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On the instability of Pope-Warner solutions

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On the instability of Pope-Warner solutions

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Content On the Instability of Pope-Warner Solutions Dissertation Isaiah Yoo isaiahyo@usc.edu Dissertation Committee Chair: Dr. Krzysztof Pilch (pilch@usc.edu) Department of Physics and Astronomy University of Southern California Los Angeles, CA 90089, USA Contents 1 Introduction 4 1.1 Motivation from holographic superconductors . . . . . . . . . . . . . . . . . 8 1.2 Prior relevant results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3 Summary of main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2 Page-Pope-like construction 22 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2 Spinor covariant derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3 The D = 11 eld equations and their linearization . . . . . . . . . . . . . . . 29 2.4 Spinor bilinears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.5 Pope-Warner background solution . . . . . . . . . . . . . . . . . . . . . . . . 32 2.6 Perturbation ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.7 AdS equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.8 Eigenmodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.9 Comparison to known results . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3 The solutions 46 4 The linearized analysis 51 4.1 Linearized Einstein equations . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.2 Linearized Maxwell equations . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.3 The master harmonic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 1 4.4 The metric harmonic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.4.1 Proof of transversality . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.4.2 Proof of (4.32) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.5 The ux harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.6 The masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.7 Additional bosonic modes in the Z multiplet . . . . . . . . . . . . . . . . . . 68 5 Examples 75 5.1 Tri-Sasakian manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.2 Homogeneous Sasaki-Einstein manifolds . . . . . . . . . . . . . . . . . . . . 78 5.2.1 S 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.2.2 N 1;1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.2.3 M 3;2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.2.4 Q 1;1;1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.2.5 V 5;2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.3 Orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6 Conclusion 95 Appendices 97 A Conventions 97 B Sasaki-Einstein manifolds: denitions and relevant information 98 2 C Sasaki-Einstein identities 105 D Some harmonics on V 5;2 107 E Volume of V 5;2 110 F Linearized bosonic eld equations of D = 11 supergravity 113 G Conventions of [56] and [50] 121 H Toric homogeneous Sasaki-Einstein manifolds via K ahler quotient 126 3 1 Introduction The research presented in this dissertation is of interest due to the important result in string theory and M-theory known as the AdS/CFT correspondence. The AdS/CFT correspon- dence rst came to light in 1998 [1], and has since been a very active area of research. In [1] the AdS/CFT correspondence was proposed in the context of D3-branes in type IIB superstring theory. In this case of the AdS/CFT correspondence, type IIB superstring theory in the background geometry AdS 5 S 5 is conjectured to be dual to the four-dimensional U(N) N = 4 super-Yang Mills theory that is known to live on the worldvolume of N coincident D3-branes. When g s N, where g s is the string coupling constant, is very large, it can be shown that the gauge theory becomes strongly coupled, and the string theory can be well- approximated as a classical gravity theory, which means that stringy eects need not be included. This case of the AdS/CFT correspondence is reviewed in detail in [2]. A concrete realization of the AdS/CFT correspondence in the context of M2 branes in M-theory was given in [3]. This correspondence is known as ABJM theory. In this case of the AdS/CFT correspondence, M-theory in the geometry AdS 4 S 7 =Z k is conjectured to be dual to the three-dimensional U(N) U(N) Chern-Simons-matter theory at level k withN = 6 or 8 supersymmetry, that is known to live on the worldvolume of N coincident M2-branes. This duality holds in the limit whereN is very large. Additionally, whenNk 5 , M-theory can be well-approximated as a classical gravity theory. In light of ABJM theory, a natural question to ask was whether the correspondence could be generalized to the case of Chern-Simons theories with less supersymmetry. See, e.g. [4, 5]. 4 In the case of Chern-Simons theories withN 2 supersymmetry it is known that if such a theory has an M-theory dual, then the geometry of such a dual theory must be of the form AdS 4 SE 7 , where SE 7 is a type of compact manifold known as a `Sasaki-Einstein manifold'. The denition of a Sasaki-Einstein manifold and some important facts about them are given in Appendix B. It is known that supergravity theory on the background geometry AdS 4 SE 7 , which is called the (`skew-whied') `Freund-Rubin' background, can be continuously deformed to supergravity theory on another AdS product space background known as the `Pope-Warner' background, see, e.g., [6, 11]. In chapter 3 we give the Freund-Rubin, skew-whied Freund- Rubin, and Pope-Warner background solutions explicitly. Whereas the supergravity theory on the Freund-Rubin background was known to be stable [44], it was unknown whether the theory on the Pope-Warner background was stable. The purpose of the research in this disser- tation is to study the stability of Pope-Warner solutions on Sasaki-Einstein manifolds, which in light of the AdS/CFT correspondence should correspond to vacua of 2 + 1-dimensional eld theories. A major motivation for studying the AdS/CFT correspondence is its possible application to condensed matter physics, see e.g. [7, 8, 9]. In this vein, it was found that in `top-down' constructions of holographic superconductors, the Pope-Warner solution corresponds to a zero-temperature quantum critical phase of a 2 + 1-dimensional superconductor [10, 11]. In light of this promising nd, it was of strong interest to determine the stability of Pope- Warner solutions on Sasaki-Einstein manifolds. In section 1.1 of this introductory chapter we further discuss the relevance of the Pope-Warner solution to superconductor solutions. 5 Having discussed the broader context in which the research presented here is of interest, we now go more directly into the research itself. The eld equations of eleven-dimensional supergravity [14, 15] in the bosonic sector are: 1 R MN +g MN R = 1 3 F MPQR F N PQR ; (1.1) d?F (4) +F (4) ^F (4) = 0; (1.2) where g MN is the metric,F (4) =dA (3) is the four form ux, and ? denotes the Hodge dual in eleven dimensions. A simple and important class of solutions are the ones in which the eleven-dimensional space time is a product AdS 4 M 7 , where M 7 is a seven-dimensional Sasaki-Einstein (SE) manifold. Those manifolds are characterized by the existence of two real Killing spinors (see, e.g., [23, 24, 26], Appendix B) and the corresponding Freund- Rubin (FR) solutions [18] areN 2 supersymmetric. Solutions in which M 7 is one of the homogeneous SE manifolds: S 7 ; N 1;1 ; M 3;2 ; Q 1;1;1 ; V 5;2 ; (1.3) were classied in the 1980s [22], but it is only quite recently that new solutions with nonho- mogeneous SE metrics have been discovered [27, 28, 29]. It has also been known since the 1980s that given a SE manifold, M 7 , there are, in addition to the supersymmetric FR solution, three non-supersymmetric solutions: the skew- whied FR solution [18] obtained by the change of orientation on M 7 , and the Englert [19] 1 We summarize our conventions in appendix A. 6 and Pope-Warner (PW) [20, 21] solutions with nonvanishing internal uxes constructed from the geometric data on M 7 . Quite generally, non-supersymmetric solutions in gauged supergravity tend to be unsta- ble. Indeed, while the stability of an AdS-type solution is guaranteed if there are some unbroken supersymmetries [36, 37], in non-supersymmetric backgrounds one expects to nd scalar uctuations, (2 AdS 4 m 2 )' = 0; (1.4) whose masses violate the Breitenlohner-Freedman (BF) bound [38] m 2 L 2 9 4 ; (1.5) where L here is the radius of AdS 4 . For the solutions above, the perturbative stability of the skew-whied FR solution in eleven-dimensional supergravity was proved in [39]. It follows by a simple observation that the mass spectrum of uctuations that might produce an instability is invariant under the change of orientation of M 7 and hence is the same for the skew-whied and the supersym- metric backgrounds. The Englert solutions are more dicult to analyze because the background ux couples the scalar and pseudoscalar uctuations. The resulting perturbative instability for any SE background, M 7 , can be shown by an explicit construction of unstable modes in terms of the two Killing spinors [42]. The same instability is also visible in the massive truncation of the eleven-dimensional supergravity onM 7 [31, 11], and whenM 7 is the round seven-sphere, S 7 , it corresponds to the instability of the SO(7) critical point ofN = 8, d = 4 gauged 7 supergravity [40, 41]. Prior to the research presented in this dissertation, it was known from [32] that the PW solution onS 7 is in fact unstable. The question of the stability of the PW solution on other SE manifolds is the main concern of the research presented here. In the rest of this introductory chapter, we will do three things. First, we will discuss the motivation for looking at the stability of PW solutions. Given that the PW solution was rst constructed in 1984, one may ask why its stability is a concern now, many years later. The answer to this question lies in the context of AdS/CFT [2] and \top down" constructions of holographic superconductors [10, 11, 12], as will be discussed in the following section. Second, we will discuss prior results that are relevant to the research presented here, and nally, we will discuss and summarize the main results of this research. 1.1 Motivation from holographic superconductors AdS/CFT It was proposed in [1] that supergravity in the background geometry AdS 4 is dual to a d = 3 dimensional CFT in at space. As previously mentioned, such a duality was concretely realized in [3]. Here, we discuss a basic aspect of the AdS/CFT correspondence. Namely, we identify the space-time of the CFT theory with the radial slices of the AdS geometry, and we identify the value of the radial coordinate in the AdS geometry as the energy scale of the CFT. The AdS 4 metric can be written as ds 2 =r 2 dt 2 + 1 r 2 dr 2 +r 2 (dx 2 +dy 2 ): (1.6) 8 Under the change of variable r = 1 z it becomes ds 2 = 1 z 2 (dt 2 +dz 2 +dx 2 +dy 2 ): (1.7) With this form of the metric it is easy to see that for each xed value of the radial variable z there is a copy of at d = 3 Minkowski space. So AdS 4 can be regarded as copies of d = 3 Minkowski space along a radial variablez. The Minkowski space variablest,x, andy of the AdS 4 space can be identied with the time and space variables of thed = 3 CFT. The radial variablez of theAdS 4 space is to be identied as the energy scale of thed = 3 CFT [16, 17]. This identication of z as the energy scale of the CFT simply follows from the fact that thed = 4 andd = 3 theories are dual, and from the fact that thed = 3 theory is conformally invariant [17]. To see how this identication follows from these facts, consider what happens when changing the length scale of the d = 3 theory. Changing the length scale of the d = 3 theory amounts to making the transformation (t;x;y)! (t;x;y). Since the d = 3 theory is conformally invariant this transformation has no eect. However this transformation will clearly change the AdS 4 metric. In order to maintain the equivalence, i.e. duality, of the d = 4 and d = 3 theories, it is necessary to also simultaneously make the transformation z ! z. With this additional transformation it is clear that the AdS 4 metric stays the same. Therefore, since energy goes as inverse length, changing the energy (length) scale of the d = 3 theory amounts to moving along the radial direction as z!z. And hence, the radial direction of the d = 4 theory should be identied with the energy scale of the d = 3 theory. Since energy goes as inverse length, larger should correspond to a smaller energy scale and a smaller should correspond to a larger energy scale. In conjunction, taking the limit 9 z!1 (or r! 0) corresponds to owing to the IR of the d = 3 theory, and taking the limit z! 0 (or r!1) corresponds to owing to the UV of the d = 3 theory. For more discussion on the identication of the radial variable as the energy scale, see section 12.3 of [17]. It is important to note that in general, the supergravity solution to the kind of set-up discussed below is only asymptotically AdS 4 , with the geometry in the bulk of the space- time being more complicated. In this case it is dicult to rigorously prove that the radial direction is to be identied with the eld theory energy scale, however, it is nonetheless taken to be so. Holographic superconductors Here, we discuss the basic idea of holographic superconductors. For more detailed dis- cussions of this topic, see [7, 8, 9]. In the IR, many condensed matter systems of interest become strongly coupled and therefore dicult to study using standard condensed matter techniques. The AdS/CFT correspondence provides a way to possibly obtain valuable information about such systems. Even though the AdS/CFT correspondence is only valid for a large number of gauge degrees of freedom N, it is nonetheless hoped that by working at very large N it wil be possible to obtain valuable information about strongly-coupled systems that is independent of N, or that it will be possible to at least gain some hints as to how to proceed for small N. Using the AdS/CFT correspondence one can consider the dual gravity theory of the system of interest. In the dual gravity theory one can derive the equations of motion for the relevant elds from an action and solve them, at least numerically. 10 Since the energy scale of the condensed matter sytem is identied with the radial variable z in the dual gravity system, to obtain the IR stongly coupled behavior of the condensed matter system, one need only take the z!1 limit of the gravity solution that was found, and then use the AdS/CFT correspondence to obtain the IR strongly coupled condensed matter system. A holographic superconductor setup involves at least a U(1) gauge eld and a complex scalar eld that is charged with respect to it. Above a critical temperature, the scalar eld has no expectation value, and is said to have `no hair'. If the scalar eld develops a non-zero expectation value, i.e. hair, below the critical temperature, then it comes to possess a denite phase, thus breaking the U(1) symmetry. In such a case the gravity solution describing this behavior is said to be a `holographic superconductor'. In order to have a non-zero temperature and nite chemical potential, an electrically charged black hole is placed at the center of the space-time. This black hole solution describes the unbroken phase of the superconductor. The gravity theory at the near-horizon limit of the black hole solution corresponds to the IR of the eld theory, and the z! 0 or r!1 limit of the black hole solution corresponds to the UV of the eld theory. The space-time at thez! 0 orr!1 limit isAdS 4 , and the gravity theory at this limit is dual to the UV of the eld theory. The goal is to obtain the behavior of the system as the temperature is decreased from above the critical temperature, where the superconductor is in an unbroken phase, to below it, where the superconductor is in a broken phase. One is especially interested in what happens at the z!1 or r! 0 limit. The gravity theory at this limit corresponds to the 11 IR of the eld theory, which is dicult to study using standard condensed matter techniques. The system is set up in such a way that for all temperatures the spacetime at the z! 0 orr!1 limit, which corresponds to the UV of the eld theory, isAdS 4 . The system is set up in this manner so that the AdS/CFT correspondence can be used. Holographic superconductors from M-theory In [8] the authors showed that many M-theory vacua corresponding to Freund-Rubin compactications on seven-dimensional Sasaki-Einstein manifolds provide holographic grav- ity duals of d = 3 CFTs that exhibit superconductivity. Holographic superconductor solu- tions are given in [8] for the linearized equations of motion of d = 11 supergravity. Much information can be obtained from solutions to the linearized supergravity equations of mo- tion, e.g., critical temperatures [8], but it is of course desireable to construct solutions for the full nonlinear equations of motion. In order to construct a holographic superconductor solution, one needs at least a metric, a U(1) gauge eld, and a charged scalar that can condense and break the U(1) symmetry. However, nding holographic superconductor solutions for the full supergravity equations of motion, involving at least these three elds, is in general a dicult task. Finding such solutions is in general dicult because of the many types of couplings that can occur between the few `desired' elds and various other `undesireable' elds that exist in the theory. A way to avoid this diculty is by working within a consistent truncation of the full theory. Working within a consistent truncation of the full theory guarantees that the many elds in the theory that are `undesireable' can be consistently set to 0, without being sourced in the course of the evolution of the system. 12 Nonlinear D = 11 super uid black brane solutions of [10] and [11] Indeed, using the universal Sasaki-Einstein consistent truncation found in [31], the au- thors of [10] and [11] were able to construct nonlinear black brane solutions of d = 11 su- pergravity whose corresponding four-dimensional gravity theories are holographic supercon- ductors. These black-brane solutions are particularly elegant because they apply universally for all Sasaki-Einstein manifolds. A notable feature of the holographic superconductor solutions found in [10] and [11] is that the T! 0 limit of these solutions are charged domain wall solutions that interpolate between the skew-whied Freund-Rubin vacuum in the UV and the Pope-Warner vacuum in the IR. Since the Pope-Warner solution is a compactication to AdS 4 , it follows that this T = 0 domain wall solution corresponds to a d = 3 CFT that has emergent conformal symmetry in the far IR. However, in the case that the Pope-Warner vacuum is unstable for a particular Sasaki-Einstein manifold, the viability of the corresponding superconductor solution is put into question. In such a case, it is reasonable to conclude that the Pope-Warner vacuum can not be used as a viable ground state for a CFT atT = 0. Indeed, one would expect that a quantum uctuation of an unstable mode would grow exponentially and cause the system to ow to a stable vacuum. However, it is not clear what to conclude for the superconductor solutions at T > 0, whose T ! 0 limits are the Pope-Warner vacuum. Perhaps it is possible that thermal uctuations could serve to stabilize the vacuum. Instability of the Pope-Warner solution and its implications for holographic su- perconductors 13 At the time [11] was written it was unknown whether the Pope-Warner solution on any Sasak-Einstein manifold is unstable. It was later found in [32] that the Pope-Warner solution on S 7 is unstable, and more recently in [33] it was shown that, in fact, the Pope-Warner solution on any of the homogeneous Sasaki-Einstein spaces is unstable. A consequence of the results of [32] and [33] on the program of constructing non-linear super uid black brane solutions is clear: the Pope-Warner solution likely cannot be used as a viable T = 0 ground state if the compactifying manifold is taken to be a homogenous Sasaki-Einstein manifold or an orbifold of one that is discussed in [33]. In conjuction, the results of [32] and [33] indicate that in constructing super uid black brane solutions, especially forT! 0, one should also utilize consistent truncations other than the universal Sasaki-Einstein truncation, and perhaps focus on particular compactication manifolds or restricted classes of them. Nonlinear D = 11 super uid black brane solutions of [13] Interestingly, in [13] it was pointed out that the critical temperatures that were obtained in [10] and [11] from using the universal Sasaki-Einstein truncation are not as high as those discussed in [8], and that, therefore, the superconductor solutions of [10] and [11] are not thermodynamically relevant. Motivated, at least in part, by this fact, and perhaps also by the instability results of [32], the authors of [13] used several dierent consistent truncations specic to S 7 to construct a variety of super uid black brane solutions. Among these solutions are ones that in the T = 0 limit are domain wall solutions that interpolate between the SO(8)AdS 4 xed point in the UV and the SU(3) U(1)AdS 4 xed point in the IR. Unlike the SU(4) xed point that uplifts to the Pope-Warner solution on 14 S 7 , the SU(3) U(1) xed point in the IR is stable because it is supersymmetric. 