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Supergravity solitons: topological contributions to the mass and the breaking of supersymmetry
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Supergravity solitons: topological contributions to the mass and the breaking of supersymmetry
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Content
SUPERGRAVITYSOLITONS:
TOPOLOGICAL CONTRIBUTIONS TO THE MASS AND THE BREAKING
OF SUPERSYMMETRY
by
Patrick A. Haas
______________________
A Dissertation presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In partial ful llment of the
requirements for the degree
DOCTOR OF PHILOSOPHY
(PHYSICS)
December 2016
c
Copyright 2016 by Patrick A. Haas
To my beloved mother, Karla, who made me become the person I am through her
strong and in nitely heartful support, care and love, ever since she gave me this
life.
ii
In the fabric of space and in the nature of matter, as in a great work of
art, there is, written small, the artist s signature.
Carl Sagan, Contact
iii
Acknowledgements
Many months before I made my rst step onto USCs campus, I got onto the
website of the physics department and fairly quickly noticed one particular faculty
member whose appearance resembled the one of a rock star. I was immediately
caught on his research on black holes and the quantum structure of spacetime and
knew with certainty that Nick Warner is the guy I want to work with. When I
o¢ cially started my phd program at USC, I took his general relativity class and
gured out immediately that he is, indeed, a rock star both as a scientist and
as a teacher. Needless to say, I felt blessed when he eventually accepted me as his
graduate student. His forthright slogan under me you learn to compute, which I
found to be taken literally, andmultiple furtherchallenges of independent working
helpedmetogrowsigni cantly. Ithuswouldliketospeakoutmyhonestgratitude
to my doctoral advisor, Nick Warner, not only for all the academic growth spurt
but also for his honarable human support outside physics.
My gratitude also goes out to professors Werner Däppen and Stephan Haas
who always had an open ear and open door for me, no matter for what issues or
concerns I called on them; even if it happened just out of the mood of having a
little chat in our common native language. I am glad and proud to say that both
of them have become friends of mine during my time at USC.
In all the years I have met quite a few great people among faculty, sta¤ and
students,andwouldliketoespeciallythankthehighenergyphysicsgroup,amongst
them Ignacio Jesus Araya-Quezada and Scott MacDonald.
Also from o¤-campus I have received friendly help and cooperation. I would
like to say thanks to Guillaume Bossard and Stefanos Kadmadas for brisk and
productive email exchanges, and also to Iosif Bena for hosting me at the CEA
iv
Saclay.
Averyspecialthank-youisdirectedatmyclosestphysicsbuddy,WalterVictor
Unglaub,whohasbecomeoneofmybestfriendsandwhosekindtechnicalsupport
in computer software-based accomplishments was essential for my progress.
I shall not forget to dearly thank Tricia Tahara-Stoller, an awesome lady who
wasoncemylaboratorystudent,andsincethenenrichedmylifewithamagni cent
singing voice, a lovely character, and good friendship.
The greatest of all gratitude is dedicated to my wonderful mother, Karla Haas,
who has always supported me in any way within human power and beyond with
the utmost care and love. It is for her that I am the one I am, go the way I am
going, and write these lines among many other accomplishments in the past, the
present and the future.
v
Abstract
In this thesis, we provide new insights to how mass arises from cohomology
of spacetime-solitons using Smarrs formula. We introduce the mechanism with a
historical review of its evolution to more generalized forms and move on to later
important applications in the framework of the Komar integral formalism. In
particular, this means presenting in detail its resulting forms and implications
for di¤erent solutions of supergravity in topologically distinct spacetimes. The
rst situation in which we outline that non-zero mass for smooth and horizonless
solutions can only be provided by cohomology, is an eleven-dimensional spacetime
compacti edonageneralsix-dimensional,Ricci-atmanifold;wecon rmtheresult
that there are no solitons without topology and prove the fact that Chern-Simons
terms in the mass formula only appear in order to generate a purely topological
integral. In the next case we derive the Smarr formula for a ve-dimensional
spacetime which has a magnetic boltin its center and is asymptoticallyR
1;3
S
1
. We show that supersymmetry and so the BPS-bound is broken by the
holonomy and how each topological feature of a space-like hypersurface enters the
mass formula, with emphasis on the ones that give rise to the stated violation
of the BPS-bound. In this light, we question if any violating extra-mass term in
a spacetime with such asymptotics is only evident in the ADM mass while the
Komar mass per sé triesto preserve BPS. Finally, we derive the cohomological
uxesforatwo-centersolutionofsupergravityina ve-dimensionalspacetimeand
examine in a more general fashion how the breaking of supersymmetry and so
theBPS-boundviolationisassociatedwiththistopology. Weespeciallyfocuson
the compact cycle linking the centers, and the contribution of non-vanishing bulk
terms in the mass formula to the breaking of supersymmetry.
