + k) 1F2 (k + 1 2 ; 1 2 , 1 2 − p; (x − y) 2 ζ 4h̵2 ) . (B.7) The first several of the Bs are B0,0 = 1, B1,1 = 0, B2,0 = u, B2,2 = 0, B3,1 = 2iu′ , B3,3 = 0, B4,0 = −u ′′ + u 2 , B4,2 = −4u ′′, B4,4 = 0. (B.8) The goal is to inverse Laplace transform the resolvent to obtain an expression for an off-diagonal matrix element of e −βH. All of the ζ-dependence is in the coefficients r: L −1 ζ [r2k,1+k+p](β) = β p √ 2πβh̵ Γ(k + 1 2 ) Γ(1 + k + p) 1F1 (k + 1 2 ; 1 2 ;− 1 β ( x − y 2h̵ ) 2 ) , (B.9) So, the inverse Laplace transform of the off-diagonal resolvent is L −1 ζ [Rˆ(x, y; ζ)](β) = 1 √ 2πβh̵ ∞ ∑ p=0 p ∑ k=0 (−1) p+kB2p,2k Γ(k + 1 2 ) Γ(1 + k + p) 1F1 (k + 1 2 ; 1 2 ;− 1 β ( x − y 2h̵ ) 2 ) (B.10) It is straightforward to show that B2p,0 = u p and that B2p,2k with k > 0 contributes starting at O(h̵2 ) using the recursion relation (B.5). Therefore the leading order contribution to the matrix element will occur at k = 0 with u ≈ u0. Using the fact that 1F1(a; a; z) = e z for any a, z, we get L −1 ζ [Rˆ(x, y; ζ)](β) ≈ 1 h̵ √ 2β e −βu0− 1 β ( x−y 2h̵ ) 2 (B.11) Hence in this normalization, we have 208 ⟨x∣e −βH∣y⟩ ≈ 1 √ 2π L −1 ζ [Rˆ(x, y; ζ)](β). (B.12) By mimicking the procedure in [47] we can also take the opportunity to define a “physcial” generalization of the Gelfand-Dikii polynomials. After expanding the hypergeometric function in the definition of r2k,1+k+p we have Rˆ(x, y; ζ) = ∞ ∑ p,n=0 p ∑ k=0 (−1) p+kB2p,2k Γ(k + n + 1 2 ) Γ(n + 1 2 )Γ(n − p + 1 2 ) (x − y) 2n ζ n−p− 1 2 (4h̵2) n . (B.13) Introduce the differential polynomials Rp,n[u] = (−1) p Γ(n − p + 1 2 ) p ∑ k=0 (−1) k Γ(k + n + 1 2 ) Γ(1 + k + p)Γ(n + 1 2 ) B2p,2k[u]. (B.14) Then Rˆ(x, y; ζ) = ∞ ∑ p,n=0 [ (x − y) 2 ζ 4h̵2 ] n Rp,n ζ p+ 1 2 . (B.15) On the diagonal x = y the only term in the sum over n that survives is n = 0. One can show using properties of the gamma function that Rp,0 matches the definition of the Gelfand-Dikii polynomials given in [47], and hence Rˆ(x, x; ζ) is the expected diagonal part of the resolvent. To conclude, we will make some remarks about the “unphysical” terms that were left out. The coefficients of the terms in the Fourier transform of (B.4) with l and m both odd are r2k+1,2+k+p = i∣x − y∣ √ 2πh̵2 ζ −p− 1 2 Γ(k + 3 2 )Γ(p + 1 2 ) Γ(2 + p + k) 1F2 (k + 3 2 ; 3 2 , 1 2 − p; ( σ 2h̵ ) 2 ) , (B.16) where we have let l = 2p + 1 and m = 2k + 1. It appears that the portion of the resolvent coming from summing over these coefficients will contribute at a different power of h̵ than the portion we previously 209 deemed physical. However, using the recursion relation (B.5) we can see that B2p+1,2k+1 contributes at O(h̵) for all p and k, which is shifted by one relative to B2p,2k. Hence we would receive an undesirable contribution to the matrix element ⟨x∣e −βH∣y⟩ at leading order if we included the (l, m) odd portion of the sum. Nevertheless, we can still define a further generalization of the Gelfand-Dikii polynomials by expanding the hypergeometric function and rearranging: R˜ p,n[x, y; u] = Rp,n + (−1) p ∣x − y∣ Γ(n − p + 1 2 ) p ∑ k=0 (−1) k Γ(k + n + 3 2 ) Γ(2 + k + p)Γ(n + 1 2 ) B˜ 2p+1,2k+1[u] h̵ , (B.17) where B˜ 2p+1,2k+1 = Im[B2p+1,2k+1]. These polynomials are explicitly dependent on x − y, unlike the Gelfand-Dikii polynomials or the generalization defined in (B.14). The full off-diagonal resolvent is therefore Rˆ(x, y; ζ) = ∞ ∑ p,n=0 [ (x − y) 2 ζ 4h̵2 ] n R˜ p,n(x, y; u) ζ p+ 1 2 . (B.18) Notice that R˜ p,n[x, x; u] = Rp,n, and hence the diagonal of this version of the resolvent produces the expected diagonal piece as well. 210 References [1] Roman Jackiw. “Lower dimensional gravity”. 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Creator Lowenstein, Ashton Camill (author)

Core Title Branes, topology, and perturbation theory in two-dimensional quantum gravity

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School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program physics

Degree Conferral Date 2024-08

Publication Date 06/19/2024

Defense Date 06/12/2024

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Tag gravity,matrix models,OAI-PMH Harvest,topology

Format theses (aat)

Language English

Advisor Haas, Stephan (committee chair), Bars, Itzhak (committee member), Benjamin, Nathan (committee member), Fulman, Jason (committee member), Johnson, Clifford (committee member)

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Abstract (if available)

Abstract Random matrix theory offers a powerful toolset to study a variety of two-dimensional quantum gravity and string theories both perturbatively and nonperturbatively. In this thesis we study three classes of matrix models describing general two-dimensional oriented gravity theories, supergravity theories, and unoriented gravity theories. Chapters 1 through 4 contain a review of necessary background material, including string theory, integrable hierarchies and intersection theory, and random matrix theory. In Chapter 5 we study perturbative solutions of two classes of differential equations, known as string equations, describing $\beta = 2$ Wigner-Dyson class matrix models and $(1 + 2\Gamma, 2)$ Altland-Zirnbauer class matrix models. In Chapter 6 we apply these results to operator correlation functions in the open and closed string sectors of the theories. We consider a duality between the open and closed string sectors of these models consisting of a relationship between correlation functions in one sector and the free energy in the other. We use this duality to deepen the understanding of the connection between D-branes and Weil-Petersson volumes, which are important in algebraic geometry and certain gravity theories. Specific examples are considered and compared to existing results in the literature. In Chapter 7 we develop a double scaling procedure in $\beta = 1$ Wigner-Dyson class matrix models, which are dual to certain unoriented gravity theories. We compute the leading density of states in two models, and form a conjecture for how to define unoriented JT gravity in a minimal model decomposition.