1.2 Prior relevant results Having discussed why the stability of the PW solution became a topic of interest, we now want to discuss prior results that were relevant in carrying out the research presented in this dissertation. The rst major result on the stability of the PW solution was given in [32]. In this paper, the authors showed that the PW solution on S 7 was in fact unstable. The authors showed that in the S 7 case the minimal sector of the SE truncation of [11, 31] coincides with the SU(4) sector ofN = 8, d = 4 gauged supergravity. Then, expanding theN = 8 potential to quadratic order about the critical point corresponding to the PW point, the authors found that there are unstable scalars that transform in 20 0 of SU(4) . The authors were able to uplift these unstable modes to eleven-dimensional supergravity, where the SU(4) symmetry becomes the isometry of CP 3 , which is the KE base ofS 7 . They give a metric perturbation and 3-form perturbations that yield the unstable scalars under reduction to four dimension. It is the discussion of the D = 11 picture in [32] that most pertains to the research presented in this dissertation. In particular, looking carefully at the structure of the metric perturbation given in [32] helped guide us toward a way to generalize the results of [32], as will be discussed in chapter 3. Given the results of [32], it was natural to ask what happens for PW solutions in the case of SE manifolds other than S 7 . It is in fact this question that is the main topic of the research presented here. A possible way to generalize the results of [32] was hinted at by the 15 contents of the paper [42], on the instability of the Englert solution. The Englert solution on the roundS 7 was rst shown to be unstable in [41]. The results of [41] were then generalized to Englert solutions on internal manifolds with two or more Killing spinors in [42]. The key idea of [42] was to construct metric and 3-form perturbations about the Englert background using two or more Killing spinors, and to see what the masses of the resulting scalars were after dimensionally reducing toAdS 4 . Carrying out this procedure yielded scalar masses that violate the BF bound. Since the construction of [42] only utilizes two or more Killing spinors, its results apply universally to all SE manifolds. In the case of the PW solution, we carried out an analogous construction using three or more Killing spinors, and we likewise found that the construction yielded scalars whose masses violate the BF bound. In this way we were able to generalize the results of [32] to all tri-Sasaki manifolds, which are SE manifolds with three or more Killing spinors (see Appendix B). This calculation is presented in chapter 2. It should be mentioned that this calculation was carried out before the release of [34], which contained the same results. In [34] the authors carried out a universal consistent truncation of eleven-dimensional supergravity on tri-Sasakian manifolds. After presenting a solution to eleven-dimensional supergravity based on seven-dimensional tri-Sasakian structure, the authors dimensionally reduced the theory to four dimensions. The process of dimensional reduction yields a poten- tial for the four-dimensional theory that has the PW solution as a critical point. Expanding the potential about the PW critical point to quadratic order, the authors found that a scalar contained in the truncation has a mass-value that violates the BF bound. As mentioned, we were able to obtain the same unstable scalar using a construction analogous to that in [42]. 16 Having established that the PW solution on tri-Sasaki manifolds is unstable, one would like to know what the situation is for non-tri-Sasaki SE manifolds, i.e., for SE manifolds with exactlyN = 2 supersymmetry. A possible way to proceed is by looking at consistent trun- cations on SE manifolds, and expanding the corresponding potentials about the PW point to see whether there exist any unstable modes. Indeed, additional consistent truncations on N = 2 SE manifolds that generalize the consistent truncation of [11, 31] were carried out in [35]. These truncations, however, do not yield unstable modes at the PW point. Another possible way to proceed for theN = 2 case was provided by the key observations we made that the metric perturbation that led to instability in the S 7 case had components only along the KE base, and furthermore, that it could be expressed in terms of a transverse, primitive (1,1)-form and a certain canonical SE object. These observations led us to focus our attention on transverse, primitive (1,1)-forms that are eigenfunctions of the Hodge-de Rham Laplacian. In particular, we used such objects together with canonical SE objects to construct metric and 3-form perturbations on SE manifolds. See chapter 4 for the details of our construction. An analogous construction is found in [58]. In this paper the authors examined the sta- bility of AdS 5 solutions of eleven-dimensional supergravity compactied on six-dimensional K ahler-Einstein (KE) spaces. The main result of the paper is that the solution suers a bosonic instability if and only if there exists a transverse, primitive (1,1)-form that is an eigenfunction of the Hodge-deRham Laplacian with eigenvalue within a certain given range of values. In particular, such a (1,1)-form can be used to construct metric and 4-form perturbations that reduce to unstable AdS 5 scalars. 17 Even though [58] deals with compactication on six-dimensional KE spaces, it contains results that are applicable to the case of interest to us because regular SE manifolds can be seen as U(1)-bundles over KE bases, see e.g. Appendix B. In particular, [58] contains explicit (1,1)-forms that also exist on SE manifolds that we look at, and that are eigenfunctions of the Hodge-de Rham Laplacian with eigenvalues that lead to instability. See subsection 1.3 and chapter 5 for details. 1.3 Summary of main results In this research we identify a potential source of perturbative instability of the PW solution on an arbitrary (regular) SE manifold. We show that starting with a basic, primitive, transverse (1,1)-form! onM 7 , which is an eigenform of the Hodge-de Rham Laplacian, 2 , with the eigenvalue ! 0, one can construct explicitly one metric and two ux harmonics, which after diagonalization of the linearized equations of motion give rise to three modes in the scalar spectrum with the following masses: (i) supersymmetric FR m 2 L 2 : ! 4 2; ! 4 + p ! + 1 1; ! 4 p ! + 1 1; (1.8) (ii) skew-whied FR m 2 L 2 : ! 4 2; ! 4 + 2 p ! + 1 + 2; ! 4 2 p ! + 1 + 2; (1.9) (iii) PW m 2 L 2 : 3 8 ! ; 3 8 ! + 3 p ! + 1 + 3; 3 8 ! 3 p ! + 1 + 3: (1.10) 18 M 7 ! m 2 L 2 # of modes KK spectra S 7 24 3 20 [43, 44] N 1;1 24 3 1 [45, 46, 47] M 3;2 16 9 3 p 17 8 [48, 49, 50] Q 1;1;1 16 9 3 p 17 9 [51] V 5;2 32=3 7 p 105 5 [52] Table 1: Unstable modes for the PW solution on homogeneous SE manifolds. For the rst two solutions, all modes in (1.8) and (1.9) are stable with the lowest possible masses saturating the BF-bound (1.5) when ! = 3 and ! = 15, respectively. However, for the PW solution, the last mode in (1.10) becomes unstable when ! lies in the range 2(9 4 p 3)< ! < 2(9 + 4 p 3): (1.11) In principle, all that remains then is to determine which SE manifolds admit such stability violating (1; 1)-forms. Unfortunately, this appears to be a dicult problem since no general bounds on the low lying eigenvalues of 2 on an arbitrary SE manifold are known. In the absence of general results, we look at the homogeneous SE manifolds (1.3) for which the spectra of the Hodge-de Rham Laplacians, k , and of the Lichnerowicz operator, L , have been calculated in the references listed in Table 1, either as part of the Kaluza- Klein program in the 1980s, 2 or, more recently, to test the AdS/CFT correspondence for 2 For a review, see, e.g., [44] and [49]. 19 M 2 -branes at conical singularities [53, 54, 55]. Specically, the eigenvalues of the Hodge-de Rham Laplacian, 2 , can be read-o from the masses of Z-vector elds that arise from the Kaluza-Klein reduction of the three-form potential along two-form harmonics. By examining the mass spectra ofZ-vector elds, we conclude that on each homogeneous SE manifold there are two-forms with the eigenvalues of 2 within the instability range (1.11). One must then determine whether any of those forms are basic, transverse and primitive. We found that, given the KK data for the two-form harmonics, which include the representation and the R-charge, it is actually the easiest to construct those forms explicitly and then verify that they indeed satisfy all the required properties. Our results are summarized in Table 1, which shows that there are unstable modes for the PW solution on all homogeneous SE manifolds. The three harmonics for the scalar elds in (1.8)-(1.10) are related to the master (1; 1)- form by operations (contractions and exterior products) that involve canonical objects of the SE geometry: the metric and the forms, which can be expressed in terms of Killing spinors on the SE manifold. From a general analysis of harmonics on coset spaces with Killing spinors [56], it is reasonable to expect that, at the supersymmetric solution, the three scalar elds and the Z-vector eld should lie in the sameN = 2 supermultiplet. Indeed, the pattern of masses in (1.8) and the presence of the Z-vector eld with the correct mass, and their R-charges, suggest that it is a longZ-vector supermuliplet [57]. Ultimately, this observation explains why we can diagonalize the mass operator for uctuations around the PW solution on such a small set of modes { the mixing due to the background ux involves only harmonics within a single supermultiplet. It also suggests where to look for an instability of the PW 20 solution on a general SE manifold. A regular SE manifold, M 7 , is a U(1) bration over its KE base, B 6 , so any (1,1)- form, !, as above is a pull-back of a transverse, primitive (1,1)-form on B 6 with the same eigenvalue of the corresponding Hodge-de Rham Laplacian, (1;1) . This shows that the potential instability of the PW solution that we have identied resides in the spectrum of (1;1) on KE manifolds. It also provides a link to a dierent class of solutions whose stability has been analyzed recently. As discussed in the previous section, precisely the same type (1,1)-forms, albeit with a dierent \window of instability," were shown in [58] to destabilize the AdS 5 B 6 solutions [59, 60] of eleven-dimensional supergravity. Two results in [58] are directly applicable to our analysis. The rst one is an explicit construction of a (1; 1)-form !, with ! = 16, on S 2 S 2 S 2 , which is the KE base for Q 1;1;1 . The second one is more general and concerns the spectrum of (1;1) on a product of two K ahler manifolds, B 6 =B 2 B 4 . It is shown that if B 4 admits a continous symmetry, then there exists a transverse, primitive (1; 1)-form ! on B 6 with the eigenvalue ! = 16. In particular, the unstable modes on M 3;2 , which is a U(1) bration over S 2 CP 2 , arise in this way. Another KE manifold that is covered by this construction is S 2 dP 3 , where dP 3 is the del Pezzo surface. This gives us an example of an inhomogeneous SE manifold with an unstable PW solution. The rest of the dissertation is organized as follows. In chapter 2 we present the calculation analogous to what was done in [42] showing that the PW solution on tri-Sasakian manifolds is unstable. In chapter 3, we review the FR and PW solutions together with some pertinent SE geometry. Even though the PW solution is given in chapter 2, we present it again in 21 chapter 3, because the conventions in chapter 2 are dierent than they are in the rest of this dissertation. We then in chapter 4 present the details of our calculation leading to the mass formulae (1.8)-(1.10). In chapter 5 we construct explicitly the unstable modes for all homogeneous examples. We conclude with some comments in chapter 6. Our conventions and some useful identities are summarized in appendices. 2 Page-Pope-like construction 2.1 Introduction In [42] Killing spinors on S 7 were used to construct linearized modes about the Englert solution of the bosonic eld equations of d = 11 supergravity. In the Englert solution, the 4-form ux has two parts. One part is taken to be the volume form of AdS 4 , and the other part is an internal ux that has components only along the compact S 7 directions. The internal 4-form ux is constructed as a spinor bilinear with four legs, using one of the eight Killing spinors on the round S 7 . This single spinor is invariant under an SO(7) subgroup of SO(8), the symmetry group of the round S 7 , while the other seven spinors transform as the 7 of this SO(7). Since the internal 4-form is constructed from an SO(7)-invariant spinor, it itself is an SO(7)-invariant form. Therefore, since the round S 7 is also SO(7)-invariant, the Englert solution is SO(7)-invariant. The S 7 in the Englert solution is the round S 7 , which has SO(8) symmetry. Therefore, all the spinors on the S 7 are Killing spinors, satisfyingr a = a , where is a constant. Perturbations of the metric and 4-form are constructed from the 8 Killing spinors and d = 7 22 gamma matrices. Since Killing spinors are used, thed = 11 linearized eld equations reduce to a much simpler set of dierential equations on AdS 4 , from which the masses of AdS 4 scalars can be easily obtained. Here, we carry out an analogous procedure on the Pope-Warner solution on the stretched S 7 [20]. The Pope-Warner solution has SU(4) (SO(6)) invariance. Like the Englert solution it has an internal 4-form ux, however in the Pope-Warner case it is constructed from two, rather than one Killing spinor. These two spinors are singlets of the SU(4) invariance group, while the other six spinors transform as the 6 of this SU(4). Analogously to the Englert case, perturbations to the metric and 4-form ux are constructed from the eight spinors and d = 7 gamma matrices, and thed = 11 linearized eld equations are reduced to a simple set of dierential equations on AdS 4 , from which masses of AdS 4 scalars are easily obtained. However, the procedure in the Pope-Warner case is complicated by the fact that spinors on the stretched S 7 do not satisfy the equationr a = a . We are able to deal with this complication by rearranging the spinor covariant derivative in a convenient way. This rearrangement of the spinor covariant derivative is carried out in section 2.2. In section 2.3 the d = 11 bosonic eld equations and their linearization are given. In section 2.4 we discuss spinors and spinor bilinears. In section 2.5 we obtain the Pope-Warner background solution. In section 2.6 we present the perturbation ansatz. In section 2.7 we plug the Pope-Warner solution and our perturbation ansatz into the linearized eld equations and obtain simple dierential equations inAdS 4 for scalar elds. In section 2.8 we diagonalize the AdS 4 equations to obtain the masses of the scalar elds. Finally, in section 2.9 we compare our results to known results. 23 It is important to mention that the conventions used in this chapter are dierent from those used in the rest of this work. The Dirac matrices are taken to be real and antisym- metric, satisfying f a ; b g =2 ab : Also, in the metric for thed = 11 compactied solution, the size of the KE base is held xed in going from the FR to the PW points. In the rest of this work, the size of the KE base varies, whereas the AdS radius squared is taken to be L 2 . 2.2 Spinor covariant derivative The spinor covariant derivative is [44] r m = (@ m 1 4 ! m bc bc ): (2.1) The index `m' is for the curved coordinates, and the ! m ab are the spin connections. The spinor covariant derivative can be expressed in terms of the frame coordinates simply by contracting with the inverse frame e m a . So r a = (@ a 1 4 ! a bc bc ); (2.2) where @ a = e m a @ m (2.3) ! a bc = e m a ! m bc : (2.4) We are interested inS 7 as a U(1) bration over the KE spaceCP 3 . In this case the metric can be written as [44] ds 2 =d s 2 +c 2 (dA) 2 ; (2.5) 24 where d s 2 is the metric on CP 3 , and A is a 1-form potential on CP 3 that gives rise to the complex structure J. In the case that c = 1, the sphere is round. Otherwise the sphere is said to be `stretched'. For frames we take e i = e i ; i = 1;:::; 6 (2.6) e 7 = c(dA) (2.7) = c e 7 ; (2.8) where the e i are frames on the CP 3 , and e 7 is the frame for the ber in the case that the sphere is round. With this choice of frames, it is found that the spin connections are given by ! ij = ! ij +cJ ij e 7 (2.9) ! 7i =cJ i j e j ; (2.10) where J ij = (dA) ij is the complex structure on CP 3 . Rearranging the spinor covariant derivative for the stretched sphere In the case that the sphere is stretched, i.e. c 6= 1, we would like to rearrange the spinor covariant derivative in such a way that the contribution made to it from stretching is manifest. First, we express the inverse frames on the stretched sphere in terms of those of the round sphere. To do so we write e m a e m b = b i e m a e m i +c b 7 e m a e 7 m (2.11) = b a : (2.12) 25 From the above equation, it must be that e m i = e m i (2.13) e m 7 = 1 c e m 7 ; (2.14) where e m a are the inverse frames on the round sphere. Next, we express the partial derivatives on the stretched sphere in terms of those on the round sphere. To do so we write @ a = e m a @ m (2.15) = i a e m i @ m + 7 a e m 7 @ m (2.16) = i a e m i @ m + 1 c 7 a e m 7 @ m : (2.17) From the above equation we see that @ i = @ i (2.18) @ 7 = 1 c @ 7 ; (2.19) where @ a is the partial derivative for the round sphere, i.e. for c = 1. Now, we express the spin connections on the stretched sphere in terms of those on the round sphere. The spin connections are ! ij = ! ij +cJ ij e 7 (2.20) ! 7i = cJ i j e j : (2.21) 26 From these we see that ! k ij = ! ij k (2.22) ! k 7j = cJ j k (2.23) ! 7 ij = cJ ij : (2.24) So along the CP 3 we have 1 4 ! k bc bc = 1 4 ! ij k ij + 1 2 ! k 7j 7j (2.25) = 1 4 ! ij k ij 1 2 cJ j k 7j (2.26) = 1 4 ! ij k ij 1 2 J j k 7j + 1 2 (1c)J j k 7j (2.27) = 1 4 ! bc k bc + 1 2 (1c)J j k 7j ; (2.28) giving 1 4 ! k bc bc = 1 4 ! bc k bc + 1 2 (1c)J j k 7j : (2.29) Along the ber we have 1 4 ! 7 bc bc = 1 4 cJ ij ij (2.30) = 1 4 J ij ij + 1 4 (c 1)J ij ij (2.31) = 1 4 ! bc 7 bc + 1 4 (c 1)J ij ij ; (2.32) giving 1 4 ! 7 bc bc = 1 4 ! bc 7 bc + 1 4 (c 1)J ij ij ; (2.33) where ! bc are the spin connections on the round sphere. The rearranged spinor covariant derivative 27 Finally, putting together equations (2), (18), and (29) we have along the CP 3 r k = @ k 1 4 ! k bc bc (2.34) = r k + 1 2 (c 1)J j k 7j ; (2.35) giving r k = r k + 1 2 (c 1)J j k 7j ; (2.36) where r a is the spinor covariant derivative for the round sphere. And putting together equations (2), (19), and (33) we have along the ber r 7 = @ 7 1 4 ! 7 bc bc (2.37) = 1 c @ 7 1 4 ! bc 7 bc 1 4 (c 1)J ij ij (2.38) = 1 c r 7 1 4 (1 1 c ) ! ij 7 ij 1 4 (c 1)J ij ij (2.39) = 1 c r 7 1 4 c 2 1 c J ij ij ; (2.40) giving r 7 = 1 c r 7 1 4 c 2 1 c J ij ij : (2.41) In summary we have r a = a r a +E a ; (2.42) where i = 1 (2.43) 7 = 1 c ; (2.44) 28 E i = J j i 7j (2.45) E 7 = J jk jk ; (2.46) and = 1 2 (c 1) (2.47) = 1 4 c 2 1 c : (2.48) So if is a Killing spinor on the S 7 , then r a = a ; (2.49) where = 1 2 for unit radius. 