vi
Contents
Contents vii
1 Introduction 1
2 Review of Smarr s formula 9
2.1 Smarrs original idea . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Generalizations of the Smarr formula . . . . . . . . . . . . . . . . . 12
2.3 Later applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Supergravity solitons in eleven dimensions 19
3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.1 Theeleven-dimensionalSupergravityactionandequationsof
motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.2 Invariances . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 Komar integrals in eleven-dimensional supergravity . . . . . . . . . 25
3.4 Recap of the ve-dimensional case . . . . . . . . . . . . . . . . . . . 27
3.4.1 Incorporating 1-forms. . . . . . . . . . . . . . . . . . . . . . 29
3.4.2 Dimensional reduction . . . . . . . . . . . . . . . . . . . . . 31
3.5 Summing-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
vii
4 An almost-BPS spacetime with a magnetic running bolt 35
4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2 ADM versus Komar. . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.4 Metric and equations of motion . . . . . . . . . . . . . . . . . . . . 44
4.5 The Komar integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.6 Topological data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.6.1 Deriving the uxes . . . . . . . . . . . . . . . . . . . . . . . 49
4.6.2 Intersection homology . . . . . . . . . . . . . . . . . . . . . 51
4.7 Mass, charges, and the breaking of supersymmetry . . . . . . . . . 53
4.8 Summing-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5 TopologicalcontributionstotheBPS-boundviolationina2-center
solution 56
5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.2 The 2-center solution . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.3 The Komar integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.4 Deriving the uxes . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.5 Topological data and intersection homology . . . . . . . . . . . . . 68
5.6 Mass, charges, and BPS-bound breaking from cohomology . . . . . 75
5.7 Summing-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6 Conclusion 82
A Functions and constants 86
B Asymptotic limits 88
viii
B.1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
B.2 Fields and uxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Bibliography 90
ix
Chapter 1
Introduction
The very rst concept of what was later to be called a black holegoes back to
the year 1783, in which John Michell wrote a letter to Henry Cavendish proposing
the idea of light captured by a su¢ ciently massive star [1].
In1796thesameideaaboutdarkstars,astheyusedtobecalledatthistime,
was brought up by Pierre-Simon Laplace [2, 3].
However, many scientists in England at that time treated light as part of New-
tonian physics and thus thought it to be made of particles which the gravity of
such a star would slow down and make fall back like any other kind of matter. As
of 1801, treating light as a wave invalidated Michells original idea, and the idea of
ablackholewasonlyrevisitedafter1915,whenAlbertEinsteinderivedhisgeneral
theory of relativity whose eld equations provided the appropriate mathematical
platform to describe such a phenomenon for the rst time ever [4]:
R
1
2
Rg
=kT
;
whereEinsteinsoughttoputgravity,sourcedbydynamicalmassandenergydistri-
1
butions, into a geometric context by equating the respective tensorial expressions.
(Considering the Newtonian limit, he xed the parameter on the right-hand side
tok =
8G
c
4
).
Throughout the decades thereafter, Karl Schwarzschild [5, 6] and others ex-
plored the theoretical end products of dying stars with masses above their Chan-
drasekhar limits [7, 8]. White dwarfs, for example, would undergo an enormous
explosion to a type I supernova once their Chandrasekhar limit ( 1:39M
) is
exceeded; neutron stars (Chandrasekhar limit 3:0M
), on the other hand, are
found to collapse unstoppably until all their matter would nally crunch to an
in nitely small and dense point with in nitely high curvature a singularity [9].
ThispathologicalresultofaphysicalsingularitycouldonlyshowthatEinsteins
eld equations, and hence general relativity as a whole, albeit being immensely
powerful in correctly describing the very major part of the macrocosm, must nec-
essarilybreakdownat the veryscaleof this singularity. Whateverthe truthabout
thisanomalymightbe,itwasbelievedthatnaturebasicallytendstocausallyshield
the outside-world from it by a cosmic censorshipin virtue of an event horizon
[10], where the enormously strong gravitational eld prevents any sort of matter
even light fromescaping the region of space emcompassd by this horizon, once it
got crossed. As a direct conclusion of not even light being able to ever get out of
it, the phenomenon received the name black holefor the rst time by the scien-
ti cjournalistAnnEwing[11], whicheventuallygainedfameafterJohnArchibald
Wheeler, one of the main pioneers on the very eld [12, 13], used it in one of his
lectures in 1967.
Inparticular,theeventhorizonisamathematicallimitfollowingfromthesolu-
tions of the eld equations; in the simplest special case of a static and electrically
2
neutral black hole of massm, the Schwarzschild-solution,
ds
2
=c
2
1
2Gm
c
2
r
dt
2
+
1
2Gm
c
2
r
1
dr
2
+r
2
d
2
+sin
2
d
2
;
this limit lies atr =
2Gm
c
2
, the Schwarzschild-radius.