2.3 The D = 11 eld equations and their linearization The bosonic sector of d = 11 supergravity consists of a metric g AB and a 3-form potential A ABC . The exterior derivative of the 3-form potential gives a 4-form uxF ABCD . Classically these elds must satisfy thed = 11 supergravity bosonic eld equations. These eld equations consist of an Einstein equation, a Maxwell equation, and the Bianchi identity for F ABCD . The Einstein equation is R AB = 1 3 F ACDE F CDE B 1 36 g AB F CDEF F CDEF ; (2.50) the Maxwell equation is r A F ABCD = 1 576 BCDEFGHIJKL F EFGH F IJKL ; (2.51) 29 and the Bianchi identity is r [A F BCDE] = 0: (2.52) We would like to perturb the elds g AB and F ABCD , in such a way that the perturbed elds still satisfy the equations of motion. Let h AB and f MNPQ be the perturbations to the metric and ux, respectively. The perturbed elds are then g AB = g AB +h AB (2.53) F ABCD = F ABCD +f MNPQ : (2.54) We would like to put these perturbed elds into the equations of motion and determine the equations the perturbations h AB and f ABCD must satisfy to rst order in order for g AB andF ABCD to be solutions. The equations h AB and f ABCD must satisfy to rst order are the `linearized eld equations'. The d=11 linearized eld equations The linearized bosonic eld equations ofd = 11 supergravity are derived in Appendix F. The linearized Einstein equation is 1 2 ^ h AB + r (A r C h B)C 1 2 r A r B h C C = F CNP A F M B NP h MC 1 36 h AB F CDEF F CDEF + 1 9 g AB h CM F CDEF F DEF M + 2 3 F MNP (A f B)MNP 1 18 g AB F MNPQ f MNPQ ; (2.55) the linearized Maxwell equation is r A f ABCD + 4 r A (F M[ABC h D] M ) 1 2 F BCDR r R h A A = 1 288 BCDEFGHIJKL F EFGH f IJKL 1 1152 Tr(g 1 h) BCDEFGHIJKL F EFGH F IJKL ; (2.56) 30 and the linearized Bianchi identity is r [A f BCDE] = 0: (2.57) We want to plug the Pope-Warner background solution obtained in section 2.5 and the perturbation ansatz given in section 2.6 into the linearized eld equations and obtain eld equations for scalars in AdS 4 . 2.4 Spinor bilinears The roundS 7 , i.e. that withc = 1 in the metric (5), has symmetry group SO(8). When the sphere is stretched, so that c6= 1, the symmetry group SO(8) is broken to the symmetry group of the CP 3 , which is SU(4). The group SO(8) has two 8-dimensional spinor irreps. Under the subgroup SU(4) that is the symmetry group of the CP 3 one of these spinor irreps breaks as [80] 8! 6 + 1 + 1: (2.58) Let the singlet spinors be denoted by and , and the spinors that transform in the 6 be denoted by i , i = 1;:::; 6. Since and are invariant under SU(4), spinor bilinear forms constructed from these spinors are invariant under SU(4). Due to the antisymmetry of the gamma matrices in 7 dimensions, there is only one spinor bilinear 1-form and only one spinor bilinear 2-form that can be constructed from and , namely a and ab , respectively. In fact the former ise 7 , where e 7 is the seventh frame in section (I), and the latter isJ, where J is the complex structure. The 3-forms abc and abc are used in the construction of the Pope-Warner back- 31 ground solution in the next section. Their exterior derivatives abcd and abcd along with the volume form of AdS 4 make up the background 4-form ux. Spinor bilinear forms can also be constructed from the spinors i that transform in the 6. These forms are clearly not SU(4)-invariant. It is convenient to dene K ij abcd = i abcd j ; i;j = 1;:::; 6: (2.59) By antisymmetry of the gamma matrices, theK ij are symmetric under interchange ofi and j. It is also convenient to dene the 4-forms K a abcd = abcd + abcd (2.60) K ij 7;abcd = 4 [a K ij bcd]7 : (2.61) The latter is simply K ij abcd with all component set equal to 0, except for those components that have a direction along the ber, i.e. along the `7' direction. The covariant derivatives of the various spinor bilinears are needed. They are obtained using the Leibnitz rule with the spinor covariant derivative. 2.5 Pope-Warner background solution The Pope-Warner solution is a compactication solution of the d = 11 supergravity bosonic eld equations with SU(4)-invariance [20]. The metric part of it is a product of the AdS 4 metric and of the stretched S 7 metric: ds 2 =l 2 ds 2 (AdS 4 ) +ds 2 (S 7 ); (2.62) 32 where ds 2 (AdS 4 ) is the metric for unit radius AdS 4 and ds 2 (S 7 ) is the metric (5) with the size of the CP 3 set so that R(CP 3 ) ij = 8 ij . The background 4-form uxes are taken to be F = 2m (2.63) F abcd = s abcd abcd ; (2.64) where Greek indices are used to label coordinates ofAdS 4 and Latin indices are used to label coordinates of S 7 . All other components of the 4-form ux are 0, i.e. there is no mixing of the AdS 4 and S 7 components. The Einstein equation is R AB = 1 3 F ACDE F CDE B 1 36 g AB F CDEF F CDEF : (2.65) The Riemann tensor for the stretched S 7 is given in [3]. It is R ijkl = ik jl il jk + (1c 2 )(J ik J jl J il J jk + 2J ij J kl ) (2.66) R 7i7j = c 2 J i k J jk =c 2 ij : (2.67) Contracting the Riemann tensor gives the Ricci tensor, which can then be input into the Einstein equation. It is R ij = (8 2c 2 ) ij (2.68) R 77 = 6c 2 : (2.69) 33 Putting the above metric and uxes into the Einstein equation gives the equations 3 l 2 = 8 3 (2m 2 + 4s 2 ) (2.70) 8 2c 2 = 4 3 (2m 2 + 4s 2 ) (2.71) 6c 2 = 8 3 (m 2 + 8s 2 ); (2.72) where l is the AdS radius. The Maxwell equation is r a F abcd = 1 6 m bcdefgh F efgh : (2.73) Putting the ux (32) into the Maxwell equation gives, using the spinor covariant derivative, sB bcd bcd = 2ms bcd bcd ; (2.74) where B =( 7 + 3) + 3( + 2): (2.75) So the Maxwell equation gives B = 2m: (2.76) Solving the Einstein and Maxwell equations gives: c = p 2 (2.77) m = 1 p 2 (2.78) s = 1 p 2 (2.79) l = r 3 8 : (2.80) 34 2.6 Perturbation ansatz We want to perturb the Pope-Warner background solution obtained in the previous section. We use the following perturbation ansatz: h ab = 2G ij (x)W ij ab (2.81) a abc = ij (x)X ij abc +(x)Y abc +! ij (x)Z ij abc ; (2.82) where W ij ab = m (i n j) m (i n j) (2.83) X ij abc = 3 [m (i np] j) + [m (i np] j) (2.84) Y abc = 3 [m np] (2.85) Z ij abc = 6 [m (i n j) p] : (2.86) Taking the exterior derivative of a gives f abcd = (da) abcd = ij (x)(dX ij ) abcd +(x)(dY ) abcd +! ij (x)(dZ ij ) abcd (2.87) f bcd = r a bcd = r ij (x)X ij abc +r (x)Y abc +r ! ij (x)Z ij abc ; (2.88) where dX ij = 4( 2)K ij + 2 [( 1) + 2]K ij 7 + 2(2 +) ij K a (2.89) dY = 4( +)K a (2.90) dZ ij = 2( +)K ij 2(3 +)K ij 7 ( +) ij K a : (2.91) 35 We want to plug this perturbation ansatz, together with the Pope-Warner background solution, into the linearized eld equations given in section 2.3. To begin with, the linearized eld equations are unwieldy and contain many terms. However, due to the following facts, which are straightforward to verify, they simplify considerably. 1. The metric uctuation ansatz h ab is traceless, i.e. h a a = 0. 2. F abcd f abcd = 0. 3. F abc n a abc = 0. 4. efg efg a efg = 0. 5.r m f mnp = 0. This is because the divergence operator is?d?, and d 2 = 0. 6.r n h n a = 0. h ab is transverse. 7. F a cnp F b m np h mc = 0. 8. F cdef F m def h cm = 0. This follows from 7 by contracting the indices a and b. 2.7 AdS equations of motion Using the perturbation ansatz and the facts given above, the linearizedd = 11 eld equations reduce to 1 2 ^ h mn = 2 3 F abc (m f n)abc 1 36 (384s 2 96m 2 )h mn (2.92) and r f npq +r m f mnpq + 4r m (F a[mnp h q] a ) = 1 6 m abcdnpq f abcd : (2.93) These equations come from the Einstein and Maxwell eld equations, respectively. The linearized Bianchi identity is trivially satised by the fact that f = da in the perturbation ansatz, so it yields no new information. 36 We want to further simplify the linearized eld equations so that all dependence on the internal 7-dimensional coordinates disappers and we are left with equations that are only on the AdS space. Einstein equation The Einstein equation is 1 2 ^ h mn = 2 3 F abc (m f n)abc 1 36 (384s 2 96m 2 )h mn : (2.94) ^ is the Lichnerowitz operator and is dened as [44] ^ h mn =2h mn 2R mpnq h pq + 2R p (m h n)p ; (2.95) where 2h mn =2 4 h mn +2 7 h mn ; (2.96) and 2 7 =r a r a : (2.97) The various terms in ^ h mn are found to be 2 7 h mn = 8[ 2 (2 2 + 5) + 2( + 4) + 2 + 8 2 ]h mn (2.98) R mpnq h pq = (3c 2 4)h mn (2.99) R p (m h n)p = 2(c 2 4)h mn ; (2.100) giving ^ h mn =2 4 h mn +lh mn ; (2.101) 37 where l = 8[ 2 (2 2 + 5) + 2( + 4) + 2 + 8 2 ] 10c 2 + 24: (2.102) The other term in the Einstein equation is found to be F abc (m f n)abc =48s (( + 3) 2) ij 2! ij W ij mn : (2.103) Now the Einstein equation is 2 4 h mn = G h mn + 64s (( + 3) 2) ij 2! ij W ij mn ; (2.104) giving 2 4 G ij = G G ij + 32s (( + 3) 2) ij 2! ij ; (2.105) where G =l + 1 18 (384s 2 96m 2 ): (2.106) So the linearized Einstein equation is 2 4 G ij =M 11 G ij +M 12 ij +M 14 ! ij ; (2.107) where the coecientsM 1j are constants that will be part of a 44 matrix M called the `mass matrix'. The rest of the elements of M will come from the linearized Maxwell equation. Maxwell equation The Maxwell equation is r f npq +r m f mnpq + 4r m (F a[mnp h q] a ) = 1 6 m abcdnpq f abcd : (2.108) Expanded, the rst term on the left hand side of the Maxwell equation is r f npq = (2 4 ij )X ij npq + (2 4 )Y npq + (2 4 ! ij )Z ij npq ; (2.109) 38 and the second term on the left hand side is r m f m npq = (?d?f) npq (2.110) = g ij 1 X ij npq +g 2 Y npq +g ij 3 Z ij npq ; (2.111) where g ij 1 = 4 2 (( + 2) + 13) 2( + 5) 4( + 1) + ( + 2) 2 ij +8(( 3) 3 2)! ij (2.112) g 2 = 16 2 () + 2 + 7 + 2 + 2 2 Tr 64( +) 2 +16 2 + 2 + 2 Tr! (2.113) g ij 3 = 8(( 1) + 2)(( 3) + 3 + 2) ij 16 2 ( 10) +(5 + 2) + 2 ! ij : (2.114) In g 2 , Tr = P 6 i=1 ii and Tr! = P 6 i=1 ! ii . One nds that in the third term on the left hand side of the Maxwell equation 4F a[mnp h q] a = 4sG ij K ij 7;mnpq : (2.115) So, 4r m (F a[mnp h q] a ) = 4sG ij (?d?K ij 7 ) npq (2.116) = ( ij 1 X ij npq + 2 Y npq + ij 3 Z ij npq ); (2.117) (2.118) 39 where ij 1 = 8s(( 1) 2)G ij ; (2.119) 2 = 32sTr G; (2.120) ij 3 = 16s(( 7) 2)G ij : (2.121) The term on the right hand side is abcdnpq f abcd = 4!(?f) npq (2.122) = 4! h ij 1 X ij npq +h 2 Y npq +h ij 3 Z ij npq ; (2.123) where h ij 1 = 2 (( + 3) 2) ij 4! ij (2.124) h 2 = 8Tr + 8( +) 4( +)Tr! (2.125) h ij 3 = 4 (( 1) + 2) ij 4(3 +)! ij : (2.126) Plugging the expansions of each of the terms into the linearized Maxwell equation yields three equations for the scalar elds, one equation for each of X ij , Y , and Z ij . They are 2 ij = g ij 1 1 G ij 4mh ij 1 (2.127) 2 = g 2 2 Tr G 4mh 2 (2.128) 2! ij = g ij 3 3 G ij 4mh ij 3 : (2.129) 40 Expanding these equations further in terms of the ansatz scalars G ij , ij , , ! ij gives 2 ij = M 21 G ij +M 22 ij +M 23 ij +M 24 ! ij (2.130) 2 = M 31 Tr G + M 32 Tr + M 33 + M 34 Tr! (2.131) 2! = M 41 G ij +M 42 ij +M 43 ij +M 44 ! ij : (2.132) The M ij in the above equations, combined with the M 1j in the linearized Einstein equation give the 4 4 mass matrix M. 2.8 Eigenmodes Plugging our background and uctuation ansatze into the linearized Einstein and Maxwell equations yielded four equations for the four AdS scalars that were part of the uctuation ansatz. These four equations can be conveniently written as (1 4 2 M) 0 B B B B B B B B B B @ G ij ij ! ij 1 C C C C C C C C C C A = 0: (2.133) The matrix M is called the `mass matrix'. If S is the matrix that diagonalizes M then we have (1 4 2 D) 0 B B B B B B B B B B @ ij 1 ij 2 3 ij 4 1 C C C C C C C C C C A = 0; (2.134) 41 where D = SMS 1 ; (2.135) and 0 B B B B B B B B B B @ ij 1 ij 2 3 ij 4 1 C C C C C C C C C C A = S 0 B B B B B B B B B B @ G ij ij ! ij 1 C C C C C C C C C C A : (2.136) The matrix D is diagonal and its entries are the eigenvalues of M, which are the squared masses of the scalar elds ij 1 , ij 2 , 3 , and ij 4 . Explicitly, the mass matrix is M = 0 B B B B B B B B B B @ 24 32 p 2 0 16 p 2 0 24 16 p 2 0 8 p 2 8 p 2 ij 16(1 + p 2) ij 16 8 ij 24 p 2 16(2 + p 2) 0 8(5 + 2 p 2) 1 C C C C C C C C C C A : (2.137) By diagonalizing M one can nd the ij a , 3 , and the squared masses. The ij a and 3 are found to be ij 1 = 1 10 (2 + 3 p 2)G ij + 1 5 (4 + p 2) ij + 1 5 (3 + p 2)! ij (2.138) ij 2 = 1 2 G ij + ij (2.139) 3 = 2 3 Tr + 1 3 Tr! (2.140) ij 4 = 3 10 (1 + p 2)G ij 1 5 (1 + p 2) ij + 1 5 (2 p 2)! ij : (2.141) The squared mass values are 72, 24, 16, and8, respectively. Multiplying the squared mass values by the AdS radius squared gives the dimensionless squared mass values. The AdS 42 radius squared was found to bel 2 = 3 8 in section 4, so the dimensionless squared mass values are 27, 9, 6, and3. The Breitenlohner-Freedman bound is [32] m 2 l 2 = 9 4 , so the eigenmode ij 4 , which has dimensionless squared mass value3, is unstable. Peeling o the dierent mass modes Inverting the above equations gives G ij = 1 7 (2 3 p 2) ij 1 + ij 2 + (1 + p 2) ij 4 (2.142) ij = 1 14 (3 p 2 2) ij 1 + 1 2 ij 2 1 2 (1 + p 2) ij 4 (2.143) = 1 7 (3 p 2)Tr 1 + 3 + 1 3 (2 + p 2)Tr 4 (2.144) ! ij = ij 1 + ij 2 + ij 4 : (2.145) These expressions can be plugged into the perturbation ansatz of section 5, and the dierent mass modes can be `peeled o'. The metric parts of the `peeled-o' mass modes are h a;mn = 2c a;1 ij a (x)W ij mn ; a = 1; 2; 4 (2.146) h 3;mn = 0; (2.147) and the 3-form potential parts of the `peeled-o' mass modes are a a;mnp = ij a (x)A ij a;mnp ; a = 1; 2; 4 (2.148) a 3;mnp = 3 (x)Y mnp ; (2.149) where A ij a;mnp =c a;2 X ij mnp +c a;3 ij Y mnp + 2c a;4 Z ij mnp : (2.150) 43 The c a;i are the coecients in equations (141)-(144), e.g. c 4;1 = 1 + p 2 (2.151) c 4;2 = 1 2 (1 + p 2) (2.152) c 4;3 = 1 3 (2 + p 2) (2.153) c 4;4 = 1: (2.154) Degeneracies of the masses The spinors i ,i = 1;:::; 6, are used to construct the tensors W ij ,X ij , andZ ij that are dened in section 5. W ij is symmetric in i and j, so there are at most 21 of them that are linearly independent. In fact, Tr W = P 6 i=1 W ii = 0, so there are actually at most 20 linearly independent W ij . Likewise, theA ij a are symmetric in i and j, and TrA a = 0, so there are at most 20 linearly independentA ij 1 ,A ij 2 , andA ij 4 . The i realize the 6 of SU(4), soW ij andA ij a each realize the symmetric product 6 s 6. This symmetric product breaks as [80]: 6 s 6! 20 0 + 1; (2.155) where 1 is the trace part of 6 s 6 and 20 0 is the symmetric traceless part. Since Tr W = TrA a = 0, it follows that W ij andA ij a each realize the 20 0 . So there are exactly 20 linearly independent W ij andA ij a . Therefore, the squared mass values 27, 9, and3 each have degeneracy 20, and the squared mass value 6 has degeneracy 1. 44 2.9 Comparison to known results The instability of the Pope-Warner solution on S 7 was rst demonstrated in [32]. There it was found that there are unstable modes with dimensionless squared mass3 that realize the 20 0 of SU(4). These modes are recovered here. The result of [32] was extended to tri-Sasakian manifolds in [34]. There a consistent truncation of d = 11 supergravity was carried out on a 7-dimensional tri-Sasakian manifold to give a d = 4 supergravity theory. In carrying out the truncation, a scalar potential for the d = 4 theory was extracted. This potential has the Pope-Warner solution as a xed point. By computing the second derivatives of this potential at the Pope-Warner xed point, the squared masses of the scalars at the Pope-Warner xed point can be found. These masses are not given in [34], but we found them to be (27; 18 2 ; 9 2 ; 6; 2:25 2 ;3; 0 7 ), where the superscripts denote the degeneracies. Therefore, the mass values found here are a subset of those found in [34]. In the special case where i and j are set equal to a xed value, the perturbation ansatz given in section 5 is contained in the consistent truncation ansatz of [34]. To go between the ansatz of [34] and the one here it suces to express the canonical 1-forms and 2-forms of the tri-Sasakian structure used in [34] in terms of the spinors used here. In terms of the spinors used here, the 1-forms of [34] are # 1 a = a (2.156) # 2 = a (2.157) # 3 = a ; (2.158) 45 and the 2-forms of [34] are J 1 ab = ab 2 [a b] (2.159) J 2 ab = ab 2 [a b] (2.160) J 3 ab = ab + 2 [a b] ; (2.161) where is an arbitrary linear combination of the i with real coecients and unit norm. In [34] a single unstable mode with squared mass3 was found. There it was supposed that on S 7 this single mode was one of the 20 unstable modes found in [32]. Here we have explicitly shown that this is indeed the case. 3 The solutions In this chapter we obtain the FR, skew-whied FR, and PW solutions of eleven-dimensional supergravity. Even though we obtained the PW solution in chapter two, we obtain it again here because the conventions used in chapter 2 are dierent from those used in the rest of this dissertation. The FR and PW solutions of eleven-dimensional supergravity on a SE manifold,M 7 , can be derived from the following general Ansatz, ds 2 11 =ds 2 AdS 4 (L) +a 2 ds 2 7 ; (3.1) F (4) =f 0 vol AdS 4 (L) +f i (4) : (3.2) A regular SE manifold,M 7 , is the total space of a U(1) bration over a K ahler-Einstein (KE) 46 base, B 6 , and the internal metric in (3.1) can be written locally as ds 2 7 =ds 2 B 6 +c 2 d +A 2 ; (3.3) were A is the K ahler potential on B 6 , is the angle along the ber and c is the squashing parameter. The potential for the internal ux in (3.2) is given by the real part of a canonical complex three-form, , on M 7 , such that (4) =d( + ): (3.4) The constants, a, c, f 0 and f i in (3.1)-(3.2) are xed by the equations of motion in terms of the AdS 4 radius, L, which sets the overall scale of the solution: Supersymmetric and skew-whied FR solutions a = 2L; c = 1; f 0 = 3 2L ; f i = 0; (3.5) where =1 and +1, respectively. PW solution a = 2 r 2 3 L; c = p 2; f 0 = p 3 2L ; f i = 4 3 r 2 3 L 3 : (3.6) In (3.1), we have factored out the overall scale, a 2 , of the internal metric so the KE metric, g B 6 , and the SE metric, g M 7 , obtained by setting c = 1 in (3.3), are canonically normalized with Ric B 6 = 8g B 6 ; Ric M 7 = 6g M 7 : (3.7) In the following, we will refer to the SE metric on M 7 as the \round" metric. 47 The one form# =d +A, called the contact form, is globally dened onM 7 , and is dual to the Reeb vector eld, =@ , which is nowhere vanishing and has length one. The other two globally dened forms of the SE geometry are the real two form,J, and a complex three form, , with its complex conjugate, . They satisfy d# = 2J; d = 4i#^ : (3.8) Note that the ansatz (3.1)-(3.4) is in fact written in terms of globally dened objects of the SE geometry. It is convenient to choose special frames, e a , a = 1;:::; 7, on M 7 , that are orthonormal with respect to the round metric and such that J = i 2 e z 1 ^ e z 1 + e z 2 ^ e z 2 + e z 3 ^ e z 3 ; =e 4i e z 1 ^ e z 2 ^ e z 3 ; # = e 7 ; (3.9) where e z 1 = e 1 +i e 2 ; e z 2 = e 3 +i e 4 ; e z 3 = e 5 +i e 6 ; (3.10) is a local holomorphic frame on the KE base. This shows that J is the pull-back of the Kahler form, while is, up to a phase along the ber, the pull-back of the holomorphic (3; 0)-form on B 6 . We will denote the components of the round metric by g ab = ab and of the squashed metric (3.3) by g ab . Then the components of the eleven-dimensional metric (3.1) along the internal manifold are g ab =a 2 g ab . One can also express #, J and as bilinears in Killing spinors, , D a = i 2 a ; = ; ; = 1; 2; (3.11) 48 that are globally dened onM 7 and whose existence is equivalent toM 7 being a SE manifold. In terms of 's we have (see, e.g., [61]) # a =i 1 a 2 ; J ab = 1 ab 2 ; abc = 1 2 ( 1 +i 2 ) abc ( 1 +i 2 ): (3.12) Using this realization together with Fierz identities, it is straightforward to prove a number of useful identities summarized in appendix B. To verify the solutions (3.5) and (3.6), we note that the covariant derivatives for the squashed and round metric are related by D a V b = D a V b 2 (c 2 1)# (a J b) c V c ; (3.13) where we have adopted a convention to raise and lower indices with the round metric, g ab . For the Ricci tensors, using identities in appendix B, we have R ab = R ab + 2(1c 2 ) g ab + 2(3c 4 +c 2 4)# a # b : (3.14) These are also the components of the Ricci tensor,R ab , along the internal manifold. The Ricci tensor for AdS 4 of radius, L, is Ric AdS 4 = 3 L 2 g AdS 4 : (3.15) so that the eleven-dimensional Ricci scalar is R = 12 L 2 + 6 a 2 (8c 2 ): (3.16) The energy momentum tensor in (1.1) has only diagonal contributions from the ux along AdS 4 andM 7 that are straightforward to evaluate. Then the Einstein equations (1.1) reduce 49 to three algebraic equations: a 2 = 4 3 (c 2 4)L 2 ; f 2 0 = 3(7c 2 16) 4L 2 (c 2 4) ; f 2 i = 2 27 c 2 (c 2 4) 3 (c 2 1)L 6 ; (3.17) for the size of the internal part of the metric and the parameters of the ux. We now turn to the Maxwell equations (1.2). Let us denote by the Hodge dual on M 7 with respect to the round metric with the volume form vol M 7 = 1 6 J^J^J^# = 3 8 i ^ ^#; (3.18) The volume form for the squashed metric is then c vol M 7 , while ca 7 vol AdS 4 ^ vol M 7 is the volume form in eleven-dimensions. It follows from (3.8) and (3.18) that d = 4 ; = 1 4 d : (3.19) Then for the ux,F (4) , in (3.2) and (3.4), we have ?F (4) =f 0 ? vol AdS 4 4f i ac vol AdS 4 ^ ( + ); (3.20) so that d?F (4) = 4f i ac vol AdS 4 ^ (4) : (3.21) The second term in (1.2) yields F (4) ^F (4) = 2f 0 f i vol AdS 4 ^ (4) ; (3.22) which shows that the Maxwell equations reduce to a single equation f i f 0 2 ac = 0: (3.23) 50 Assuming that both a and c are positive, one veries that (3.5) and (3.6) exhaust all solutions to (3.17) and (3.23). Note that the only dierence between the supersymmetric and skew-whied FR solutions is the sign,, of the ux alongAdS 4 . Equivalently, one could reverse the orientation of the internal manifold, which changes the sign of the Hodge dual in (1.2). Here, we will keep the orientation of M 7 xed as in (3.18). 4 The linearized analysis We will not attempt here a complete analysis of the Kaluza-Klein spectrum around the PW solution, but instead will identify a small set of harmonics for the low lying scalar modes on which the scalar mass operator in the linearized expansions around both the FR and PW backgrounds can be diagonalized. In doing that, we will be guided both by the explicit structure of the linearized equations of motion and by the properties of unstable modes on S 7 that were identied in [32]. The scalar modes we want to consider correspond to uctuations of the internal metric and the internal three-form potential, g ab ='(x)h ab ; A (3) ='(x) (3) ; (4.1) where'(x) is a scalar eld onAdS 4 , whileh ab and (3) are, respectively, a symmetric tensor and a three form harmonic on M 7 . 51 4.1 Linearized Einstein equations We begin with the metric harmonic and the linearization of the Einstein equations (1.1). Following a crucial obervation in [32] for the unstable modes on S 7 , we will assume that metric harmonic, h ab , corresponds to a deformation of the internal metric along the (2; 0) and (0; 2) components on the KE base. Specically, h ab is horizontal, that is # a h ab = 0, and its only nonvanishing components in the basis (3.10) are h z i z j and h z i z j . It is then automaticaly traceless. Finally, we will assume that it is transverse with respect to the round metric, D a h ab = 0. It follows then from (3.13) that it is also transverse with respect to the internal metric with any value of the squashing parameter, c. With those assumptions, the metric uctuation (4.1) is both transverse and traceless in eleven dimensions, so that the expansion of the Ricci tensor in (1.1) yields only one term with the Lichnerowicz operator (see, e.g., [62]), and there are no terms from the Ricci scalar to linear order. The eleven-dimensional Lichnerowicz operator becomes then a sum, 3 2 AdS 4 1 a 2 c L ; (4.2) where c L is the Lichnerowicz operator onM 7 with respect to the squashed metric. Then on the metric harmonics, h ab , as above, c L h ab = L + 4(1c 2 ) + 1 1 c 2 L 2 h ab ; (4.3) where L is the Lichnerowicz operator for the round metric, andL is the Lie derivative along the Reeb vector. 3 We use2 AdS4 =g r r , but L =g ab D a D b +::: 52 Let's denote the combination on the left hand side in (1.1) byE MN . After collecting all the terms in the expansion and using (3.16), we obtain E ab = 1 2 2 AdS 4 2 '(x)h ab ; (4.4) where 2 = 1 a 2 L + 84 a 2 24 L 2 1 a 2 1 1 c 2 (16c 2 L 2 ): (4.5) Let us now turn to the expansion of the energy momentum tensor,T MN , on the right hand side in (1.1). For the metric variation as above, the only terms that contribute to the linear expansion of the energy momentum tensor come from the ux, T ab = f i 3a 6 g cf g dg g dh (d acde bfgh +d bcde afgh )'(x): (4.6) Assuming that metric harmonic is an eigentensor of the Lichnerowicz operator in (4.5), we see that in order to diagonalize the linearized Einstein equations we must nd a ux harmonic such that the symmetric tensor in (4.6) is proportional to h ab . 4.2 Linearized Maxwell equations The expansion of the Maxwell equations is d?F (4) + 2F (4) ^F (4) +d(?)F (4) = 0; (4.7) where F (4) =d'^ (3) +'d (3) ; (4.8) and (?) is the variation of the Hodge dual due to uctuation of the metric. Dene a four form (gF) MNPQ = g M 0 M 00 g MM 0F M 00 NPQ +::: +g Q 0 Q 00 g QQ 0F MNPQ 00: (4.9) 53 Then for a traceless uctuation of the metric, (?)F (4) =? (gF (4) ): (4.10) Specializing to the background ux (3.2) and the uctuations (4.1), the linearization (4.7) splits into terms that are one, three, and four forms along AdS 4 , respectively. They yield the following equations d c = 0; ^ (4) = 0; (4.11) and (2 AdS 4 ') c (3) +' 2f 0 a d (3) 1 a 2 d c d (3) + f i a 4 d c (h (4) ) = 0; (4.12) where c denotes the dual with respect to the squashed metric and h (4) is dened as in (4.9) using the round metric. The various factors of the internal radius, a, in (4.12) are consistent with the overall 1=L 2 dependence of the mass terms on the AdS 4 radius. 4.3 The master harmonic Our task now is to identify the smallest set of harmonics on which we can diagonalize the Maxwell equation (4.12). The rst step will be to streamline the evaluation of the Hodge duals. Any k-form, k , on M 7 can be uniquely decomposed into the sum, k =! k +#^! k1 ; (4.13) where ! k and ! k1 are horizontal forms, that is, { ! k ={ ! k1 = 0. Then c k =c! k + 1 c (#^! k1 ); (4.14) 54 where is the Hodge dual with respect to the round metric. We may simplify this further by introducing another Hodge dual,, in the space perpendicular to the ber, or, equivalently on the KE base, B 6 . Then for a horizontal form, !, using vol M 7 = vol B 6 ^#, we have 4 ! =#^!; (!^#) =!; (4.15) and hence c k =c#^! k + 1 c (1) k1 ! k1 : (4.16) To further restrict the Ansatz for the ux harmonic, let us look at the last term in (4.12), which is already constrained by the conditions we have imposed in section 4.1 on the metric harmonic, h ab . Since h ab has nonvanishing components only along the KE base, we have h (4) = 4i#^h ( ); (4.17) where all contractions between the metric harmonic and the background ux form are with the round metric. We can now evaluate the forms (4.17) on S 7 using the metric harmonics given in [32]. It turns out that h ( ) is closed (!) and it is both horizontal and invariant along the ber. This means that it is a closed basic three-form with nonvanishing (2; 1) and (1; 2) components. OnS 7 , it is then a pull-back of the corresponding closed form on CP 3 and thus is exact. Indeed, we nd that h ( ) =64d!; (4.18) where! is a basic, primitive, transverse (1,1)-form, and an eigenform of the Laplacian, with 4 Note that on a k-form, 2 = 2 c = 1, while 2 = (1) k . 55 the eigenvalue 24. In the following we will show that a similar construction can be carried out on a general SE manifold, M 7 . We start with a primitive, (1; 1)-form, ! on the KE which is a transverse eigenform of the Hodge-de Rham Laplacian with the eigenvalue ! . Its pull-back to M 7 is then a basic form, satisfying { ! = 0; L ! = 0; (4.19) which we also denote by !. We will now discuss the conditions on ! and derive some identities that are used later. (i) The condition that ! is a primitive (1; 1)-form means that J ab ! ab = 0; J a c J b d ! cd =! cd ; (4.20) where the rst condition can be equivalently written as J^! = 0 or J^J^! = 0: (4.21) It follows from (4.20) that on B 6 and M 7 , respectively, 5 J^! =! and (J^!) =#^!: (4.22) (ii) Transversality on the KE base d! = 0; (4.23) 5 The operatorJ^ on a six-dimensional Kahler manifold maps two-forms into two-forms. It has eigen- values1, 1 and 2 with degeneracies 8, 6 and 1, respectively, corresponding to the primitive (1; 1)-forms, (2; 0) + (0; 2)-forms and (1; 1)-forms proportional to J. 56 implies transversality on the SE manifold, d! =d(#^!) = 2J^!#^d! = 0; (4.24) where the last step follows from (4.22) and (4.23). By taking the exterior derivative of (4.22), we get J^d! = 0; (4.25) and a somewhat less obvious J^d! = 0: (4.26) Since the last identity is on the KE base, upon taking a dual we obtain a 1-form with components proportional to 2J r ! +J r ! : (4.27) On a K ahler manifold, J is covariantly constant and the rst term can be written as J r ! =r (J ! ) =r (J ! ) = 0; (4.28) where we used that! is a transverse (1; 1)-form. The vanishing of the second term in (4.27) is shown similarly. (iii) Finally, ! is an eigenfunction of the Hodge-de Rham Laplacian operator, (1;1) on B 6 , which for a transverse form is simply, (1;1) !dd! = ! !: (4.29) 57 Then the Laplacian on M 7 , after using (4.15) and (4.26) is !dd! =d(#^d!) =(2J^d!#^dd!) = ! !: (4.30) Hence ! is also an eigenfunction of the Laplacian on M 7 with the same eigenvalue, ! . 4.4 The metric harmonic We now take the following Ansatz for the metric harmonic in terms of a pure imaginary (1; 1)-form, !, h ab = (d!) acd ( b cd b cd ) + (a$b): (4.31) This tensor is manifestly horizontal and has only (2; 0) and (0; 2) components as we have required in section 4.1. It also satises (4.18), as one can verify using identities in section 4.3 and appendix A. We will now show that h ab is a transverse eigentensor of the Lichnerowicz operator on M 7 , L h ab = h h ab ; h = ! + 4; (4.32) with the eigenvalue, h , xed by ! . Before we present a somewhat lengthy proof, let us note that the same relation between the eigenvalues of the Hodge-de Rham Laplacian and the Lichnerowicz operator has been derived in [56] through a general analysis of the fermion/boson mass relations on manifolds with Killing spinors, see Appendix G. In particular, it was shown that if a two-form and 58 a symmetric tensor harmonics arise from the same spin-3/2 harmonic by a supersymmetry transformation generated by Killing spinors, the resulting shift of the eigenvalues is precisely the one given in (4.32). While we have not derived the intermediate spin-3/2 harmonic in general, some explicit checks on S 7 (or, more generally on tri-Saskian manifolds), where all the forms in (4.31) can be realized in terms of Killing spinors, 6 have convinced us that our construction here and in the following sections yields a subset of harmonics in a single N = 2 supermultiplet as in [56]. We will discuss it further in chapter 5, where we identify this supermultiplet as the long Z-vector multiplet [57]. We also note that a similar construction for tensor harmonics on a ve-dimensional SE manifolds has been recently carried out in [63] and it follows a much earlier construction for four-dimensional Kahler manifolds in [64]. 4.4.1 Proof of transversality There are four types of terms in the transversality condition, 7 D a h ab = 0. First, we have D a (d!) acd b cd = ! ! cd b cd = 0; (4.33) since ! is a (1; 1)-form. Secondly, (d!) acd D a b cd = 4i (d!) acd # [a bcd] = 0; (4.34) since d! is horizontal, and hence d! abc # a = 0. Similarly, the full contraction between d!, which is a sum of a (2; 1) and a (1; 2) form, and the (3; 0) form, , must vanish. The third 6 See, section 5.1. 7 Throughout this section, D a is the covariant derivative with respect to the round metric. 59 type of terms are D a (d!) bcd a cd =D a (d!) bcd acd : (4.35) Since D [a (d!) bcd] = 0, we have 3D a (d!) bcd acd =D b (d!) acd acd =D b (d!) acd acd (d!) acd D b acd =4i (d!) acd # [b acd] = 0; (4.36) as d! is either contracted with # or fully contracted with . Finally, the last type of terms are (d!) bcd D a a cd = 0; (4.37) since is itself transverse, see, e.g., (C.12). Transversality of the terms with is veried similarly. 4.4.2 Proof of (4.32) The Lichnerowicz operator, 8 L , on k-forms coincides with the Hodge-de Rham Laplacian, =d +d; = (1) k d : (4.38) We have assumed that ! = ! !. Using (3.19) we also nd = 16 : (4.39) For an arbitrary tensor, the Lichnerowicz operator is dened by L T a 1 :::a k =2T a 1 :::a k + (R a a 1 T aa 2 :::a k +:::) 2(R a a 1 b a 2 T aba 3 :::a k +::: ); (4.40) 8 For a list of properties of the Lichnerowicz operator, see, e.g., [62]. 60 where there are k-terms in the rst bracket and 1 2 k(k 1) in the second. An important property, which we are going to exploit in the following, is that L commutes with the contraction. Consider the tensor t acdbef = (d!) acd bef ; (4.41) from which the (2; 0)-part ofh ab is obtained by contracting over the pairsce anddf and then symmetrizing over ab. It follows from the denition (4.40) that ( L t) acdbef = ( L d!) acd bef + (d!) acd ( L ) bef 2D g (d!) acd D g bef +R-terms; (4.42) where the R-terms involve split contractions with both d! and , R-terms =2 (d!) gcd hef R g a h b + 8-terms (4.43) We will now show that all terms in (4.42) give contributions to L h ab that are proportional to h ab and evaluate the proportionality constants. From the rst two terms we get ( ! + 16)h ab . Next, we consider the R-terms, which can be traded for covariant derivatives acting on using [D a ;D b ] cde = fde R f cab ::: : (4.44) This gives R-terms = 2 (d!) gcd [D g ;D a ] + (d!) agd [D g ;D c ] + (d!) acg [D g ;D d ] bef : (4.45) The covariant derivatives acting on can be evaluated using (C.12). This yields terms that are products of the form d! J or d! # # : (4.46) 61 Performing the contractions as in the denition of h ab , see (4.31), we are left with two free indices with all other ones contracted. Because of the symmetrization, the free indices in the terms of the rst type in (4.46) must be on two dierent tensors. In particular, this implies that J is always contracted with either or d! or both. All terms in which J is contracted with are simplied using (C.7) and yield terms proportional to h ab . This leaves terms in which J is contracted with d!. By inspection, in all those terms d! is doubly contracted with , which means that the contraction with J is once more a multiplication by i. The second type terms in (4.46) all vanish except when the two#'s are contracted. Collecting all the terms we nd that the total contribution from the R-terms to L h ab is10h ab . Finally, we consider the third term in (4.42). Since d! is closed, we rewrite this term as 2D g (d!) acd D g b cd +(a$b) =2D a (d!) gcd D g b cd 4D c (d!) agd D g b cd +(a$b): (4.47) Let's start with the rst term in (4.47). Since (d!) gcd D g b cd = 4i (d!) g cd # [g bcd] = 0; (4.48) we have D a (d!) gcd D g b cd =(d!) gcd D a D g b cd : (4.49) Expanding the covariant derivatives using (C.12), we nd that all terms involving # vanish. The remaining terms haved! contracted withJ and twice contracted with , which reduces the contraction with J to the multiplication by i. Then the net contribution from this term to L h ab is6h ab . This leaves us with the second term in (4.47), which we once more rewrite using the Leib- nitz rule. However, now the total derivative term does not vanish, but yields the derivative 62 D c of the following terms, (d!) agd D g bc d =i(d!) agd (2# g bc d +# b c gd # c b gd ): (4.50) The rst term on the rhs vanishes as d! is horizontal. The second term can be rewritten as i(d!) agd # b c gd =ih ac # b i(d!) cgd gd a # b : (4.51) Acting with D c and using 3D a D [a ! bc] =D a (d!) abc = ! ! bc ; (4.52) and the transversality of h ab , we get ih ac J c b i(d!) cgd D c a gd # b i(d!) cgd a gd J c b =h ab + 0 (d!) bgd a gd ; (4.53) which gives4h ab contribution in L h ab . The last term in (4.50) is iD c (# c (d!) agd b gd ) =i# c D c (d!) agd b gd i# c (d!) agd D c b gd : (4.54) Using d 2 ! = 0, the rst term on the right hand side above can be simplied using i# c D c (d!) agd =i# c D a (d!) c gd i# c D g (d!) a c d i# c D d (d!) ag c =iJ ac (d!) c gd +iJ gc (d!) a c d +iJ dc (d!) ag c = (d!) agd ; (4.55) where the second line follows using the Leibnitz rule, horizontality of d! and (C.12). The second in term (4.54), using (C.12), is i# c (d!) agd D c b gd = (d!) agd b gd : (4.56) 63 Hence the last term in (4.50) is2(d!) agd b gd , and by (4.47) it contributes8h ab to L h ab . Finally, using the Leibnitz rule, we are left with 4(d!) agd D c D g bcd = 16h ab : (4.57) Hence all terms in (4.42) are indeed proportional to h ab , with the net result h = ! + 16 10 6 4 8 + 16 = ! + 4: (4.58) This concludes the proof of (4.32). 4.5 The ux harmonics We take as internal ux harmonic the linear combination (3) =t 1 #^! +t 2 d (#^!); (4.59) where t 1 and t 2 are arbitrary pure imaginary parameters. 9 The harmonics that arise in the expansion of the Maxwell equation (4.12) are: d , c , and d c d. We will now show that for given by (4.59), each of those terms is a linear combination of the following two linearly independent harmonics: 1 =(#^!) and 2 =d(#^!): (4.60) Specically, we nd d = ! t 2 1 + (t 1 2t 2 ) 2 ; (4.61) c = 1 c t 1 + 2t 2 (c 2 1) 1 +ct 2 2 ; (4.62) d c d = ! c (t 1 2t 2 ) 1 +c ( ! t 2 2(t 1 2t 2 )) 2 : (4.63) 9 In the following, we denote this harmonic simply by . 64 The rst identity follows from dd(#^!) =d (2J^!)d (#^d!) =2d(#^!) +dd! =2d(#^!) + ! ! =2d(#^!) + ! (#^!): (4.64) where we used (3.8), (4.15), (4.22) and (4.29). The second one is an immediate consequence of (4.14), (3.8) and (4.22). For the third one, we have d c d =t 2 ! d c (#^!) + (t 1 2t 2 )d c d(#^!) =c ! t 2 d(#^!) + (t 1 2t 2 ) 1 c dd(#^!) + 2(c 2 1)d(#^!) = ! c (t 1 2t 2 )(#^!)c [2(t 1 2t 2 ) ! t 2 ] d(#^!): (4.65) In evaluating the contribution from the metric uctuation to the linearized Maxwell equations (4.12) we also need the indentity h (4) =128i#^d!: (4.66) To prove it, we note that by the second identity in (3.8), (4) = 4i#^ ( ): (4.67) Since h ab is horizontal, h (4) =i(h )^# +i(h )^#; (4.68) where (h ) abc = 3h d[a bc] d . Using the denition (4.31) and the identity (C.8), we nd that only terms inh ab contribute to the contraction h . The three terms in that contraction 65 are then evaluated using (C.9) and (C.11). The result is given in (4.18), but now we have shown that it holds on any SE manifold. Including the conjugate terms yields (4.66). Finally, d c (#^d!) = ! c (#^!) = ! c 1 : (4.69) This proves that all terms in (4.12) are linear combinations of the two basis harmonics (4.60). It also follows from (4.69) that d 1 = 0. Since d 2 = 0 as well, we have d c = 0 as required by (4.11). The other equation in (4.11) is satised automatically. We must also evaluate the linearized energy momentum tensor (4.6). To this end we note that the two basis harmonics (4.60), using (3.8) and (4.22), can be written as 1 =J^!; 2 =2 1 #^d!: (4.70) Hence d in (4.61) is a linear combination of a horizontal (2; 2)-form J^! and a mixed form #^d!. Given (4.67), the contraction in (4.6) with J^! must vanish. Similarly, the only nonvanishing terms in the contraction with the second form are those in which the free indices are along the base and the two #'s are contracted. This gives g cf g dg g eh (#^!) acde fgh = 12i c 2 (d!) ade ( b de b de ); (4.71) where the indices on the right hand side are raised with the round metric. The full expansion of (4.6) is then T ab = 4i a 6 c 2 f i (t 1 2t 2 )'(x)h ab ; (4.72) and is indeed proportional to the metric harmonic. 66 4.6 The masses For a scalar eld,'(x), satisfying (1.4) with mass, m, and the metric and ux harmonics as above, the linearized Einstein equations (4.4)-(4.6) become diagonal, 1 2 m 2 1 a 2 ( ! + 4) + 24 L 2 + 4 a 2 4c 2 4 c 2 21 h ab =4i f i a 6 c 2 (t 1 2t 2 )h ab : (4.73) To evaluate the left hand side, we have used (4.32) andL 2 h ab =16h ab . The latter follows from the observation that the R-charge of the metric harmonic is q = 4 and is the same as of the background ux. The contraction in the uctuation of the energy momentum tensor on the right hand side has been evaluated in (4.72). The linearized Maxwell equation (4.12) can be simplied using (4.61)-(4.63). After pro- jecting onto the basis harmonics, 1 and 2 , it yields two equations 1 c m 2 ! a 2 t 1 + 2 c (c 2 1)m 2 + 2 ! 1 a 2 c + f 0 a t 2 =128i f i a 4 ! c ; 2 c a 2 + f 0 a t 1 + c m 2 ! a 2 4 a 2 4 f 0 a t 2 = 0: (4.74) For the FR solutions (3.5) there is no internal ux,f i = 0, and the Einstein and Maxwell equations decouple. From the rst one we get the same mass, m 2 1 L 2 = ! 4 2; (4.75) for both the supersymmetric and skew-whied solution. The other two masses in (1.8) and (1.9) are then obtained by setting the determinant of the homogeneous system of equations (4.74) for t 1 and t 2 to zero. This yields a quadratic equation for m 2 , whose solutions are either m 2 2 L 2 = ! 4 + p ! + 1 1; m 2 3 L 2 = ! 4 p ! + 1 1; (4.76) 67 for the supersymmetric or m 2 2 L 2 = ! 4 + 2 p ! + 1 + 2; m 2 3 L 2 = ! 4 2 p ! + 1 + 2; (4.77) for the skew-whied solutions, respectively. For the PW solution, all three equations are coupled by the non-vanishing internal ux. Solving (4.74) for t 1 and t 2 and plaguing into (4.73) yields a cubic equation for m 2 , whose solutions are m 2 1 L 2 = 3 8 ! ; m 2 2 L 2 = 3 8 ! + 3 p 1 + ! + 3; m 2 3 L 2 = 3 8 ! 3 p 1 + ! + 3: (4.78) For each of the masses there is a uctuation of the metric and the ux that together di- agonalize the linearized equations of motion around the PW solution. As we have already discussed in section 1.3, the last mass will violate the BF bound when ! lies in the range (1.11). One may note that the massesm 2 2 andm 2 3 for the PW solution are simply 3/2 of the masses for the ux modes in the skew-whied FR solution. 