The interest and fascination about these densest objects in the known universe
was growing and inspiring the following generations of physicists, amongst them
StephenW.HawkingfromCambridge. Besidesotherremarkableaccomplishments
onthe eld,hewasthe rsttodiscoverthepropertyofablackholetoemitradiation
(Hawking-radiation) in 1975 [14]. This was an intriguing result, since black holes
werebelievedtobeastringentone-wayroadforwhatevercrossestheeventhorizon.
It was these ndings and the considerations by Jabob Bekenstein in 1973 [15] to
assign an entropy to the event horizons surface area, that led to the concept of
blackhole thermodynamics. More precisely, Bekensteinunderstoodthat a particle
falling into a black hole must both decrease the external entropy and increase the
horizon area and concluded that, in order to avoid a violation of the second law
of thermodynamics, the areas growth must come along with an entropic increase
overcompensatingthementionedexternaldecrease. AlthoughBekensteinsoriginal
idea, inspired by Larry Smarrs mass formula for a KerrNewman-black hole from
the same year [16], of combining information theory with black hole dynamics was
intellectually outstanding, his entropy value was far o¤ the correct result later
found by Hawking, on account of the purely classical assumption of no emission.
Inparticular,Hawkingconsideredquantume¤ectsintheproximityoftheevent
horizon, that is, he made a quantum eld theoretical approach in a massive black
holes spacetime background. That led to the prediction of pair-creation of virtual
particlesfromthevacuumuctuations,whereoneparticlefallsinsideandtheother
3
managestoescapeandcarryawayenergyfromtheblackhole. Thelatterparticles
giverisetotheradiationwhichHawkingassociatedwithablack-bodytemperature
in virtue of the Boltzmann relation:
T
H
=
~c
3
8GMk
B
:
Now, within a consistent thermodynamical theory,
dS
H
=
dQ
T
H
=
dM
c
2
T
H
=
8Gk
B
c
5
~
MdM;
he received the right answer for the entropy by integration, so the Bekenstein-
Hawking entropy amounts to:
S
BH
=
4Gk
B
M
2
c
5
~
=
k
B
c
3
A
4~G
;
whereA is the surface area of the horizon.
Nevertheless, this developement highlighted some big problems.
First, there was the already known issue of the singularity, a pathological con-
sequenceofgeneralrelativitysinvaliditywhenapproachingtheblackholescentral
region.
Second, there was no room for embedding a microstate structure which would
giverisetothepredictedentropy,sinceclassicallyablackholeisuniquelyspeci ed
by its mass, angular momentum and electric charge, as required by the no-hair
theorem[17]; and this information manifestly renders the entropy to be S /
log(1) = 0.
Solvingthisinformationparadox[18,19,20]hasbeenoccupyingthemindsof
4
physicistseversince. Wideagreementprevailsthoughthatasolutioncouldonlybe
foundbysearchingforaconsistentquantumdescriptionofblackholes,whichwould
allowfortheincorporationofmicrostatestructurereectingallinfalleninformation
andthuscountinguptoexactlytherightvalueofthepredictedentropy. Themain
question thereby is: Where are the microstates located?
Quantum gravity should not become signi cant unless one is very close to the
center, where the problematic singularity is supposed to be smoothened out. So,
one rstassumedalltheinfalleninformationtobefullyconcentratedintheimme-
diatevicinityofr = 0. However,thisdirectlybroughtupthemajorissueofhowthe
thermalprocesses,whichmanifestlyhappenintheregionofthehorizon,physically
correspondwiththemicrostategeometriesinthecorethroughthehugevoidofthe
blackholesinterior. Allowingforamacroscopicallyextendedquantumscenarioin
which the empty space in between would be lled with some substrate that might
beabletoemergethemicrostategeometries,appearedtobeinconsistentaccording
to Buchdahls theorem [21].
In 2002, Samir Mathur proposed a concept within the framework of string the-
ory: Thefuzzballprogram[22,23]. Inparticular,theideaofcentrallycompressed
microstatesenclosedbyaneventhorizonwithnothingbutvaccuminbetweengets
replaced by a fuzzystructure comprised of strings and branes at the horizon
scale; and since this structure is not a classical uid, it does not disagree with
Buchdahls theorem.
It might be added that the topological issues discussed in this thesis go to the
heart of how such an object is supported against gravity.
The right value of the entropy would be precisely summed up to by properly
counting the degenerate energy states of this string-brane constellation. Conse-
5
quently, both the issue of the singularity and the information paradox might be
resolved in this way. Also, in this picture there are no horizons anymore either.
Consistency is still preserved though, since the extension of this fuzzball exactly
equalstheclassicalradiusoftheeventhorizon. Ontopofallthat,Mathurssolution
limits to the known results for spacetime outside a black hole. Another intrigu-
ing fact: The time it would take a particle to decohere into the fuzzball structure
exactly matches its classical infalling time, and escaping the fuzzball would need
precisely the Hawking-evaporation time [24].