4.7 Additional bosonic modes in the Z multiplet In addition to the three scalar elds in theZ-vector multiplet, there are two additional scalar elds that are associated with symmetric tensor harmonics, as seen in Table 2 below. It is reasonable to ask what happens to these two scalar elds. In particular it would be nice to know what the masses of these elds are at the FR and PW points. In order to determine the masses of these two scalars, one needs to know the symmetric tensor harmonics associated with them, and their eigenvalues under the Lichnerowitz opera- tor. Since these two elds lie in the same supermultiplet as elds whose associated harmonics 68 we know, it is in fact possible to construct their associated harmonics. Key results useful for carrying out such a construction are provided in the paper [56]. In [56] the authors provide a formula that gives a spinor-vector harmonic in terms of a 3-form harmonic and a Killing spinor. In table I of [56] they give =a Y +b Y +c D Y ; (4.79) where is a spinor-vector harmonic, the are Dirac matrices, is a Killing spinor,Y is a 3-form harmonic, anda,b, andc are given constants. The relation between the eigenvalue of the spinor-vector under the Rarita-Schwinger operator and the eigenvalue of the 3-form under the `square root of the Hodge-de Rham operator' is given to be M (3=2)(1=2) 2 =4(M (1) 3 + 1); (4.80) where M (3=2)(1=2) 2 is the eigenvalue of the spinor-vector, and M (1) 3 is the eigenvalue of the 3-form. In the same table, the authors provide a formula that gives a symmetric tensor harmonic in terms of a spinor-vector harmonic. It is given by Y () =a f g +b D f g ; (4.81) where Y () is a symmetric tensor harmonic. The relation between the eigenvalue of the symmetric tensor under a Lichnerowicz-like operator and the eigenvalue of the spinor-vector harmonic is given by M (2)(0) 2 = (M (3=2)(1=2) 2 + 4)(M (3=2)(1=2) 2 + 8): (4.82) 69 In general, one can see that using these formulas to obtain harmonics from known har- monics will give objects that will be unwieldy to deal with. However, in the special case where the internal manifold is tri-Sasaki and the known harmonics are, as in Chapter 2, con- structed only in terms of Killing spinors and Dirac matrices, it is expected that the resulting objects will be easier to deal with. In particular, in Chapter 2 we constructed two 3-form harmonics at the FR point, H 1 = X 2 3 Y andH 2 =X +Y + 5Z. LetH denote either of these 3-form harmonics. Then using the formula from [56], one obtains the spinor-vector harmonics r = rmnp H mnp r = rmnp H mnp : (4.83) One nds, at least for the tri-Sasaki case, that in the formula provided in [56], all three terms are proportional to each other, and so it is sucient to keep only the rst term. In turn, one can use the formula given by [56] to obtain symmetric tensor harmonics from these spinor-vectors. One obtains the symmetric tensor harmonics h (1) (mn) = fm ng fm ng h (2) (mn) = fm ng h (3) (mn) = fm ng + fm ng (4.84) As in the formula for the spinor-vectors, one nds that the derivative terms are proportional to the non-derivative terms, and so can be dropped. The constructed symmetric tensor h (1) is actually proportional to the symmetric tensor harmonic that we have already constructed in terms of the (1; 1)-form ! and the 3-form . 70 In fact, the other two symmetric tensorsh (2) andh (3) are actually proportional to symmetric tensor harmonics that are simply constructed from ! and J or . It is found that, up to constants, h (1) (ab) = (d!) a cd ( bcd bcd ) + (a$b) h (2) (ab) = (d!) a cd ( bcd + bcd ) + (a$b) h (3) (ab) = ! c(a J c b) : (4.85) Even though these relations were found in the particular case where the internal manifold is tri-Sasaki, it would not be surprising if these relations were true in general. That is, it should be expected that, in general, the symmetric tensors h (2) and h (3) given above are indeed the symmetric tensor harmonics that lie in the same supermultiplet as h (1) . To verify this expectation, one can rst note that the U(1) R charges are in agreement with what is given in table 2. In addition, one can show that the tensors are transverse, and that they are eigenfunctions of the Lichnerowicz operator, with the correct eigenvalues, i.e. ! + 4. Sinceh (2) is the same ash (1) except for a sign, the calculations for h (2) are the same as those forh (1) , which have already been done. Hence, we need only do the calculations for h (3) . First, we show h (3) is transverse. D a W ab = 1 2 D a (! ca J b c +! cb J a c ) (4.86) The rst term on the right-hand-side is D a (! ca J b c ) =! ca D a J b c =! ca ( a b # c + g ac # b ) =! cb # c +! ca g ac # b = 0: (4.87) 71 The rst equality follows from transversality of !, the second equality follows from (B.12), and the third equality follows from the fact that ! is horizontal and traceless. The second term on the right-hand-side is D a (! cb J a c ) =D a (J c e J b f ! ef J a c ) =D a ( a e J b f ! ef ) =D a (! fa J b f ) = 0: (4.88) The last equality follows from the calculation for the rst term on the right-hand-side. Therefore, h (3) is indeed transverse. Now we want to obtain the eigenvalue of h (3) under the Lichnerowicz operator. This calculation follows in similar fashion to the same calculation for h (2) . Letting t abcd = ! ab J cd , so that h (3) is obtained by the appropriate contraction and sym- metrization, the action of the Lichnerowitz operator on this tensor is given by ( L t) abcd = (!)J cd +! ab (J) cd 2(D e ! ab )(D e J cd ) + R-terms: (4.89) One nds that the R-terms are given by R-terms = 2(! eb [D e ;D a ] +! ae [D e ;D b ])J cd : (4.90) Computing the second derivatives of J, one nds [D e ;D a ]J cd = 2J e [c g d]a 2J a[c d] e : (4.91) Plugging this into the expression for the R-terms gives R-terms = 4(! eb J e [c g d]a J a[c ! d]b ) (a$b): (4.92) Carrying out the appropriate contraction and symmetrization to obtain the symmetric tensor h (3) from t gives that the contribution to L h (3) from the R-terms is6h (3) . 72 Next, we want to get the contribution to L h (3) from the term2(D e ! ab )(D e J cd ). Using (B.12) gives (D e ! ab )(D e J cd ) =(D c ! ab )# d + (D d ! ab )# c (4.93) In contracting the indices a and d, one uses the fact that ! is transverse and horizontal, metric compatibilty, and (B.12) to nd that (D e ! ab )(D e J c a ) =! ab J c a : (4.94) So symmetrizing the indices to obtainh (3) fromt gives that the contribution to L h (3) from the term2(D e ! ab )(D e J cd ) is2h (3) . To obtain the contribution to L h (3) from the term! ab (J) cd , one simply needs to know the eigenvalue of J under the Hodge-de Rham Laplacian. The Hodge-de Rham Laplacian on J is given by J =dJ. One uses (B.12) to nd that (J) m =D l J lm = 6# m ; (4.95) so that J = 6d# = 12J. Hence, the contribution to L h (3) from the term ! ab (J) cd is 12h (3) . Adding up the contributions from all the terms gives L h (3) = ( ! + 12 2 6)h (3) = ( ! + 4)h (3) ; (4.96) as expected. The symmetric tensor harmonics h (2) and h (3) are associated with AdS 4 scalars (2) and (3) , so that the eleven-dimensional metric uctuations are (2) h (2) ab and (3) h (3) ab . The 73 2-form harmonic! is associated with anAdS 4 vector eldZ, so that the appropriate eleven- dimensional object is the 3-form uctuation Z ! mn . We would like to plug the metric uctuations (2) h (2) ab and (3) h (3) ab and the 3-form uctuation Z ! mn into the linearized eld equations to obtain the masses of the scalars (2) and (3) at the PW point. Plugging the uctuations into the linearized Einstein equation 1 2 ^ h AB + r (A r C h B)C 1 2 r A r B h C C = F CNP A F M B NP h MC 1 36 h AB F CDEF F CDEF + 1 9 g AB h CM F CDEF F DEF M + 2 3 F MNP (A f B)MNP 1 18 g AB F MNPQ f MNPQ (4.97) gives the two equations 1 2 ^ h (2) AB = 1 36 h (2) AB F CDEF F CDEF 1 2 ^ h (3) AB = F CNP A F M B NP h (3) MC 1 36 h (3) AB F CDEF F CDEF ; (4.98) one for each of the metric uctuations. The rst term on the right-hand-side of the equation for h (3) is 0 for h (2) because h (2) has terms of type (2; 0) and (0; 2), whereas h (3) is of type (1; 1). From the rst equation one nds that at the FR point the scalar (2) has mass m 2 L 2 = 1 4 ( ! 8), as expected, and at the PW point it has mass 3 8 ! . From the second equation one nds that at the FR point the scalar (3) has mass m 2 L 2 = 1 4 ( ! 8), as expected, and at the PW point it has mass 3 8 ( ! 8). Note that the masses of the scalars are larger at the PW point than at the FR point, so their values are stable. 74 The linearized Maxwell equation is r A f ABCD + 4 r A (F M[ABC h D] M ) 1 2 F BCDR r R h A A = 1 288 BCDEFGHIJKL F EFGH f IJKL 1 1152 Tr(g 1 h) BCDEFGHIJKL F EFGH F IJKL (4.99) Plugging the uctuations into the linearized Maxwell equations, one nds that only the rst two terms on the left-hand-side are non-zero. The rst term on the left-hand-side yields the free massive vector eld equation for Z. As expected, it gives that the mass of Z is given by the eigenvalue of! under the Hodge-de Rham operator. The rst and second terms together give that the divergence of Z, D Z is 0 at the FR point but proportional to the scalar (2) away from it. 5 Examples In this chapter we will construct explicitly the (1; 1)-form(s), !, leading to an instability of the PW solutions for two classes of SE manifolds: the tri-Sasakian manifolds and the homogeneous manifolds (1.3). Throughout this section we take ! to be real. The unstable modes in chapter 4 are then constructed using the form i!. 5.1 Tri-Sasakian manifolds The eleven-dimensional supergravity admits a consistent truncation on an arbitrary tri- Sasakian manifold to aN = 3, d = 4 gauged supergravity [34]. As shown in [34], the instability of the PW solution follows then from the existence of a single scalar mode with 75 the mass m 2 =3 in the spectrum of uctuations around the corresponding critical point of the scalar potential. Starting with that unstable scalar mode in the four-dimensional theory, one can follow the truncation and reconstruct the unstable mode in eleven-dimensions. However, it is simpler to look directly for a (1; 1)-form, !, in terms of the geometric data on a tri-Sasakian manifold. A tri-Sasakian manifold admits three globally dened orthonormal Killing spinors, i , in terms of which the three one-forms, K i , dual to the SU(2) Killing vectors, are given by K i a = i 2 ijk j a k : (5.1) Dene M i = 1 2 dK i ; M i ab = 1 2 ijk j ab k : (5.2) The forms K i and M i satisfy [65] D a K i b =M i ab ; (5.3) D a M i bc = 2 g a[b K i c] ; (5.4) K i a K ja = ij ; (5.5) M i ab K jb = ijk K k a ; (5.6) M i ac M jc b =K i a K j b ij g ab + ijk M k ab : (5.7) Using those identities we show that the two-forms ! i = 1 2 ijk K j ^K k + 1 3 M i ; (5.8) are transverse eigenforms of the Hodge-de Rham Laplacian, ! i = 24! i : (5.9) 76 Indeed, the transversality follows directly from (5.3)-(5.6), which imply that D a M i ab = 6K i b ; D a (K j [a K k b] ) = jki K i b : (5.10) To prove (5.9), we note that on a transverse form, ! i , ! i ab = D c (d! i ) abc ; (5.11) where d! i =2 ijk M j ^K k : (5.12) The divergence in (5.11) is then evaluated by rst using (5.3) and (5.4) and then simplifying the resulting contractions using (5.5)-(5.7). The PW solution is now obtained by choosing any two orthonormal Killing spinors that x a particular SE structure. Given the SU(2) isometry, we may simply take ( ) = ( 1 ; 2 ) and set = 3 to be the additional Killing spinor. Then # =K 3 and J =M 3 , see (3.12). Consider the two form ! =K 1 ^K 2 1 3 J; (5.13) with components ! ab =2( 1 [a )( 2 b] ) 1 3 1 ab 2 : (5.14) It follows from (5.5) and (5.6) that ! is horizontal. Similarly, the form d(K 1 ^K 2 ) =2(M 1 ^K 2 K 1 ^M 2 ); (5.15) is horizontal, so that ! is in fact basic. Finally, by contracting with J, we check that ! is a primitive (1,1)-form. 77 We have checked that the unstable mode arising from ! in (5.13) reproduces precisely the unstable mode in the truncation in [34]. We also note that a more complete construction and classication of harmonics onN 1;1 in terms of Killing spinors, including the forms above, can be found in [66]. While the construction above gives an unstable mode on any tri-Sasakian manifold, there will be additional modes if the manifold admits more than three Killing spinors. 10 In par- ticular, to construct the unstable modes onS 7 found in [32], we can generalize the foregoing as follows. Let j be the additional six Killing spinors and let K j a =i a j ; = 1; 2; j = 1;:::; 6: (5.16) Then ! ij = 1 2 (K 1i ^K 2j +K 1j ^K 2i ) 1 3 J ij ; (5.17) are symmetric, ! ij = ! ji , and traceless, ! ij ij = 0, and transform in 20 0 of SU(4), which is the isometry of the KE base, CP 3 . In the same way as above, one checks that ! ij are basic (1,1)-forms and that the diagonal forms,! jj , are transverse and ! jj = 24! jj . By the SU(4) symmetry, the same holds for the remaining forms. 5.2 Homogeneous Sasaki-Einstein manifolds The homogeneous SE manifolds (1.3) are given by G=H coset spaces, which is a conve- nient realization for a calculation of the KK spectrum of the corresponding AdS 4 M 7 compactication of the eleven dimensional supergravity. However, one can also realize any 10 In fact, the only regular manifold with more than three Kiling spinors is S 7 [23]. 78 Spin Field Energy U(1) R m 2 L 2 1 Z E 0 + 1 q 4E 0 (E 0 1) 0 E 0 + 2 q (E 0 + 2)(E 0 1) 0 E 0 + 1 q + 4 (E 0 + 1)(E 0 2) 0 E 0 + 1 q (E 0 + 1)(E 0 2) 0 E 0 + 1 q 4 (E 0 + 1)(E 0 2) 0 E 0 q E 0 (E 0 3) Table 2: The bosonic sector of a Z-vector multiplet. homogeneous SE manifold as a hypersurface in some C N , in some cases modded out by a continous Abelian symmetry. This has been discussed in detail in [53, 54, 55, 46, 52, 67]. In this section we use the latter construction to nd explicitly stability violating (1; 1)-forms, !, on each of the spaces (1.3). An advantage of this method is that the required properties of ! are either manifest or easy to verify. In principle, one could try to identify (1; 1)-forms leading to instabilities of PW solutions by examining the KK spectra that have been studied for all homogeneous SE manifolds in references in Table 1. Indeed, in the KK reduction of the three-form potential,A (3) , a transverse two-form harmonic gives rise to a vector eld whose mass is given by the eigenvalue of the Hodge-de Rham Laplacian [68, 39]. In the terminology of [56], the vector eld is called the Z-vector eld and it is present in the KK towers of the followingN = 2 supermultiplets [57, 50]: the long and/or semi-long graviton multiplet, the two long and/or 79 semi-long gravitino multiplets and the Z-vector multiplet. However, the mere presence in the KK spectrum of a two-form harmonic,!, whose mass, ! , lies in the instability range (1.11), is not yet sucient to conclude that the PW solution is unstable. One must also show that! is a transverse, primitive, basic, (1; 1)-form, which is by no means obvious. For that reason, we rst construct explicitly stability violating (1; 1)- forms,!, and then check whether both ! and the supersymmetric FR scalar masses (4.75) and (4.76) agree with the known KK spectra, in particular, whether the corresponding elds: theZ-vector eld, the scalar and the two pseudo-scalar elds lie in a longZ-vector multiplet. The comparison works perfectly forS 7 ,N 1;1 andM 3;2 , but reveals missing multiplets in the published KK spectra for Q 1;1;1 and V 5;2 . The bosonic elds of a long Z-multiplet are listed in Table 2 , with the R-charge in the second column and the masses in the last column given in the conventions used in this paper. Specically, theR-charge is twice the charge in the original tables in the KK literature (see, e.g., Table 3 in [50]). We dene the mass of aZ-vector as the eigenvalue of the corresponding Hodge-de Rham operator. This agrees with the usual denition used in the references in Table 1, except that our normalization of the metric for the FR solution introduces a factor of four dierence, m 2 Z L 2 = 1 4 M 2 Z e 2 : (5.18) The masses of the scalar elds are related by m 2 ; L 2 = 1 16 M 2 ; e 2 32 ; (5.19) where e 2 = 1=(16L 2 ) is usually set to one. 80 5.2.1 S 7 We represent S 7 as the unit sphere in C 4 , ju 1 j 2 +::: +ju 4 j 2 = 1: (5.20) The U(1) R symmetry is the rotation by the phase. Let ij k l be a constant complex tensor in C 4 that is antisymmetric in [ij] and [ k l], primitive with respect to the canonical complex structure inC 4 , and satises the reality condition ij k l =( kl i j ) . Then the pull-back onto S 7 of ! = ij k l u i u k du j ^d u l ; (5.21) yields 20 basic (1; 1)-forms with ! = 24, which give rise to the unstable modes obtained in [32]. Our calculation agrees with the general result for the spectrum of the Hodge-de Rham Laplacian on CP 3 [43], conveniently summarized in Table 2 in [58]. There we nd that there is a single tower of (1; 1)-forms in [k; 2;k] irrep of SU(4) with the eigenvalues (1;1) = 4(k + 2)(k + 3); k = 0; 1; 2;::: : (5.22) The forms (5.21) lie at the bottom of the tower with k = 0. The higher level forms with k 1 have (1;1) 48 and thus lie outside the instability bound (1.11). One may note that those forms are not the lowest lying transverse two-forms on S 7 . Indeed, the spectrum of the Laplacian on two-forms on S 7 is [44] (2) = (p + 2)(p + 4); p = 1; 2; 3;::: ; (5.23) 81 of which (5.22) is a subset. Forp = 1 and 2, the eigenvalues are 15 and 24, respectively, and satisfy (1.11). However, the two-forms with (2) = 15 are trilinear in u i and u i , hence have a nonzero R-charge and are not basic. 5.2.2 N 1;1 The (hyper-)K ahler quotient construction for N 1;1 [46, 67] starts with C 3 C 3 with coor- dinates (u j ;v j ), j = 1; 2; 3, that transform as 3 and 3 under SU(3), respectively, and with (u j ; v j ) transforming as doublets under SU(2). The N 1;1 manifold is then the surface ju j j 2 =jv j j 2 = 1; u j v j = 0; (5.24) modded by the U(1) action (u i ;v i ) (e i u i ;e i v i ). The standard metric [69, 70] is obtained by a reduction from the at metric inC 6 . We refer the reader to [67] for a detailed discussion of the metrics and for explicit angular coordinates. The three Killing forms in section 5.1 can be taken as K 1 = 1 2 (u j dv j + u j d v j ); K 2 = i 2 (u j dv j u j d v j ); K 3 = i 2 (u j d u j +v j d v j ); (5.25) in terms of which the form, !, is given by (5.13). It is now manifest that ! is a (1; 1)-form, which is invariant under the U(1) action of the K"ahler quotient, and hence a well-dened form onN 1;1 . It is also a singlet of SU(3) and is invariant under the U(1) R SU(2) isometry, (u j ;v j )! (e i u j ;e i v j ), along the SE ber. Evaluating it in angular coordinates, we verify that it is basic and primitive. The complete KK spectrum on this space was obtained in [45] (see, also [71, 46, 47, 66]), where one nds 21 towers of two-form harmonics. Specifying to the (1,3) irreducible 82 representation of SU(3) SU(2), M 1 = M 2 = 0 and J = 1 in the notation in [45], leaves two possible eigenvalues (2) 12 = 96 and (2) 21 = 48 lying in the series E 8 with j = 0. The rst three forms are the ones constructed above in (5.8), one of which is the sought after (1; 1)-form,!, with ! = 24. The remaining three are the three canonical two-forms,M i , on the tri-Sasakian manifold, one of which is the complex structure, and hence is not primitive, while the other two are not basic. In this example theN = 2 long Z-vector multiplet is a part of a longN = 3 gravitino supermultiplet, see Table 4 in [66]. Following [56], all harmonics in this multiplet can be constructed in terms of the three Killing spinors on N 1;1 [66]. 5.2.3 M 3;2 TheN = 2 supersymmetry of the FR solution on M 3;2 was proved in [72]. The complete Kaluza-Klein spectrum was obtained in [48] (see also [49]) and further analyzed more recently in [50]. The KE base of M 3;2 is CP 2 CP 1 and the SE metric in the form (2.3) is given by [73, 74] ds 2 = 3 4 ds 2 CP 2 + 1 2 ds 2 CP 1 + (d + 3 4 A CP 2 + 1 2 A CP 1) 2 ; (5.26) where the ds 2 CP k is the Fubini-Study metric and A CP k is the K ahler potential with dA CP k = 2J CP k. The K ahler quotient construction for this SE manifold is explained in Appendix H. See also the references [54, 55]. The construction starts with C 3 C 2 with coordinates, u i and v . In terms of these coordinates, M 3;2 is the surface dened by the equations 2u j u j = 3v v = 1; (5.27) 83 modded by the U(1) symmetry, (u i ;v ) (e 2i u i ;e 3i v ). Once more the U(1) R symmetry is (u i ;v )! (e i u i ;e i v ). In light of what was discussed in Chapter 4, one would like to nd or construct (1,1)-forms on the space that are basic, primitive, and transverse, and that are eigenfunctions of the Hodge-de Rham Laplacian with eigenvalues in the range between 2(94 p 3) and 2(9+4 p 3). The condition that the (1; 1)-forms be basic amounts to imposing that they are invariant under the U(1) R symmetry, or in other words, that they live on the KE base, and the condition that the eigenvalues lie in the given range means that one should look for modes of the Laplacian that are low-lying. As discussed, the existence of such a primitive and transverse (1; 1)-form means that there is a scalar that causes instability. Modes that satisfy the desired conditions were in fact constructed in [58], and the con- struction of [58] relies upon an important result that was found in [75]. In [75] it is shown that if there exists a Killing vector K a on a KE space, then there exists a scalar on the space such thatK a =J ab @ b . The scalar is shown to be an eigenfunction of the Laplacian with eigenvalue 2, i.e. 2 + 2 = 0, where is such that R ab = g ab . The converse of this statement is also shown to be true. That is, if a KE space has a scalar that is an eigenfunction of the Laplacian, with eigenvalue 2, then the objectK a =J ab @ b is a Killing vector on the space. Therefore, if a KE space has a Lie group symmetry, then it is guaranteed to possess as many scalar harmonics with eigenvalue 2 as there are dimensions in the group. For example, sinceCP 2 has symmetry group SU(3), it has eight scalar harmonics with eigenvalue 2. Each of the eight generators of SU(3) corresponds to a Killing vector on CP 2 , and each of these 84 Killing vectors can be expressed in terms of a scalar harmonic in the way explained above. For the KE space CP n it is shown in [75] that the scalar harmonics with eigenvalue 2 that generate the Killing vectors are = 1 Z A Z A T A B Z A Z B ; (5.28) where the Z A are the homogeneous coordinates on CP n and T A B is an arbitrary Hermitian traceless tensor. In [58] the authors study the stability of AdS 5 solutions of M-theory compactied on six-dimensional KE spaces. Among other spaces, they look at the KE space CP 2 CP 1 , which is the KE base of the SE manifold M 3;2 . On this space, they note that given a scalar harmonicY onCP 2 , one can construct a primitive transverse (1; 1)-form! from it that is an eigenfunction of the Hodge-de Rham Laplacian with the same eigenvalue as the scalar Y . This (1; 1)-form ! is a linear combination of the forms @ B @ B Y , YJ (4) , and YJ (2) . It is straightforward to see that each of these forms is an eigenfunction of the Hodge-de Rham Laplacian =d +d on the KE space, with the same eigenvalue as Y . For the rst form, the Dolbeault operators commute with , so clearly it is an eigenfunction with the same eigenvalue as Y . For the second and third forms, one notes that d and of the complex structures are zero because their covariant derivatives are 0, and so only acts onY . So it is clear that the second and third forms are also eigenfunctions of with the same eigenvalue as Y . The coecients in ! are xed by imposing that it be primitive and transverse. In the special case where Y is taken to be a scalar harmonic that generates a Killing 85 vector on CP 2 , which is given explicitly above, and where = 8, one nds ! to be ! = 2i@ B @ B Y + 16YJ (4) 16YJ (2) ; (5.29) where the SE two form is J = J (4) +J (2) with J (4) = 3 4 J CP 2 and J (2) = 1 2 J CP 1, and @ B and @ B are the Dolbeault operators (see, e.g., [6]). In the notation and conventions used in this work, Y =t i j u i u j ; (5.30) where t ij is a constant hermitian, traceless matrix and Y = 16Y: (5.31) As discussed, this (1; 1)-form is primitive, transverse, and basic, and it is an eigenfunction of with eigenvalue 16. It is thus associated with an unstable scalar at the Pope-Warner point. Furthermore, there are eight such (1; 1)-forms because there are eight scalar harmonics with eigenvalue 16, and thus there are eight unstable scalar modes at the Pope-Warner point transforming in (8; 1) of SU(3) SU(2). These eight unstable scalar modes have the mass m 2 3 L 2 = 9 3 p 173:3693: (5.32) In the KK spectrum for the supersymmetric solution, we should nd aZ-vector multiplet with the masses m 2 Z L 2 = 16; m 2 L 2 = 2; m 2 L 2 = 3 p 17: (5.33) 86 Indeed, there is such a multiplet given by eqs. (3.23) and (3.24) in [55]. In [55] an irrep of G = SU(3)SU(2)U(1) is specied by (M 1 ;M 2 ;J;Y ), whereM 1 is the number of columns in theSU(3) tableau with one box,M 2 is the number of columns in theSU(3) tableau with two boxes,J is theSU(2)-irrep `quantum' number, andY is theU(1) charge. In (3.23) and (3.24) we must set M 1 =M 2 = 1 and J = 0. Doing so gives E 0 = 1 2 (1 + p 17); (5.34) which reproduces the masses (5.33) using formulae in table 2. The space M 3;2 and its complete KK spectrum were treated in much detail in [55]. It is therefore appropriate to present the harmonics associated with the unstable scalars in the language and conventions used in that paper. Let = 1 3 2 i = 1 i 2 i = ij j 3 3 ; 12 = 1 ij = 1 2 i j 3 + j i 3 : (5.35) The two 3-form harmonics and the metric harmonic associated with the unstable scalars will be constructed from these tensors. The two 3-form harmonics are constructed from the ve 3-forms A j given below. These 87 3-forms are pieces of the 3-form H decomposition given in [55]. A 1 ABC = ABCD D 3i i D i3 i A 2 Amn = mn A 3i i + A i3 i A 3 AB3 = [A i3 B] 3j ik kj A 4 AB3 = [A i3 B] 3i A 5 mn3 = mn : (5.36) The remaining components of the A j are 0. The A j close under the action of Q =?d. (Note that d in [55] is dened dierently than usual, so that on 3-forms it is 1 4 of the usual one.) QA 1 = i 4 p 3 A 5 QA 2 = i 4 p 3 A 3 + 2 p 3 A 4 QA 3 = i p 3 2 A 2 +A 3 QA 4 = 1 p 3 A 2 A 4 + 2A 5 QA 5 = A 4 +i 1 p 3 A 1 : (5.37) For A =c j A j we can use the above to determine the c j and such that QA =A: (5.38) Solving these equations gives the eigenvalues = 1 2 (1 p 17), 2, 0, and3. These eigenvalues agree with those listed in equation (6.37) of [55]. In particular the modes corresponding to = 1 2 (1 p 17) are the ones associated with 88 unstable scalars. For these the constants are found to be (xing c 1 = 1) c 1 = 1 c 2 = i c 3 = 1 6 p 3 p 51 c 4 = i p 3 4 1 p 17 c 5 = i 1 2 p 3 p 51 : (5.39) The corresponding masses, in the conventions of [37], of these two modes are obtained using the 3-form mass formula (equation B.3 of [37]) m 2 f = 16(Q 2)(Q 1): (5.40) They are m 2 f = 16 5 p 17 . Note that these are the masses in the supersymmetric, non-skew-whied case. In going from the skew-whied Freund-Rubin solution to the Pope-Warner solution, the two 3-form modes `mix' with a metric mode to form a new mode. We obtain this metric mode by contracting the 4-formf =dA =?QA with the background internal ux over three indices and symmetrizing over the remaining two indices. Y =F ( f ) ; (5.41) giving, without worrying about the overall constant, Y 1A = ij ( A j3 i + A 3j i ) Y 2A = i ij ( A j3 i A 3j i ); (5.42) with all other components equal to 0. 89 5.2.4 Q 1;1;1 Recall that Q 1;1;1 is a U(1) bundle over CP 1 CP 1 CP 1 , with the metric (see, e.g., [74]) ds 2 = 1 2 (ds 2 CP 1 (1) +ds 2 CP 1 (2) +ds 2 CP 1 (3) ) + d + 1 2 (A CP 1 (1) + A CP 1 (2) + A CP 1 (3) ) 2 : (5.43) The Kahler quotient construction for this manifold [54, 55], has three C 2 's, with coordinates u , v and w , respectively, one for each CP 1 factor in the KE base. Then Q 1;1;1 is the surface in C 6 , u u =v v =w w = 1; (5.44) modded by two U(1) symmetries, (u ;v ;w ) (e i u ;e i v ;e ii w ). In terms of the projective coordinates, z i , on CP 1 (i) , and the ber angle, , we have u 1 = z 1 e 2i =3 (1jz 1 j) 1=2 ; v 1 = z 2 e 2i =3 (1jz 2 j) 1=2 ; w 1 = z 3 e 2i =3 (1jz 3 j) 1=2 ; u 2 = e 2i =3 (1jz 1 j) 1=2 ; v 2 = e 2i =3 (1jz 2 j) 1=2 ; w 2 = e 2i =3 (1jz 3 j) 1=2 : (5.45) The SU(2) Killing vectors on each CP 1 yield triplets of scalar harmonics, Y (1) =t (1) u u ; Y (2) =t (2) v v ; Y (3) =t (3) w w ; (5.46) which are eigenfunctions of the Laplacian with the eigenvalue 16 [74]. The two forms ! (1) =Y (1) (J CP 1 (2) J CP 1 (3) ); ! (2) =Y (2) (J CP 1 (3) J CP 1 (1) ); ! (3) =Y (3) (J CP 1 (1) J CP 1 (2) ); (5.47) are primitive, transverse (1; 1)-eigenforms of the Hodge-de Rham operator with the same eigenvalue [58]. This gives nine unstable modes for the PW solution on Q 1;1;1 in the adjoint representation of SU(2) SU(2) SU(2). 90 Clearly, numerical values of all the masses of the Z-vector eld and the scalar and pseu- doscalar elds at the FR solution are the same as for M 3;2 , and one expects to nd a similar structure ofN = 2 supermultiplets as well. Hence it is surprising that the KK spectrum in section 4 in [51] does not contain a long Z-vector multiplet in the adjoint of SU(2) SU(2) SU(2) with the energy (5.34). In fact, there is also no graviton multiplet corresponding to the scalar harmonics (5.46). However, a closer examination of the allowed harmonics onQ 1;1;1 and their masses, which are listed in section 3 of the same paper, shows that the Z-multiplet we are looking for should have been included in the nal \complete classication." 5.2.5 V 5;2 As discussed in [53] (see, also [52, 76, 77]), the Stiefel manifold, V 5;2 , is the intersection of the Kahler cone in C 5 , (u 1 ) 2 + (u 2 ) 2 + (u 3 ) 2 + (u 4 ) 2 + (u 5 ) 2 = 0; (5.48) with the unit sphere, ju 1 j 2 +ju 2 j 2 +ju 3 j 2 +ju 4 j 2 +ju 5 j 2 = 1: (5.49) 91 Writing u j = x j + iy j , the real and imaginary part vectors (x j ) and (y j ) in R 5 can be parametrized by the Euler angles of the coset space SO(5)=SO(3), 11 0 B B B B B B B B B B B B B B @ x 1 y 1 x 2 y 2 x 3 y 3 x 4 y 4 x 5 y 5 1 C C C C C C C C C C C C C C A = 0 B B @ R 3 ( 1 ; 2 ; 3 ) 0 0 R 2 () 1 C C A 0 B B B B B B B B B B B B B B @ cos 0 0 cos 0 0 sin 0 0 sin 1 C C C C C C C C C C C C C C A R 2 4 3 ; (5.50) where 0 1 ; 3 < 2; 0 2 ;<; 2 ;< 2 ; 0 < 3 8 ; (5.51) andR 2 andR 3 are rotation matrices. In terms of the coordinates on the cone and the angles, the SE metric on V 5;2 is ds 2 = 3 2 du j d u j 3 16 ju j d u j j 2 = 3 8 h d 2 + cos 2 2 1 +d 2 + cos 2 2 2 + 1 2 sin 2 ()( 3 +d) 2 + 1 2 sin 2 ( +)( 3 d) 2 i + h d + 3 8 cos()( 3 +d) + 3 8 cos( +)( 3 d) i 2 ; (5.52) where i are the SO(3)-invariant forms, d i = j ^ k . The metric (5.52) is the canonical SE form of the U(1) bration over the KE base, which is the Grassmannian, Gr 2 (R 5 ). The volume of the space is computed from this metric in Appendix E. The harmonics onV 5;2 are obtained by the pullback of tensors in C 5 and decompose into SO(5) U(1) R . Here SO(5) acts on u j in the real vector representation, while U(1) R is the 11 A somewhat dierent explicit parametrization of V 5;2 is given in [76, 77]. 92 phase rotation, u j !e i u j . The lowest lying scalar harmonic that is invariant under U(1) R is ij = u i u j u j u i . It is an eigenfunction of the Laplacian with the eigenvalue 16 [52]. Similarly, the lowest lying (1; 1)-forms that are not proportional to the Kahler form are ! i = ijklm u j u k du l d u m : (5.53) They transform as 5 of SO(5) and are invariant under U(1) R . Expanding those forms using (5.50) conrms that they are basic. They satisfy ! i = 32 3 ! i ; (5.54) and hence give rise to ve unstable modes of the PW solution with the mass m 2 3 L 2 = 7 p 1053:2469: (5.55) The masses for the supersymmetric solution are m 2 Z L 2 = 32 3 ; m 2 L 2 = 2 3 ; m 2 L 2 = 5 3 r 35 3 : (5.56) The Z-vector multiplet has then E 0 = 1 6 (3 + p 105): (5.57) While such a multiplet is not listed in the tables in [52], the authors note at the end of section 2 that there might be an additional vector supermultiplet with this energy. 12 In appendix C, we list all bosonic harmonics on V 5;2 that transform in 5 of SO(5) and show that they decompose unambigously intoN = 2 supermultiplets including a long Z-vector multiplet in agreement with our construction. 12 We thank A. Ceresole and G. Dall'Agata for correspondence, which claried this point. 93 5.3 Orbifolds Homogeneous SE manifolds also admit discrete symmetries such that the quotient manio d, M 7 = is still SE. The natural question is what happens to the master (1,1)-forms in this projection and whether the PW solution for the quotient SE manifold is stable. We will now examine this for some examples of SE discrete quotients that were considered in the literature. For S 7 , it has been shown in [32] that if the discrete symmetry group is a subgroup of SU(4), it will preserve some of the unstable modes. The same reasoning applies to the (1,1)-forms (5.21) and shows that some of them will be well-dened on the quotient. Orbifolds of M 3;2 , Q 1;1;1 , and S 7 , can be obtained as limits of the Y p;k Sasaki-Einstein manifolds [78]. Specically, when 2k = 3p and p = 2r, one has that Y 2r;3r (CP 2 ) =M 3;2 =Z r , where Z r is a nite subgroup of SU(2) acting on CP 1 . Since the master 2-forms for M 3;2 are constructed from scalar harmonics on the CP 2 , see (5.30) and (5.29), they are preserved under the orbifolding. Hence the instability persists for these orbifolds of M 3;2 . Similarly, whenk =p, one hasY p;p (CP 1 CP 1 ) =Q 1;1;1 =Z p , whereZ p is a nite subgroup of SU(2) acting on one of the three CP 1 's. Each independent master (1,1)-form on Q 1;1;1 , see (5.46) and (5.47), is constructed from a scalar harmonic on one of the CP 1 factors. For the SU(2) acting on CP 1 (i) , the forms ! (j) , j6= i, are invariant under Z p and hence are well dened on the quotient Q 1;1;1 =Z p . For k = 3p, one has that Y p;3p =S 7 =Z 3p , where Z 3p SU(4) acts by (u 1 ;u 2 ;u 3 ;u 4 ) ! (e 2i=3p u 1 ;e 2i=3p u 2 ;e 2i=3p u 3 ;e 2i=p u 4 ): (5.58) 94 The six master (1; 1)-forms onS 7 that contain precisely oneu 4 or u 4 are not invariant under (5.58). This yields fourteen unstable modes on that space. The orbifolds V 5;2 =Z k have been discussed in [77]. The nite group here is Z k U(1) b , where U(1) b is a diagonal subgroup of the SO(2)SO(2) rotation in the (12) and (34) planes inC 5 . Clearly, the master 2-form! 5 , see (5.53), is invariant under this action and yields one unstable mode on V 5;2 =Z k . 6 Conclusion In the research presented in this dissertation we have analyzed a subset of scalar modes in the linearized spectrum of eleven-dimensional supergravity around the Pope-Warner solution on an arbitrary SE manifold and derived a condition under which the solution becomes perturbatively unstable. Specically, we have shown that when the manifold admits a basic, transverse, primitive (1; 1)-form within a certain range of eigenvalues of the Hodge-de Rham Laplacian, then there are scalar modes violating the BF bound. We have also constructed such destabilizing (1; 1)-forms on all homogenous SE manifolds, and on their orbifolds, and found that when viewed as harmonics for uctuations around the supersymmetric solution, those forms give rise to a long Z-long vector supermultiplet in the KK spectrum. Using this fact it would be straightforward to rephrase the stability condition in terms of spinor-vector harmonics on the SE manifold. To do so one could use the formulae given in, e.g., [56], since by the construction in [56], spinor-vector harmonics give rise to longZ-vector supermultiplets. 95 Throughout this work we have assumed that the SE manifold was quasi-regular, i.e. regular or non-regular, and the quasi-regularity was used explicitly in some of the proofs. In particular quasi-regularity was used in establishing the shift between the eigenvalues of the symmetric tensor and 2-form harmonics under their respective mass operators. However, since this proof is local, one would expect that our construction should hold for an arbitrary SE manifold. Indeed, the fact that this same shift was proven in [56] (see Appendix G) for any internal manifold with a Killing spinor indicates that this expectation does actually hold. It remains an open problem to see whether stability violating 2-forms exist on any SE manifold. In other words, even though the PW solution turned out to be unstable for all the concrete SE manifolds we looked at, it is not yet known whether there exists an SE manifold for which the PW solution is stable. It would be notable to nd such a SE manifold. As discussed, if the manifold is quasi-regular, the question of stability reduces to the problem of determining the low lying spectrum of the Hodge-de Rham Laplacian on a six-dimensional KE manifold, which in itself is a dicult problem with rather few explicit results (see, e.g., [81]). If the manifold is irregular, perhaps the results of [91] may be of use. In this paper the author presents a generalization of the identity = 2 @ to SE manifolds. There is also an analogue of the PW solution in type IIB supergravity [82], which is known to be unstable within theN = 8 d = 5 supergravity [83, 84] obtained by compactication onS 5 . It would be interesting, and perhaps simpler, to examine the stability of this type of solutions for the new class of ve-dimensional SE manifolds [27, 28] for which the spectra of the scalar Laplacian were obtained in [85, 86]. 96 As discussed in the introduction, the main motivation for recent interest in PW solutions came from the \top-down" construction of holographic models of superconductors in [10, 12, 11]. The PW solutions are then dual to zero entropy states with emergent conformal invariance at T = 0. In light of this duality to a conformal theory, it would be nice to see what the PW instability discussed here looks like in the dual CFT under the AdS/CFT correspondence. A possible starting point for such an endeavor is provided in the papers [66] and [47]. In these papers the authors discuss a special N = 3 long gravitino multiplet that, in fact, contains the tri-Sasaki modes discussed here that lead to PW instability. In particular, the authors give a composite CFT operator that they claim corresponds to this long gravitino multiplet. A Conventions We use the same conventions as in [79] and [32], with the mostly plus space-time metric and the bosonic eld equations of eleven-dimensional supergravity given in (1.1) and (1.2), and the gravitino supersymmetry transformations M =D M + 1 144 I M NPQR 8 M N I PQR F NPQR : (A.1) On a manifold with a Minkowski signature metric, g, we dene the Hodge dual, ?, by ? ^ =jjvol g : (A.2) The Hodge dual,, for a riemannian metric, g, is then dened without the minus sign. 97 The eleven-dimensional Dirac matrices in the 4 + 7 decomposition are I = 1 1; = 1;:::; 4; I a+4 = 5 a ; a = 1;:::; 7; (A.3) where 5 =i 0 1 2 3 ; 7 =i 1 ::: 6 : (A.4) Then I 1 I 2 ::: I 11 = 12:::11 1 = 1: (A.5) We use the representation in which the four-dimensional -matrices are real, while the seven- dimensional -matrices are pure imaginary and antisymmetric. For a real spinor, , on the internal manifold, we then have = T . B Sasaki-Einstein manifolds: denitions and relevant information Due to their prominence in string theory and M-theory, it is worthwhile to properly dene what a Sasaki-Einstein manifold is. In this appendix we dene what a Sasaki-Einstein manifold is, and provide information about them that is relevant in subsequent sections. The content of this section follow prin- cipally from the contents of [23] and [26]. A thorough treatment of the subject is given in [24]. Contact manifolds 98 In dening what a Sasaki-Einstein manifold is, it is natural to start by rst dening what a contact manifold is. A contact manifold is a (2n 1)-dimensional manifoldM such that there exists a 1-form on it, with the property that ^ (d) n 6= 0 (B.1) at each point of M. Such a 1-form is called a contact 1-form. Given a contact manifoldM with contact 1-form, there is a unique vector eld called the Reeb vector eld. The Reeb vector eld is dened to be the unique vector eld satisfying the conditions () = 1; i d = 0: (B.2) At each point p on the manifold M one can consider the hyperplane ker(p). This hyperplane is a (2n-2)-dimensional subspace of the tangent space TM p , and the bundle D of all such hyperplanes, D = ker, is a sub-bundle of the tangent bundle TM. In this way the contact 1-form species a distribution D of (2n-2)-dimensional hyperplanes on the manifold M. D is called the contact distribution. The contact distribution D is maximally non-integrable, which translates into the fact that d is nondegenerate, i.e. for every vector X on a hyperplane there exists a Y on the hyperplane such thatd(X;Y )6= 0. Intuitively, the 2-formd is a way to measure the failure of the parallelogram, formed by the vectorsX andY in the hyperplane, to close in the Reeb vector direction. For more details on contact manifolds see [87]. Since it is nondegenerate, the 2-form d can be regarded as a symplectic form ! on D. In addition to a symplectic form on D one would also like to have an almost complex 99 structure J, which is a type (1,1) tensor, on D. Furthermore, one would like for this J to be compatible with !. Compatibility with ! means the relations d(JX;JY ) = !(X;Y ) and !(JX;X) > 0 are satised. In terms of indices, the rst relation is equivalent to J i m J j n ! ij =! mn . This J can be used to get a Riemannian metric on D, namely g D (X;Y ) = !