However, there are a few issues involved with Mathurs original idea. In super-
stringtheory,spacetimehastendimensions;Mathurfactorizedthesixcompacti ed
oneslikeS
1
M
5
,wherethecirclegetswrappedbyastringandthe ve-dimensional
manifoldbyaD5-branethequantumnumbersof theirexcitations formasystem
of two-chargestates [23, 25, 26], which turned out to only work for Planck-scale
horizons. Furthermore, the solutions are limited purely to BPS. Strominger and
Vafa made the rst extension to a three-chargesystem [27]. However, they only
worked in the limit of vanishing gravitational coupling. Further development of
their states to nite gravitational coupling, in order to be able to consider black
holes, happened in [28, 29, 30, 31, 32, 33, 34, 35], in which Bena & Warner ex-
tended the results to non-BPS situations. In particular, the spectrum of possible
state con gurations within the system gives rise to the microstate geometries, and
their counting is reected by the degree of degeneracy of the energy states.
If one considers the fuzzball solution in the supergravity limit, the microstate
geometriesleadtosmooth,horizonless,andasymptoticallyatsolutionsrepresent-
ing solitons if time-independent. Supergravity solitons have long been doubted to
existthough,especiallysincesometheoremsappeartorulethemout[36,37,38,39,
6
40]. The predominant view was that there are no solitons without horizons; in
otherwords, foragloballysmoothfour-dimensional spacetime, freeof singularities
andinnerboundaries,theywereclearlyruledoutbythoseno-gotheorems,espe-
ciallysinceaSmarrformulawouldundersuchcircumstancesnevergiveanon-zero
result for the solitonic mass.
Gibbons and Warner showed that these no-go theoremscan be circumvented
through non-trivial topology on the solutions spatial hypersurfaces [41]. In order
to arrange non-trivial topology, in paricular, homological cycles, on space-like hy-
persurfaces without the need for singularities and inner boundaries, one needs at
least four spatial dimensions. It is the very cohomological uxes on those cycles
which give rise to a non-zero mass in virtue of a generalized Smarr formula within
the framework of the Komar integral formalism [42]. This will be elaborated in
more detail in the following chapters.
An additional important nding by Gibbons and Warner was the general con-
tribution of Chern-Simons interactions to solitonic solutions.
Sincethen, furtherworkhasbeendoneupontheabovestatedresults, exploring
the implications for the mass in various physical situations [44, 45, 46, 47, 48].
Anotherimportantinsightwasgivenbymyworkin[45]byshowingthattheChern-
Simons terms only play a secondary role in combining all the uxes in the integral
of themassformulatoacloseddi¤erentialformandthusensurethesolitonsmass
to be purely topological.
From looking at a strongly simpli ed example of a solution asymptotically be-
having like R
1;3
S
1
, one may assume that any extra-mass term violating the
BPS-bound in spacetimes with such asymptotics might be solely visible in the
ADM mass while the Komar mass per sé triesto preserve BPS. This is one as-
7
pectofthenon-BPSsituationconsideredbymein[47], whichgetsquestionedhere
within the topological analysis of the asymptotic mass and said extra-mass in a
spacetime which is asymptoticallyR
1;3
S
1
.
Thisthesiswillbeconcludedwithamuchmoregeneral2-centersolution. After
deriving the uxes and setting up the fomulae for mass and Maxwell-charges, an
extensive and explicit examination is done on the very topological origin of the
violation of the BPS-bound and thus the breaking of supersymmetry in the frame-
workofintersectionhomology;withspecialemphasisonthecorequestion,whether
supersymmetry-breaking could in general be cast as a pure boundary integral or
would additionally be supported by compact homology in the core of the solution.
8
Chapter 2
Review of Smarr s formula
2.1 Smarr s original idea
In1973,thejournalPhysicalReviewLetterspublishedashortpaperwhoseabstract
opens with the words: A new mass formula for Kerr black holes is deduced...
It was written by Larry Smarr, a doctoral student at the University of Texas at
Austin at that time [16].
ThecentralobjectofSmarrsworkisthechargedKerrsolutionorKerr-Newman
solution [49, 50, 51], which describes a rotating, charged black hole determined by
three parameters: Its mass, M, angular momentum, L =Ma, and charge, Q.
In summary, Smarr refers to known results of black hole physics in order to
deduce a new mass formula and points out through more detailed elaboration
that the total mass of even more general situations can be decomposed into three
energyterms, one of whichwouldbecome soonthereafterapivotal element forthe
establishment of black hole thermodynamics.
In particular, Smarr started o¤by applying Hawkings result of a black holes
surface area to never decrease [52], to the case of a rotating, charged black hole.