(JX;Y ), whereX andY are smooth sections ofD. This metric is compatible with the almost complex structureJ, which means thatg D (JX;JY ) =g D (X;Y ). In terms of indices, this expression is equivalent to J i m J j n g ij =g mn . Also, in terms of indices, the expressiong D (JX;Y ) =!(X;Y ) is equivalent tog ik J k j = ! ij . So ! can be thought of as J with its index lowered with the metric. J on D can be extended to a tensor of type (1,1) on TM by letting = J on D and = 0. Also, the metric g D on D can be extended to a metric g on TM by letting g(X;Y ) =g D (X;Y ) +(X)(Y ) =d(X;Y ) +(X)(Y ). It is clear that under this metric the Reeb vector is orthogonal to the vectors in D, i.e. g(X;) = 0 for any section X of D. The orthogonality follows from the denition of D, i.e. D = ker, and from the denition of the Reeb vector, which requires that i d = 0. Furthermore, this metric satises the compatibility condition g(X; Y ) = g(X;Y ) (X)(Y ). The above construction motivates the denition of a metric contact structure: If in addition to the contact 1-form and its associated Reeb vector eld, there is a tensor eld of type (1,1) and a Riemannian metric g that satisfy 2 =1 + ; g(X; Y ) =g(X;Y )(X)(Y ); (B.3) 100 then the contact manifold M is said to have a metric contact structure. Sasakian and Sasaki-Einstein manifolds Given a compact manifold M with Riemannian metric g, the metric cone over M is dened to be the space R + M with metric ds 2 =dr 2 +r 2 g. If the metric cone over M is K ahler, then M is dened to be a Sasakian manifold. A Sasakian manifold is automatically a contact manifold with a metric contact structure, and its type (1,1) tensor and metric g are as in the construction above, i.e. = J on D = ker, = 0, and g(X;Y ) =g D (X;Y ) +(X)(Y ) =d(X;Y ) +(X)(Y ). Finally, a Sasaki-Einstein manifold is dened to be a Sasakian manifold with Ric g = 2(n1)g. The metric cone over a Sasaki-Einstein manifold is Ricci- at K ahler, hence Calabi- Yau. The converse of this statement is true, i.e. a manifold whose cone is K ahler Ricci- at is a Sasaki-Einstein manifold. An interesting special type of Sasaki-Einstein manifold is a 3-Sasakian manifold. A 3- Sasakian manifold is a Sasakian manifold whose metric cone is hyper-K ahler. This means the holonomy of the cone metric is contained in Sp(p). Sp(p)SU(2p), so a hyper-K ahler manifold is a Calabi-Yau manifold, and a 3-Sasakian manifold is a Sasaki-Einstein manifold. Reeb foliation The Reeb vector eld was dened to be the unique vector eld satisfying the conditions () = 1; i d = 0: (B.4) From the rst condition it is clear that is nowhere vanishing. Since it is nowhere vanishing, it can be used to generate a 1-parameter family of dieomorphisms of the space on which it is dened. 101 Therefore, given a Sasakian manifold M with Sasakian structure (,,,g), one can par- tition M into disjoint orbits of the dieomorphism generated by the Reeb vector . Each orbit is a 1-dimensional space. PartitioningM in this way is called the Reeb foliation, and the orbits are the leaves of the Reeb foliation. Sasakian manifolds split into three dierent classes, depending on the nature of the Reeb foliation. If the leaves of the foliation close, so that they are circles, then the Sasakian man- ifold is said to be quasi-regular. For a quasi-regular manifold, the Reeb vector generates a U(1) action. This U(1) action is always locally free. If in addition the U(1) action is free overall, then there is no point on the manifold that is xed by a nontrivial element of the U(1) action. In this case the Sasakian manifold is said to be regular. If the U(1) action is not free overall, then it must `wrap around' an orbit an integer number of times, so that the orbit is xed by a discrete subgroup of theU(1) action. In this case the manifold is said to be non-regular. If the leaves of the foliation do not close, then they are noncompact. In this case the manifold is said to be irregular. Transverse Kahler structure Motivated by the denition of the distribution D = ker and the form of the metric g, the tangent bundle can be split into the direct sumTM =DL , so that at a pointp onM, TM p =D p L ;p , whereD p is a (2n-2)-dimensional hyperplane, andL ;p is the 1-dimensional line that is tangent to the Reeb vector at p. The spaces D p and L ;p are orthogonal with respect to the metric g. As discussed previously D naturally has a (almost) complex structure J = j D , a sym- 102 plectic structure d, and a metric g D (X;Y ) = d(JX;Y ). (D;J;d) gives the Sasakian manifold M what is referred to in [23] as a transverse K ahler structure. It is important to note that in general this (2n-2)-dimensional K ahler structure holds only locally. One would like to know, however, when this K ahler structure holds globally. For a Sasakian manifoldM, letZ be the space of leaves of its Reeb foliation. Then if the Reeb foliation is quasi-regular then the (2n-2)-dimensional K ahler structure holds globally. In particular, if the Reeb foliation is regular thenZ has the structure of Kahler manifold, and if the Reeb foliation is non-regular thenZ has the structure of an orbifolded Kahler manifold. For necessary details about the orbifold structure in the non-regular case see [23] and [26]. The converse of this statement holds true, i.e. given thatZ is a Kahler manifold or a proper orbifold of one, a principalU(1) bundleM overZ is a Sasakian orbifold with metric h + , where is the projection fromM toZ,h is the metric onZ, and is a 1-form on M such that d = 2 !, where ! is the symplectic structure ofZ. For necessary details see [23] and [26]. If the Reeb foliation of a Sasakian manifold is irregular, then the situtation is more complicated. However, it is known that in this case the closure of the group action generated by the Reeb vector is isomorphic to a torus T k , with k 2. Finally, ifM is 3-Sasakian, then it is anSU(2) bundle over a 4-dimensional quaternionic K ahler manifold or orbifold. Accordingly, its metric can be written as g = g O + 1 1 + 2 2 + 3 3 , whereg O is the metric of the 4-dimensional quaternionic Kahler manifold or orbifold, and the i are 1-forms that are dual to a triplet of Reeb vectors i that form an 103 su2 lie algebra. For more details see [25] and [26]. A nice fact about 3-Sasakian manifolds is that they are automatically Einstein. Homogeneous Sasaki-Einstein manifolds There is a special type of Sasaki-Einstein manifold that has been well-known to super- gravity theorists since the 1980s, namely homogeneous Sasaki-Einstein manifolds. A Sasaki-Einstein manifold is homogeneous if there is a group of isometriesG that acts transitively on it and preserves the Sasakian structure. A group action is transitive if there is a point in the space such that every other point in the space can be obtained via a group action on that point; so there is only one orbit of the group action. Hence, a homogeneous Sasaki-Einstein manifold can be expressed as a coset space. There are in fact only ve seven-dimensional homogeneous Sasaki-Einstein manifolds: S 7 , N 010 ,V 5;2 ,M 32 , andQ 111 [CRW]. They are principal U(1) bundles over the K ahler-Einstein spaces CP 3 , SU(3)=T 2 , Gr 2 (R 5 ), CP 2 CP 1 , and CP 1 CP 1 CP 1 , respectively. The seven-dimensional homogeneous Sasaki-Einstein spaces have all been used to com- pactify eleven dimensional supergravity to AdS 4 , and the complete Kaluza-Klein spectra of these compactications have been determined in [44], [45], [46], [52], [55], and [51]. Killing spinors A Killing spinor is a spinor that satises the relation r Y =Y (B.5) for any vector eld Y , where Y = Y m m and is a constant. For applications in supergravity, string theory, and M-theory, it is important to know when a Sasaki-Einstein manifold admits Killing spinors and how many of them it possesses. 104 If a seven-dimensional Sasaki-Einstein manifold is simply connected it admits at least two Killing spinors, and both of them satisfy the dening relation with the same constant , with > 0 [23, 26]. If a seven-dimensional 3-Sasakian manifold is simply connected it admits at least three Killing spinors, and all of them of them satisfy the dening relation with the same constant , with > 0 [23]. C Sasaki-Einstein identities In the local frame e 1;2;3 =e r=L dx 0;1;2 ; e 4 =dr; e a+4 = 2L e a ; a = 1;:::; 7; (C.1) on AdS 4 M 7 , cf. (3.9), the unbroken supersymmetries are given by =" , " =e r=L " 0 ; 012 " 0 =" 0 ; (C.2) and = (cos(2 ) + sin(2 ) 12 ) 0 ; 12 0 = 34 0 = 56 0 ; (C.3) where " 0 and 0 are constant spinors. We choose the two independent solutions, 1 and 2 , of (C.3) such that the components of the two SE tensors in (3.9) and (3.12) are the same. Given the Reeb vector eld of unit length, 13 a a =# a # a = 1; (C.4) the projection operator a b = a b # a # b ; (C.5) 13 All indices are raised and lowered with the SE metric, g ab . 105 is a map onto the subspace perpendicular to the Reeb vector. Any tensor H ab:::c satisfying # a H ab:::c =# b H ab:::c =::: =# c H ab:::c = 0; (C.6) will be invariant under the projection, and, modulo its dependence on the ber coordinate, , can be thought of as a tensor on the Kahler-Einstein base. We refer to such tensors as horizontal. For complex horizontal tensors of rankn there is a further decomposition into (p;q)-type tensors, wherep andq,p +q =n, refer to the number of holomorphic and anti-holomorphic indices according to the corresponding decomposition along the Kahler-Einstein base. In particular, J ab and abc , are horizontal tensors of type (1; 1) and (3; 0), respectively. A contraction ofJ with a (p; 0)-type and (0;p)-type horizontal tensor is a multiplication by +i andi, respectively. For example, J a d bcd =i abc ; J a d bcd =i abc : (C.7) Horizontal tensors (forms) that are in addition invariant along the Reeb vector eld are called basic. Using the explicit realization of the Sasaki-Einstein forms in terms of Killing spinors (3.12), one can prove additional identities, which we use frequently. First, we have the following \single contraction" identities J ac J bc = a b ; abe cde = 0; (C.8) abe cde = 4 [a [c b] d] 4J [a [c J b] d] 8i [a [c J b] d] ; (C.9) from which the higher contractions follow, J ab J ab = 6; acd bcd = 8 a b 8iJ a b ; abc abc = 48: (C.10) 106 We also need the following uncontracted identity abc def = 6 [a [d b e c] f] 18i [a [d b e J c] f] 18 [a [d J b e J c] f] + 6iJ [a [d J b e J c] f] : (C.11) and covariant derivatives of the Sasaki-Einstein forms that are given by D a # b = J ab ; D a J bc =2 g a[b # c] ; D a bcd = 4i # [a bcd] : (C.12) Identities (3.8) follow from (C.12) by antisymmetrization. D Some harmonics on V 5;2 The classication of supermultiplets in the KK spectrum on V 5;2 given in Tables 2-6 in [52] does not include any long Z-vector supermultiplet. However, the discussion in section 2 in that paper suggests that some vector multiplets might be missing from the classication. In this appendix, we use standard group theory methods (see, e.g., [49]) to list all harmonics on V 5;2 that transform in 5 of SO(5). This allows us to determine unambigously that there must be a long Z-vector supermultiplet in the KK spectrum consistent with the explicit construction in section 5.2.5. We refer the reader to [52] and the references therein for the group theoretic set-up of the harmonic analysis on this space. The V 5;2 manifold is a G=H coset space, V 5;2 = SO(5) U(1) SU(2) U(1) ; (D.1) where the embeding of H in G is dened by the branching rule 5 Q ! 3 Q + 1 Q+1 + 1 Q1 : (D.2) 107 It then follows that the embedding of H into the tangent SO(7) group is given by 1 ! 1 0 ; (D.3) 7 ! 3 1 + 3 1 + 1 0 ; (D.4) 8 ! 3 1=2 + 3 1=2 + 1 3=2 + 1 3=2 : (D.5) This shows that the embedding is through the chain SU(2) U(1) SU(3) U(1) SU(4) SO(7); (D.6) where SU(2) SU(3) is the maximal embedding. The other two embeddings are regular, except that the normalization of the U(1) charge is half the conventional one [80]. In addition to (D.4), we also need the branchings of 21, 35 and 27 of SO(7), which determine the two-form, the three-form and the symmetric tensor harmonics, respectively, 21 ! 1 0 + 3 2 + 3 1 + 3 0 + 3 1 + 3 2 + 5 0 ; 35 ! 1 3 + 1 1 + 1 0 + 1 1 + 1 3 + 3 2 + 3 1 + 3 0 + 3 1 + 3 2 + 5 1 + 5 0 + 5 1 ; 27 ! 1 2 + 1 0 + 1 2 + 3 1 + 3 0 + 3 1 + 5 2 + 5 0 + 5 2 : (D.7) We recall that each independent harmonic is completely specied by itsGH represen- tation. It follows from (D.2) that only representations 3 q and 1 q in the branchings (D.3), (D.4) and (D.7) give rise to harmonics in 5 Q of SO(5) U(1) R . Specically, each 3 q yields a single harmonic, (5 q ; 3 q ), while each 1 q yields two harmonics, (5 q1 ; 1 q ) and (5 q+1 ; 1 q ). After compiling the list of all harmonics, one must identify the longitudinal ones, which do not give rise to four-dimensional elds in the KK expansion. This can be done by looking at the representation labels of the harmonics. For example, there are two scalar 108 Q = 4 3 2 1 0 1 2 3 4 h sg + sg Z sg sg + Z sg sg + A sg + sg W W + W W + W , H W + Z, Z W W + , H W Z W + Z W Z W + W S H H Table 3: TheN = 2 supermultiplets on V 5;2 in 5 of SO(5). harmonics in (5 1 ; 1 0 ) and (5 1 ; 1 0 ), and four vector harmonics in (5 1 ; 3 1 ), (5 1 ; 3 1 ), (5 1 ; 1 0 ) and (5 1 ; 1 0 ). The last two are in the same representations as the scalar harmonics and are longitudinal. Indeed, the scalar harmonics are the functions z i and z i , respectively, and the corresponding longitudinal vector harmonics are dz i andd z i . The remaining two transverse vector harmonics are obtained from z i z j d z j and z i z j dz j . The same procedure is used to count the two-form, the three-form, and the symmetric tensor longitudinal harmonics. Using KK expansions in [56] (see also [66] for a succinct summary), it is then straightfor- ward to identify the four dimensional elds corresponding to the transverse harmonics and arrange them intoN = 2 supermultiplets, whose eld content is given, e.g., in Tables 1-9 in 109 [50]. The result is summarized in Table 3, where the rst column lists the four-dimensional elds. The remaining columns are labelled by theU(1) charges of theR-symmetry subgroup of G. The R-charge in [50] is y 0 = 2Q=3. Each entry in those columns corresponds to a transverse harmonic in the 5 Q representation of SO(5) U(1) R , with the symbol indicating theN = 2 supermultiplet that the corresponding four-dimensional eld belongs to: sg { short graviton multiplets, Z { a long Z-vector multiplet, W { long W -vector multiplets, and H { a hypermultiplet. E Volume of V 5;2 The metric is ds 2 =ds 2 (KE) + [d + 3 8 cos()( 3 + d) + 3 8 cos( +)( 3 d)] 2 ; (E.1) where ds 2 (KE) = 3 8 [d 2 + cos 2 2 1 + d 2 + cos 2 2 2 + 1 2 sin 2 ()( 3 +d) 2 + 1 2 sin 2 ( +)( 3 d) 2 ]: (E.2) The j are SO(3) left-invariant forms. 1 = cos d + sin sind 2 = sin d cos sind 3 = d + cosd; (E.3) where 0; 0 2; 0 2: (E.4) 110 The ranges of the other angles in the metric are 0<; 2 ;< 2 ; 0 < 3 8 : (E.5) We nd that the determinant of this metric is Detg = 27 512 2 sin 2 cos 2 cos 2 sin 2 () sin 2 ( +); (E.6) so that (Detg) 1=2 = 27 512 sin cos cos sin() sin( +): (E.7) Note that the quantity above will have both positive and negative (and 0) values in the coordinate patch. So in computing the volume of the space, the absolute value of it must be used. The volume of the space is Vol = Z j(Detg) 1=2 jdd dd ddd = Z dd dd Z (Detg) 1=2 ddd = 3 2 4 Z j(Detg) 1=2 jddd = 81 1024 4 Z sind Z cos cosj sin() sin( +)jdd = 81 512 4 Z cos cosj sin() sin( +)jdd: (E.8) The integrals are over the ranges of the coordinates given in (E.4) and (E.5). To do the last integral it is convenient to re-write the expression inside the absolute value as sin() sin( +) = (cos + cos)(cos cos); (E.9) 111 and let f(;) = cos cos(cos + cos)(cos cos): (E.10) Then the volume is given by Vol = 81 512 4 Z jf(;)jdd: (E.11) Computing this integral is a little tedious but straightforward. One must determine the values of and for which f is positive and for which it is negative. Since and are both in the interval [ 2 ; 2 ], we have f(;)> 0; jj<<jj < 0; 2 <<jj; jj<< 2 ; (E.12) so that jf(;)j = f(;); jj<<jj = f(;); 2 <<jj; jj<< 2 : (E.13) Therefore, Z jf(;)jdd = Z + f Z f; (E.14) where the rst integral is over the region where f is positive and the second integral is over the region where f is negative. One can see that Z + f = Z 2 2 d Z jj jj f(;)d; (E.15) and that Z f = Z 2 2 d Z jj 2 + Z 2 jj ! f(;)d: (E.16) 112 These double integrals are readily computed by Mathematica. We nd them to be Z + f = 2 3 Z f = 2 3 : (E.17) So the volume is found to be Vol = 27 128 4 : (E.18) This value for the volume is in agreement with what is calculated in [76]. F Linearized bosonic eld equations of D = 11 super- gravity (I) The bosonic eld equations of d=11 supergravity The bosonic sector ofd = 11 supergravity consists of a metricg AB and a 3-form potential A ABC . The exterior derivative of the 3-form potential gives a 4-form uxF ABCD . Classically these elds must satisfy thed = 11 supergravity bosonic eld equations. These eld equations consist of an Einstein equation, a Maxwell equation, and the Bianchi identity for F ABCD . The Einstein equation is R AB = 1 3 F ACDE F CDE B 1 36 g AB F CDEF F CDEF ; (F.1) the Maxwell equation is r A F ABCD = 1 576 BCDEFGHIJKL F EFGH F IJKL ; (F.2) 113 and the Bianchi identity is r [A F BCDE] = 0: (F.3) Let g AB and F ABCD be a solution to the eld equations. We take this solution to be the `background solution'. We would like to perturb the background elds, g AB and F ABCD , in such a way that the perturbed elds still satisfy the equations of motion. Let h AB and f MNPQ be the perturbations to the metric and ux, respectively. The perturbed elds are then g AB = g AB +h AB F ABCD = F ABCD +f MNPQ : (F.4) We would like to put these perturbed elds into the equations of motion and determine the equations the perturbations h AB and f ABCD must satisfy in order for g AB andF ABCD to be solutions. The equations h AB andf ABCD must satisfy to rst order are the `linearized eld equations'. (II) Linearizing the Einstein equation LetR AB be the Ricci tensor obtained from the perturbed metric g AB . Then the Einstein equation is R AB = 1 3 F ACDE F CDE B 1 36 g AB F CDEF F CDEF : (F.5) We want to expand each of the terms to rst order in the perturbations and obtain the linearized Einstein equation. (II.1) LinearizingR AB ExpandingR AB to rst order gives R AB =R AB +R AB ; (F.6) 114 where R AB = 1 2 ^ h AB + r (A r C h B)C 1 2 r A r B h C C : (F.7) r A is the covariant derivative for the background metric. ^ h AB is called the `Lichnerowicz operator', and its action on h AB is ^ h AB = r C r C h AB 2R ACBD h CD + 2R C (A h B)C : (F.8) (II.2) LinearizingF ACDE F CDE B ExpandingF ACDE F CDE B gives F ACDE F CDE B = g MC g ND g PE F ACDE F BMNP = (g MC h MC )::: (F BMNP +f BMNP ) = F ACDE F CDE B +O(h) +O(f); (F.9) where O(h) = (h MC g ND g PE +g MC h ND g PE +g MC g ND h PE )F ACDE F BMNP O(f) = g MC g ND g PE F ACDE f BMNP +g MC g ND g PE f ACDE F BMNP : (F.10) After some straightforward manipulations O(h) = 3F CNP A F M B NP h MC O(f) = 2F MNP (A f B)MNP : (F.11) So F ACDE F CDE B =F ACDE F CDE B 3F CNP A F M B NP h MC + 2F MNP (A f B)MNP : (F.12) (II.3) Linearizing g AB F CDEF F CDEF 115 Expanding g AB F CDEF F CDEF gives g AB F CDEF F CDEF = g AB g CM g DN g EP g FQ F CDEF F MNPQ = (g AB +h AB )(g CM h CM ) (F CDEF +f CDEF )(F MNPQ +f MNPQ ) = g AB F CDEF F CDEF +O(h) +O(f): (F.13) O(h) is the part that is rst order in the h AB . It is O(h) = h AB F CDEF F CDEF g AB (h CM g DN g EP g FQ +g CM h DN g EP g FQ )F CDEF F MNPQ g AB (g CM g DN h EP g FQ +g CM g DN g EP h FQ )F CDEF F MNPQ : (F.14) After some straightforward manipulation O(h) =h AB F CDEF F CDEF 4g AB h CM F CDEF F DEF M : (F.15) O(f) is the part that is rst order in the f ABCD . It is straighforward to obtain that O(f) = 2g AB F MNPQ f MNPQ : (F.16) So g AB F CDEF F CDEF = g AB F CDEF F CDEF +h AB F CDEF F CDEF 4g AB h CM F CDEF F DEF M + 2g AB F MNPQ f MNPQ : (F.17) (II.4) The linearized Einstein equation Putting together equations (F.5), (F.6), (F.7), (F.12), and (F.17) gives the linearlized 116 Einstein equation. It is 1 2 ^ h AB + r (A r C h B)C 1 2 r A r B h C C = F CNP A F M B NP h MC 1 36 h AB F CDEF F CDEF + 1 9 g AB h CM F CDEF F DEF M + 2 3 F MNP (A f B)MNP 1 18 g AB F MNPQ f MNPQ (F.18) (III) Linearizing the Maxwell equation The Maxwell equation is r A F ABCD = 1 576 BCDEFGHIJKL F EFGH F IJKL : (F.19) We want to expand each of the terms to rst order in the perturbations and obtain the linearized Maxwell equation. (III.1) Linearizingr A F ABCD Expandingr A F ABCD gives r A F ABCD = g AM g BN g CP g DQ r A F MNPQ = (g AM h AM )::: (g DQ h DQ )r A (F MNPQ +f MNPQ ) = r A F ABCD +O(f) +O(h) +O(@h): (F.20) O(f) is the term that results from varyingF,O(h) is the term that results from varying the 4 inverse metrics g AM , and O(@h) is the term that results from varying the Christoel symbols in the covariant derivativer A . First, we obtain O(f). It is straightforward to see that O(f) = r A f ABCD : (F.21) 117 Next, we obtain O(h). After shuing terms around it is possible to obtain O(h) in a nice, compact form. O(h) = (h AM g BN g CP g DQ +g AM h BN g CP g DQ +g AM g BN h CP g DQ +g AM g BN g CP h DQ ) r A F MNPQ = h AM r A F M BCD h BN r A F A N CD h CP r A F AB P D h DQ r A F ABC Q = 4h M[A r A F BCD] M = 4 r A F M[ABC h D] M + 4( r A h M[A )F M BCD] = 4 r A F M[ABC h D] M F BCDM r A h A M + 3F AM[BC r A h D] M (F.22) So O(h) = 4 r A F M[ABC h D] M F BCDM r A h A M + 3F AM[BC r A h D] M : (F.23) Now, we want to obtain O(@h). This term arises from varying the Christoel symbols. R AM = 1 2 g RS (@ A g MS +@ M g AS @ S g AM ) = 1 2 (g RS h RS )[@ A (g MS +h MS ) +@ M (g AS +h AS )@ S (g AM +h AM ] = R AM + R AM ; (F.24) where R AM is the Christoel symbol for the background metric and R AM = 1 2 h RS (@ A g MS +@ M g AS @ S g AM ) + 1 2 g RS (@ A h MS +@ M h AS @ S h AM ): (F.25) It is possible to express the rst term of R AM in terms of K AM . Doing so gives R AM =h RS g SK K AM + 1 2 g RS (@ A h MS +@ M h AS @ S h AM ): (F.26) Furthermore, using the fact that r A h MS =@ A h MS K AM h KS K AS h MK ; (F.27) 118 it is possible to show that R AM = 1 2 ( r A h M R + r M h A R r R h AM ): (F.28) The covariant derivative of F is r A F MNPQ = @ A F MNPQ + 4 R A[M F NPQ]R = r A F MNPQ + 4 R A[M F NPQ]R ; (F.29) so O(@h) = 4g AM g BN g CP g DQ R A[M F NPQ]R = 2 h ( r A h R [A )F BCD]R + ( r [A h AR )F BCD]R ( r R h A [A )F BCD]R i : (F.30) Expanding the antisymmetrizations and simplifying gives O(@h) =F BCDR r A h A R 3F AR[BC r A h R D] 1 2 F BCDR r R h A A : (F.31) Finally, we put the parts together to obtain r A F ABCD = r A F ABCD + r A f ABCD + 4 r A (F M[ABC h D] M ) 1 2 F BCDR r R h A A (F.32) (III.2) Linearizing BCDEFGHIJKL F EFGH F IJKL Expanding the right hand side of the Maxwell equation gives BCDEFGHIJKL F EFGH F IJKL = (g) 1=2 ~ BCDEFGHIJKL (F EFGH +f EFGH )(F IJKL +f IJKL ) = BCDEFGHIJKL F EFGH F IJKL +O(f) +O(h); (F.33) where g is the determinant of the metric, O(f) is the term that arises from varyingF, and O(h) is the term that arises from varying the determinant of the metric g. 119 It is straightforward to see that O(f) = 2 BCDEFGHIJKL F EFGH f IJKL : (F.34) To get O(h) one needs to use the fact that, to rst order, det(g AB ) =det(g AB ) +det(g AB )Tr(g 1 h); (F.35) where g 1 h is the matrix multiplication of the inverse metric and the metric perturbation. So (g) 1=2 = (ggTr(g 1 h)) 1=2 = (g) 1=2 (1 +Tr(g 1 h)) 1=2 = (g) 1=2 (1 1 2 Tr(g 1 h)) = (g) 1=2 1 2 (g) 1=2 Tr(g 1 h): (F.36) This gives O(h) = 1 2 Tr(g 1 h) BCDEFGHIJKL F EFGH F IJKL : (F.37) (III.3) The linearized Maxwell equation Putting together equations (F.19), (F.32), (F.33), (F.34), and (F.37) gives the linearized Maxwell equation. It is r A f ABCD + 4 r A (F M[ABC h D] M ) 1 2 F BCDR r R h A A = 1 288 BCDEFGHIJKL F EFGH f IJKL 1 1152 Tr(g 1 h) BCDEFGHIJKL F EFGH F IJKL (F.38) (IV) The linearized Bianchi identity Expanding the Bianchi identity gives r [A F BCDE] =r [A F BCDE] +r [A f BCDE] = 0: (F.39) 120 So the linearized Bianchi identity is r [A f BCDE] = 0: (F.40) Note that there is no need to consider the variation of the covariant derivative because the Christoel symbols vanish when taking an exterior derivative. (V) Summary So to summarize, the linearized eld equations are the linearized Einstein equation 1 2 ^ h AB + r (A r C h B)C 1 2 r A r B h C C = F CNP A F M B NP h MC 1 36 h AB F CDEF F CDEF + 1 9 g AB h CM F CDEF F DEF M + 2 3 F MNP (A f B)MNP 1 18 g AB F MNPQ f MNPQ (F.41) the linearized Maxwell equation r A f ABCD + 4 r A (F M[ABC h D] M ) 1 2 F BCDR r R h A A = 1 288 BCDEFGHIJKL F EFGH f IJKL 1 1152 Tr(g 1 h) BCDEFGHIJKL F EFGH F IJKL (F.42) and the linearized Bianchi identity r [A f BCDE] = 0: (F.43) G Conventions of [56] and [50] In this appendix we translate the denition of the scalar mass used in [56] and [50] into the denition of it used here. We also clarify the `Lichnerowicz-like' operator used in [56] and demonstrate that this paper agrees with our result that L = 2 + 4. 121 AdS Klein-Gordan equation and scalar eld mass used here Here the Klein-Gordan equation for a scalar eld in AdS space is taken to be 2 = L 2 : (G.1) 2 is the Laplacian in AdS with the metric sign convention ( + ++), and L 2 is the AdS radius squared. = m 2 L 2 is regarded as the dimensionless mass, and L 2 is regarded as the mass. So the mass is the eigenvalue of the scalar eld under the Laplacian, and the dimensionless mass is obtained from the mass by multiplying by L 2 . AdS Klein-Gordan equation and scalar eld mass in [56] and [50] In [56] and [50] the Klein-Gordan equation for a scalar eld in AdS space is taken to be (equations (3.22a) and (3.22b) of [56]) (2 f 32) =m 2 f : (G.2) 2 f is the Laplacian in AdS with the metric sign convention (+). In [56] and [50] m 2 f is regarded as the mass. This Klein-Gordan equation is obtained from the one derived in [49], which is (2 f + 1 3 R) =m 2 f ; (G.3) where R is the Ricci scalar of AdS. It is important to note that some authors have a denominator of 6 instead of 3 in the Ricci scalar term in the Klein-Gordan equation, see e.g. [6]. In the Klein-Gordan equation above the denominator is 3 because the authors dene their Riemann tensor so that it is 1 2 of what it is traditionally ((A.1.28) of [7]). 122 Using this convention, the Ricci tensor for AdS in [72] is given to be R a c = 3 2 a c : (G.4) Comparing with (G.2) we see that in [56] and [50] the size of AdS is xed so that =16. The mass here in terms of the mass of [56] and [50] Here the traditional denition of the Riemann tensor is used, and the Ricci tensor is R a c = 3 L 2 a c : (G.5) Setting the the right-hand-side of (G.4) to 1 2 the right-hand-side of (G.5) gives = 1 L 2 ; (G.6) so that the square of the AdS radius in [56] and [50] is L 2 = 1 16 : (G.7) We would like to have the mass used here in terms the mass of [56] and [50]. To do this we note that for a given AdS radius2 f =2 p because [56] and [50] uses the opposite metric sign convention used here. So the Klein-Gordan equation of [56] and [50] becomes (2 32) =m 2 f ; (G.8) which with further massaging becomes 2 = (m 2 f 32): (G.9) Comparing with (G.1), and setting L 2 = 1 16 , gives = 1 16 (m 2 f 32): (G.10) 123 D'Auria and Fre's `Lichnerowicz-like' operator In [56] the authors use what they call the `Lichnerowicz-like' operator on symmetric tensors. In equation (2.11e) of that reference they give it to be M (2)(0) 2Y () = h (2 + 40) () () 4C i Y () : (G.11) The tensor C is the Weyl tensor on the internal space, which is Einstein. In equation (2.9a) of [56] give it to be C =R 4e 2 ; (G.12) where the rst term is the Riemann tensor and the second term is the antisymmetrized product of 's, i.e. = [ ] : (G.13) Acting with the antisymmetrized 's on the symmetric tensor Y gives Y () = 1 4 ( )(Y +Y ) = 1 4 (Y +Y ) = 1 2 Y () : (G.14) To get the second equality it is assumed that the symmetric tensor Y is traceless. Therefore acting with the Weyl tensor on Y gives C Y () =R Y () + 2e 2 Y () ; (G.15) and putting this, with e = 1, into the action of the Lichnerowicz-like operator given in equation (G.11) gives M (2)(0) 2Y () =2Y () 4R Y () + 32Y () : (G.16) 124 It is important to note that the above is not the usual Lichnerowicz operator, which is given in equation (V.4.111e) of [49], but in the case of an Einstein space diers from it by a constant. The usual Lichnerowicz operator does not appear to be given in [56], but it is given in equation (V.4.111e) of [49] to be L Y (ab) = h (2 + 48) (de) (ab) 4R de ab i Y (de) : (G.17) (N.B. In equation (V.4.109) of the same publication, i.e. reference [49], the authors give the same operator as above, but with the opposite sign in front of the Riemann tensor. The operator above seems to be the correct one.) Comparing equations (G.16) and (G.17) gives that L =M (2)(0) 2 + 16; (G.18) where the operator on the left-hand-side is the usual Lichnerowicz operator and the operator on the right-hand-side is what the authors of [56] call the `Lichnerowicz-like' operator. From equations (4.27), (4.71), (3.23b), and (3.23g) of [56] one obtains M (2)(0) 2 =M (1) 2 (0) ; (G.19) where the operator on the right-hand-side is the Hodge-de-Rham operator on 2-forms. This relation together with equation (G.18) gives L =M (1) 2 (0) + 16; (G.20) where the operator on the left-hand-side is the usual Lichnerowitz operator given in [49] and the operator on the right-hand-side is the Hodge-de-Rham operator on 2-forms. 125 Equations (2.11c) of [56] and (V.4.111c) of [49] both give M (1) 2 (0) Y [] = 3D D [ Y ] : (G.21) Equation (G.20) is for the case when the size of the space is such that R = 24g , for the usual denition of the Riemann tensor. In the case when the size of the space is such that R = 6g the dierential operators get scaled down by 1 4 . Hence, in the case that size of the space is such that R = 6g , the relation that is equation (G.20) becomes L =M (1) 2 (0) + 4; (G.22) where the operator on the left-hand-side is the usual Lichnerowitz operator given in [56] and the operator on the right-hand-side is the Hodge-de-Rham operator given in [56] and [49]. H Toric homogeneous Sasaki-Einstein manifolds via K ahler quotient The K ahler quotient provides a straightforward way to construct a K ahler manifold from a higher-dimensional one. The higher-dimensional K ahler manifold is taken to be a simpler space, e.g. typicallyC n . As a result, objects in the constructed lower-dimensional space, e.g. the metric and K ahler 2-form, can be more easily described in terms of those in this simpler higher-dimensional space. In this appendix, we explain the K ahler and hyper-K ahler quotients, and obtain the toric homogeneous Sasaki-Einstein manifolds, M 3;2 , Q 1;1;1 , and N 1;1 in terms of them. 126 For more details on the K ahler and hyper-K ahler quotients, see [87], [88], [89], and [90]. K ahler quotient Suppose a Lie group G acts on a symplectic manifold M. The basis elements of the Lie algebra g are vector elds V a on M, where the index a runs from 1 to Dim(g). Each V a is a vector eld that can in turn be written in terms of the @=@x i , where the x i are local coordinates onM. These vector eldsV a can be regarded as Hamiltonian vector elds that generate Hamiltonian phase ows on the manifold M. In other words, a vector eld V a on M, which is a basis element of g, gives rise to a Hamiltonian a on M given by d a =i V a!; (H.1) where! is the symplectic form onM. The a are components of an object, which is called a moment map, and is also known as a momentum map. The moment map is to be regarded as a map from M to the dual of the Lie algebra, i.e. : M! g . In other words, is to be regarded as a 1-form on M. can be written as = a V a , where V a is the 1-form dual to the vector eld V a , so that V a (V b ) = b a . Given any element = a V a of g one has that dh;i =d( a a ) =i !; (H.2) and soh;i can be seen as a Hamiltonian on M with corresponding Hamiltonian vector eld . In the case of Euclidean three-dimensional conguration space, the phase space, i.e. the space of positions together with momenta, is six-dimensional. In this case, M = C 3 . When the group of symmetries is taken to be the group of spatial rotations about the origin, the 127 components of the moment map are the values of the angular momentum, and when the group of symmetries is taken to be spatial translations, the components of the moment map are the values of the momentum [87]. Hence the name `moment map' or `momentum map'. Given an element in the dual of the Lie algebrap2 g , one can consider the set of points in M dened by M p = 1 (p), which is the set of all the points in M that map under the moment map to the dual Lie algebra element p. Such a set of points is called a level set. In general, a group action will move a point in a level set into another level set, but it is shown in [87] that M p is xed under the action of the subgroup G p of G consisting of those elements g2G such that Ad g p =p. In the case that p = 0 one of course has that G p is the entire group G. Since M p is xed under G p one can mod out the action of G p on M p and consider the space ofG p -orbits ofM p . This quotient space is called, e.g. in [87], a reduced phase space. In the case that M is a K ahler manifold, the K ahler 2-form is the symplectic form and the quotient is called a K ahler quotient. The simplest example of the K ahler quotient construction is when the starting manifold is M = C n , the symmetry group is G = U(1) r , and the level set is taken for the dual Lie algebra elementp = 0. The subgroup that xes the level set is of courseG p =G. The action of the group G is given by z i !e iaQ a i z i ; (H.3) where = ( 1 ;:::; r ) is an element of the Lie algebrau(1) r =R r . To obtain the Lie algebra 128 vector elds, consider a scalar function F (z i ; z i ) on M. Under the group action, one has F (z i ; z i ) ! F (e iaQ a i z i ;e iaQ a i z i ) ' F ((1 +i a Q a i )z i ; (1i a Q a i ) z i ) ' F (z i ; z i ) + n X i=1 (i a Q a i z i @F @z i i a Q a i z i @F @ z i ): (H.4) From this one can see that the Lie algebra vectors are V a = @ @ a =i n X i=1 Q a i (z i @ @z i z i @ @ z i ): (H.5) Inserting this vector into the equation d a =i V a!; with K ahler form ! =i X i dz i ^d z i ; (H.6) one can solve for the moment map. The right hand side of the equation gives i V a! = i X i dz i (V a )d z i dz i d z i (V a ) = X i Q a i (z i d z i + z i dz i ) = X i Q a i d(z i z i ): (H.7) One can then see that the moment map is given by a = X i Q a i jz i j 2 t a ; (H.8) where the t a are integration constants. Setting the integration constants equal to 0 and restricting to the level set corresponding to the 0 element of g gives the set of points dened 129 by the r algebraic equations (a = 1;:::;r) X i Q a i jz i j 2 = 0: (H.9) The space of orbits obtained by further quotienting out by the group action z i !e iaQ a i z i is the K ahler quotient. An important fact is that if the charges satisfy the condition P i Q a i = 0 for each a, then the resulting space is a toric Calabi-Yau manifold. M 3;2 and Q 1;1;1 via K ahler quotients If one starts with the spaceM =C 5 parameterized by the complex coordinates (u 1 ;u 2 ;u 3 ;v 1 ;v 2 ), and takes the K ahler quotient by U(1), with charge 2 for the u i and charge3 for the v i , u i v i U(1) charges 2 3 then one obtains the space dened by the equation 2(ju 1 j 2 +ju 2 j 2 +ju 3 j 2 ) = 3(jv 1 j 2 +jv 1 j 2 ); (H.10) with the coordinates identied according to the U(1) action (u i ;v i )! (e 2i u i ;e 3i v i ). This space is in fact the Calabi-Yau cone over the homogeneous Sasaki-Einstein manifold M 3;2 . M 3;2 is obtained by further restricting to a xed radius in the cone, which is achieved by setting 2(ju 1 j 2 +ju 2 j 2 +ju 3 j 2 ) = 3(jv 1 j 2 +jv 1 j 2 ) = 1: (H.11) The homogeneous Sasaki-Einstein manifold Q 1;1;1 can also be obtained as a K ahler quo- tient. If one starts with the space M = C 6 parameterized by the complex coordinates (a 1 ;a 2 ;b 1 ;b 2 ;c 1 ;c 2 ), and takes the K ahler quotient by U(1) 2 with charges as given in the table, 130 a i b i c i U(1) 1 charges 1 0 1 U(1) 2 charges 0 1 1 then one obtains the space dened by the equations ja 1 j 2 +ja 2 j 2 =jb 1 j 2 +jb 2 j 2 =jc 1 j 2 +jc 2 j 2 ; (H.12) with the coordinates identied according to the U(1) 2 action (a i ;b i ;c i )! (e i 1 a i ;e i 2 b i ;e i 1 i 2 c i ). This space is in fact the cone over the homogeneous Sasaki-Einstein manifoldQ 1;1;1 . Q 1;1;1 is obtained by further restricting to a xed radius in the cone, which is achieved by setting ja 1 j 2 +ja 2 j 2 =jb 1 j 2 +jb 2 j 2 =jc 1 j 2 +jc 2 j 2 = 1: (H.13) Hyper-K ahler quotient Whereas a K ahler manifold looks locally like C n , a hyper-K ahler manifold looks locally likeH n , whereH is the space of quaternionsq =a+ib+jc+kd,a;b;c;d2R. The imaginary unit i in C gives rise in K ahler manifolds to the complex structure J, and analogously, in hyper-K ahler manifolds the units i, j, and k in H give rise to three complex structures, I, J, and K. Practically speaking, K ahler 2-forms are obtained by lowering the upper indices on the complex structures with the metric. So a hyper-K ahler manifold has three K ahler 2-forms as well. The space of quaternions H can be seen as R 4 , so the unit quaternions can be seen as the 3-sphere S 3 . S 3 is the same as the Lie group SU(2) when considered as a manifold, and when considering the multiplication of quaternions, the unit quaternions can be identied as SU(2). 131 More concretely, a quaternion can be represented in terms of the Pauli matrices, and in terms of two complex numbers as q =q 4 1 2 +i~ ~ q = 0 B B @ u v v u 1 C C A ; (H.14) where u = q 4 +iq 3 and v = q 2 +iq 1 . The units i, j, and k are represented as the Pauli matrices, i 1 , i 2 , and i 3 , respectively. This representation of the quaternions makes it clear that the units i, j, and k transform as a triplet, i.e. in the adjoint representation, under SU(2). Since the unit quaternions i, j, and k transform as a triplet under SU(2), the three complex structures and the three K ahler 2-forms do as well. The triplet of K ahler 2-forms for H, which transform in the adjoint representation of SU(2), is given by the relation i~ !~ = 1 2 dq^d q; (H.15) which more explicitly is i 0 B B @ ! 3 ! 1 i! 2 ! 1 +i! 2 ! 3 1 C C A = 0 B B @ du dv d v d u 1 C C A ^ 0 B B @ d u dv d v du 1 C C A : (H.16) (Note that the conjugate of a quaterion is q = aibjckd.) This relation gives the K ahler 2-forms to be ! 3 = i 2 (du^d u +dv^d v) ! 1 i! 2 = i(du^dv): (H.17) If there is a Lie group G that acts on a hyper-K ahler manifold M, then there is a construction, called the hyper-K ahler quotient, that gives a hyper-K ahler manifold of 132 lower dimension. In what follows, we assume that the starting hyper-K ahler manifold is M =H n , and that the Lie group is of the form G = U(1) r . In particular, G is taken to act as q i !q i e iQ a i 3 a ; (H.18) which in terms of the u i and v i is u i ! u i e iQ a i a v i ! v i e iQ a i a : (H.19) In the same way that they were derived in the K ahler quotient case, i.e. by Taylor expanding a scalar function to rst order, one can derive the Lie algebra vector elds in the hyper-K ahler case. They are found to be V a = @ @ a =i X i Q a i (u i @ @u i u i @ @ u i v i @ @v i + v i @ @ v i ): (H.20) In the K ahler case, each component of the moment map was a scalar. However, in the hyper-K ahler case there is a triplet of K ahler 2-forms, so each component of the moment map will be a triplet. In particular, the moment map is given by d~ a =i V a~ !: (H.21) Plugging the Lie algebra vector elds and the K ahler 2-forms into this equation, one can obtain the moment map in the same way it was derived in the K ahler quotient case. One nds a 3 = 1 2 X i Q a i (ju i j 2 jv i j 2 ) a 1 i a 2 = X i Q a i u i v i : (H.22) 133 As in the K ahler case, we care about the level set corresponding to the 0 element of the dual Lie algebra. This set of points is the solution to the equations (a = 1;:::;r) X i Q a i (ju i j 2 jv i j 2 ) = 0 X i Q a i u i v i = 0: (H.23) The space obtained by further quotienting out by the group action q i !q i e iQ a i 3 a is the hyper-K ahler quotient. 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Abstract (if available)

Abstract We show that a perturbative instability of the Pope-Warner solution of eleven‐dimensional supergravity will arise for any quasi‐regular Sasaki-Einstein manifold admitting a basic, primitive, transverse (1,1)-eigenform of the Hodge-de Rham Laplacian with eigenvalue within a certain range of values. The Pope-Warner solution on any homogeneous Sasaki-Einstein manifold is shown to be unstable.

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Creator Yoo, Isaiah (author)

Core Title On the instability of Pope-Warner solutions

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Physics

Publication Date 04/22/2014

Defense Date 03/10/2014

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Tag AdS/CFT,holographic superconducor,instability,OAI-PMH Harvest,Pope-Warner,Sasaki-Einstein,supergravity

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Advisor Pilch, Krzysztof (committee chair), Bars, Itzhak (committee member), Daeppen, Werner (committee member), Dappen, Werner (committee member), Johnson, Clifford V. (committee member), Malikov, Fedor (committee member)

Creator Email isaiahyo@usc.edu,iyoo218@gmail.com

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