9
In doing so, he received an explicit formula for the surface area,A, and inverted it
to an expression for the mass as a homogeneous function of this area, the angular
momentum and the charge [53]:
M (A;L;Q) =
q
A
16
+
4L
2
A
+
Q
2
2
+
Q
4
A
: (2.1)
Since this result is obviously an all square rooted term, the desired decomposition
couldnotbeachievedwithoutsomemorework. Inanycase,(2:1)alreadysuggested
atthatpointthatfornochargeandsu¢ cientlysmallrotation,thetotalmasswould
be related with the surface area in a rather striking way.
The rst step towards the goal was to rewrite the mass in di¤erential form, for
which Smarr referred to earlier work of Christodoulou and Bekenstein [54, 55],
dM =TdA+
dL+dQ; (2.2)
in order to assign three physical invariants: E¤ective surface tension, T, angular
velocity,
, and electrostatic potential, , computed by
T =
1
M
1
32
2L
2
A
2
Q
4
2A
2
;
=
4L
MA
; =
1
M
Q
2
+
2Q
3
A
: (2.3)
Another important and insightful step was the derivation of a bilinear mass
formula from this di¤erential form,
M = 2TA+2
L+Q; (2.4)
in virtue of Eulers homogeneous function theoremapplied to (2:1) which is homo-
geneous of degree
1
2
in (A;L;Q
2
) and hence yields the coe¢ cients in (2:4).
10
This is the original Smarr formula.
Smarrdisplayeditmainlyinordertopointouttheformalresemblancetoresults
not involving black holes [56], and hence physical applicability of (2:2)(2:4) for
more general matter con gurations, had been already shown to be possible by
Bardeen, Carter, and Hawking [57] (article not yet published by that date).
Now the crucial part of Smarrs work was ready to be done. For this he had
to integrate (2:2) termwise along a speci cally chosen path in the parameter space
(A;L;Q) in order to arrive at the decomposition of the total mass into the three
energy components in question: The surface energy,E
s
, the rotational energy,E
r
,
and the electromagnetic energy, E
em
, so that
M =E
s
+E
r
+E
em
: (2.5)
In more detail:
E
s
=
1
2
q
A
4
; E
r
=
1
2
q
A
4
1
q
1
4a
2
A
1
; E
em
=
1
2
q
A
4
4Q
2
A
q
1
4a
2
A
: (2.6)
Christodoulou already showed for an uncharged Kerr black hole [53] that the
non-rotational component of the total energy de nes an irreducible mass, M
ir
,
which Smarr, extending the situation to the charged case, concluded to be equal
to his derived surface energy:
M
ir
=E
s
=
q
A
16
: (2.7)
One can see directly that this last result would be obtained from (2:1) by setting
L andQ equal to zero.
11
Note: ItwaspreciselythisirreduciblesurfaceenergywhichinspiredBekenstein
onlyshorttimelatertohisremarkableworkonablackholesentropyasameasure
foritsirreduciblemassmoreprecisely: energythatisnotextractableinvirtueof
the Penrose process and thus irreducible surface area of the horizon [15], setting
the keystone for black hole thermodynamics.
2.2 Generalizations of the Smarr formula
A very short time after the publication of Smarrs paper [16], Brandon Carter,
in collaboration with James Bardeen and Stephen Hawking, worked in this spirit
on nding a more general mass formula by means of geometry and topology of
a space-like hypersurface centering a Kerr-Newman black hole [58]. A further
very important result of their elaborations was the derivation of a generalized
mass variation formula for the purpose of enunciating the rst law of black hole
mechanics.
In particular, the deduction of the mass formula was accomplished by applying
the Komar integral formalism [42] to a general stationary axisymmtric system:
M =
1
4
I
1
k
a;b
dS
ab
; (2.8)
where time-independence leads to a time-like Killing-vector,k
a
, and the conserved
energy/mass associated with this symmetry is the Komar mass. (The indices of
the space-like 2-sphere at in nity, S
ab
, represent the normal vector s direction).
Note, that this is precisely a Gaussian integral over a source term of the eld
in question a mass source in this case; for an electric charge, one would nd the
12
known result
Q =
1
4
I
1
A
a;b
dS
ab
; (2.9)
whereA
a
is the Maxwell-potential 1-form.
For their further elaboration, they rewrote the above expression in virtue of
the generalized Stokestheorem and geometric standard identities with regard to
spaces curvature,
r
2
k
a
=R
ab
k
b
; (2.10)
to arrive at
M =
1
4
Z
R
ab
k
a
d
b
1
4
I
H
k
a;b
dS
ab
; (2.11)
where R
ab
is the Ricci-tensor and a space-like hypersurface de ning the bulk
between the central black holes horizon 2-surface (inner boundary), denoted by
H, and spaces in nity (outer boundary).
Cases like a Kerr black hole do not have a bulk contribution to their mass
formula, since they are solutions to the vacuum Einstein equations which have
R
ab
= 0. So, their mass formula collapses to the second term of eq. (2:11), that is,
the Komar integral over their horizons.
Important note: The crucial core piece, without which no non-zero result for
the mass would happen, is the inner boundary,H, being non-trivial through, for
example,-functionsinthecenteroratthehorizon, representingthemassdensity.
Following the example of Smarrs mass decomposition (2:5), Carter and his co-
authors transformed this mass expression once more in virtue of the Killing vector
in the rigidly co-rotating frame of the black hole, l
a
= k
a
+
H
m
a
, the angular
velocity,
H
, and the geometric de nition of the surface gravity of a black hole,
13
=l
a
;b
l
b
n
a
(n
a
is a null vector on the horizons surface withn
a
l
a
= 1), to arrive at
a more general form of the Smarr formula:
1
2
M =
1
8
Z
R
ab
k
a
d
b
+
H
J
H
+TA; (2.12)
whereT =
8
was denoted as a surface tension term to meet the form of (2:4);J
H
is the angular momentum of the black hole, denoted in Smarrs paper byL, whose
index Hdenotes its component at the horizon:
J
H
=
1
8
I
H
m
a;b
dS
ab
: (2.13)
Obviously there was one more essential step missing to have a complete gener-
alization of Smarr s mass formula: The incorporation of an electromagnetic eld.
Bysplittinguptheenergy-momentumtensorintoamatteranda eldpart,and
byapplyingtheMaxwell-equations,thehorizoncomponentsoftheabovequantities
could be extended by contributions from matter and elds,
J =J
H
+J
M
+J
F
withJ
M
=
Z
T
ab
M
m
a
d
b
andJ
F
=
Z
T
ab
F
m
a
d
b
; (2.14)
Q =Q
H
Z
j
a
d
a
withQ
H
=
1
4
I
H
A
a;b
dS
ab
; (2.15)
and a more general result for the mass formula was derived to:
1
2
M =
H
J
M
+
H
J +
H
Q
H
+TA+bulk integrals. (2.16)
The authors extended (2:12) such that the original Smarr formula could be repro-
duced for zero bulk contribution.
14
Like Smarr outlined in his original work, not only the mass formula itself is of
importancebutalsoitsvariationformula. Bardeen,Carter,andHawkingrederived
theirgeneralizedSmarrformulaindi¤erentialformcompletelyviavariationalprin-
ciple:
dM =
H
(dJ
H
+dJ
F
)+
H
dQ
H
+
8
dA+integral terms. (2.17)
This formula quanti es the rst law of black hole mechanics, in direct analogy
to the rst law of thermodynamics. In light of all contributing terms depending
on the horizon, the Smarr formula could be named a mass formula summing over
horizons.
2.3 Later applications
Already in his original paper, Smarr anticipated that his mass formula could be
applied to any generic matter con guration. Utilized for scores of various physical
situations since then, the major application remained to be mass determination
of black holes, variations among which have been made abundantly in terms of
spacetime dimension number, physical properties, topology and more.
A nal brief review shall be given to new insights regarding topology after this
period. As outlined in the introduction chapter, the Smarr formula is intrinsically
tied to the existence of singularities and inner boundaries on space-like hypersur-
faces in order to give a nonzero result for the mass.
Thisholdstrueforfourspacetimedimensions,asthefollowingbriefproofshows:
15
Taking one Maxwell- eld, F =dA, and its dual, G =?
4
F, it holds
dF = 0 =dG; (2.18)
since we consider electrostatics. Assuming them to have the same symmetries
as the metric, their Lie derivative with respect to a time-like Killing vector, K,
vanishes,
L
K
F = 0 =L
K
G; (2.19)
so that with Cartans formula and (2:18) follows:
d(i
K
F) = 0,i
K
F =d+H
(1)
F
andd(i
K
G) = 0,i
K
G =d+H
(1)
G
; (2.20)
where and are scalar functions.
Note: Only harmonic uxes of degree one, H
(1)
F
and H
(1)
G
, can be constructed
in four dimensions.
Without any singularities and horizons, and under the assumption of the elds
anduxestodropo¤towardsin nity,themassformulaonlyinvolvesbulkintegrals
according to (2:11):
M =
1
4G
4
Z
R
ab
K
a
d
b
=
1
4
Z
h
r
c
(F
c
b
G
c
b
)+H
(1)
F;c
F
c
b
H
(1)
G;c
G
c
b
i
d
b
;
(2.21)
wherewerestoredNewton sconstantandusedthefour-dimensionalEinsteinequa-
tions with purely electromagnetic energy-momentum tensor,
R
ab
=G
4
(F
ac
F
c
b
G
ac
G
c
b
): (2.22)
16
Non-trivial 1-forms can only be supported by spaces that are not simply con-
nected. For a three-dimensional space this would require the existence of singular-
ities or inner boundaries/horizons [36, 37, 38, 39, 40] which we ruled out.
So, the above expression for the mass becomes with Stokestheorem:
M =
1
4
Z
r
c
F
bc
G
bc
d
b
=
1
4
I
1
F
bc
G
bc
dS
bc
= 0; (2.23)
since the elds fall o¤towards in nity and there are no horizon terms as in (2:11).
Hence, smoothnessandabsenceofnon-trivialtopologynecessarilymeanzeromass
according to Smarrs formula in four dimensions.
Spacetime solitons, smooth and horizonless con gurations in space which rep-
resent the supergravity limit of fuzzballs, a string theory concept for a possible
quantum description of black holes, were considered impossible precisely because
of their incapacity of being massive objects according to Smarrs formula.
In spacetimes of 3+1 dimensions, non-trivial 1-cycles could contribute to the
mass formula, but the solutions would be singular. However, if the number of
dimensions is increased to ve or more, then, as Gibbons and Warner could show
in a work in 2013 [41], the embedment of 2-cycles can work perfectly well, and the
Smarr formula can assign nonzero mass values to supergravity solitons in virtue of
non-trivial topology.
Inparticular,theyturnedonChern-Simmonsinteractionsinthe ve-dimensional
action and made the assumption that the elds have the same symmetries as the
metric; or, more precisely, their Lie-derivative with respect to the direction of a
time-likeKilling-vector eld,vanishes. Thisgivesasmallsetofverypowerfulequa-
tionsfromwhichtheverycohomologicaluxesofthehomological2-cyclesemerge.
17
By means of the equations of motion from varying the action and the Komar in-
tegral formalism, these uxes nally nd their way into the properly normalized
Smarr formula.
A more detailed mathematical outline of the procedure Gibbons and Warner
conducted, will be obtained in the following chapters, where exactly this principle
is applied for various physical situations.
18
Chapter 3
Supergravity solitons in eleven
dimensions
3.1 Motivation
Asmentionedinealierchapters,theso-calledno-gotheoremsstatingNosolitons
without horizons, can be circumvented by allowing non-trivial topology on the
solutions spatial hypersurfaces [41]. For this to work, spacetime is required to be
at least ve-dimensional, because the non-trivial cycles need to be embedded into
an at least four-dimensional Cauchy-surface to avoid the creation of singularities
[36, 37, 38, 39, 40].
Afterthe ve-dimensionalSmarrformula[41,44]wascomputedandinterpreted,
the natural question came up whether a generalization to higher dimensions along
with various kinds of compacti cations would implicate new interesting physics.
The homological cycles would get extended accordingly and the legsof their as-
sociatedcohomologicaluxescouldbealignedindi¤erentwayswiththedirections
19
of spacetime and thus create a spectrum of possible e¤ective ux degrees in the
non-compact spacetime in which we live and measure.
Inthis chapter, theideais studiedineleven-dimensional supergravity, theclas-
sical limit of M-Theory. In particular, we examine these matters in an eleven-
dimensional spacetime, six dimensions of which are compacti ed on a Ricci-at
manifold. WealsoshowthattheChern-Simonsterms, conjecturedinearlierworks
as necessary supplement to pure topology, in fact only play a secondary role in
combining all the uxes in the integral of the mass formula to a closed di¤erential
form and thus ensure the solitons mass to be purely topological.
Hence, for the existence of solitons in supergravity the princliple No solitons
without topologystill holds, and the Chern-Simons interactions entirely support
this circumstance without adding any extra support mechanisms.
Insection2,wewriteoutthebosonicpartoftheeleven-dimensionalsupergrav-
ity action [59] allowing an arbitrary constant in front of the Chern-Simons term,
and reexamine the work in [41] under these more general circumstances.
In section 3, we set up the eleven-dimensional Komar integral, determine its
normalization based on the known result in ve-dimensions [41], and derive the
eleven-dimensional version of Smarrs formula reecting the mass of the solitonic
solutions purely by topology.
In section 4, we recap the ve-dimensional calculations from [41], also allowing
for non-trivial 1-forms and an arbitrary Chern-Simons coe¢ cient, in order to show
boththepurelytopologicalnatureofthemassandtheprocessofgettingtherefrom
the more general eleven-dimensional case by special choices of elds and compact
geometry
1
.
1
For detailed elaboration on ux compacti cation, see [60, 61, 62].
20
3.2 Preliminaries
3.2.1 Theeleven-dimensionalSupergravityactionandequa-
tions of motion
Eleven-dimensional supergravity has in its bosonic sector the graviton, g
MN
, and
the 3-form potential, C
MNK
= C
[MNK]
.
2
The latter gives rise to a eld strengh
4-form, F =dC.
The bosonic action [59] is
S
11
=
Z
d
11
x
p
g
R
1
2
jFj
2
6
Z
C^F^F; (3.1)
wherewehaveintroducedanarbitraryconstantcoe¢ cient,,fortheChern-Simons
term. Supersymmetry corresponds to = 1 but we wish to examine the impact
of the Cherm-Simons interactions in a broader, non-supersymmetric version of the
theory.
The Einstein equations resulting from (3:1) are
R
MN
1
2
g
MN
R =
1
12
F
MRSK
F
RSK
N
1
96
g
MN
F
RSKL
F
RSKL
; (3.2)
which may be written,
R
MN
=
1
12
F
MRSK
F
RSK
N
1
144
g
MN
F
RSKL
F
RSKL
: (3.3)
2
We indicate the eleven-dimensional spacetime indices with a capital latin, the non-compact
ve-dimensional spacetime with a greek, and the compact six-dimensional space with a small
latin index.
21
The Maxwell equations resulting from (3:1) are
r
N
F
N
RSK
=J
CS;
RSK
; (3.4)
with the eleven-dimensional Chern-Simons 3-form current
3
,
J
CS;
RSK
=
1152
RSKM
4
:::M
11
F
M
4
:::M
7
F
M
8
:::M
11
: (3.5)
De ne a dual 7-form,
G =?
11
F ,G
M
1
:::M
7
=
1
24
M
1
:::M
11
F
M
8
:::M
11
: (3.6)
The equation of motion forG is simply the Bianchi identity forF and vice versa,
r
R
G
M
1
:::M
6
R
=
1
24
M
1
:::M
6
RM
8
:::M
11
r
R
F
M
8
:::M
11
= 0; (3.7)
and, with (3:4),
r
[M
4
G
M
5
:::M
11
]
=
35
8
F
[M
4
:::M
7
F
M
8
:::M
11
]
,dG =
2
F^F: (3.8)
Note that
G
MS
1
:::S
6
G
NS
1
:::S
6
=150
N
[M
F
K
1
:::K
4
]
F
K
1
:::K
4
= 120F
MK
2
K
3
K
4
F
NK
2
K
3
K
4
30
N
M
F
K
1
:::K
4
F
K
1
:::K
4
; (3.9)
3
The Levi-Civita tensor for curved spacetimes is related to the Levi-Civita symbol of
Minkowski spacetime like
M1:::M11
= (g)
1
2
M1:::M11
,
M1:::M11
= (g)
1
2
M1:::M11
.
22
which allows us to rewrite (3:3) as
R
MN
=
1
18
F
MRSK
F
RSK
N
+
1
4320
G
MS
1
:::S
6
G
S
1
:::S
6
N
: (3.10)
3.2.2 Invariances
We assume that the matter elds have the symmetries of the metric, that is, they
are invariant under a Killing vector, K,
L
K
F = 0 =L
K
G; (3.11)
whereL
K
is the corresponding Lie derivative. Cartans magic formula,
L
K
! =d(i
K
!)+i
K
(d!); (3.12)
applied to the 4-formF, yields
0 =d(i
K
F),K
M
F
MNRS
= 3r
[N
RS]
+H
(3)
NRS
; (3.13)
where are the magnetostatic 2-form potentials ofG and the electrostatic 2-form
potentials of F, respectively, and H
(3)
is a closed but not exact 3-form, that is,
H
(3)
2H
3
(M
11
).
ForG we nd
0 =d(i
K
G)+i
K
(dG)
,d(i
K
G) =
d+H
(3)
^F =d
^FH
(3)
^C
,d
i
K
G+^FH
(3)
^C
= 0; (3.14)
23
and so
K
M
G
MM
1
:::M
6
= 6r
[M
1
M
2
:::M
6
]
15
[M
1
M
2
F
M
3
:::M
6
]
+20H
(3)
[M
1
M
2
M
3
C
M
4
M
5
M
6
]
+H
(6)
M
1
:::M
6
; (3.15)
where is a generic 5-form andH
(6)
2H
6
(M
11
) a closed but not exact 6-form.
From (3:13) and (3:15) follows
K
M
F
MRSK
F
RSK
N
=3r
R
SK
F
R
SKN
+H
(3)
RSK
F
RSK
N
+
384
SK
SKNM
4
:::M
11
F
M
4
:::M
7
F
M
8
:::M
11
(3.16)
K
M
G
MS
1
:::S
6
G
S
1
:::S
6
N
= 6r
S
1
S
2
:::S
6
G
S
1
:::S
6
N
15
24
NS
1
:::S
10
S
1
S
2
F
S
3
:::S
6
F
S
7
:::S
10
(3.17)
+
20H
(3)
S
1
S
2
S
3
C
S
4
S
5
S
6
+H
(6)
S
1
:::S
6
G
S
1
:::S
6
N
;
and hence, the Einstein equations (3:10) become
K
M
R
MN
=
1
720
r
R
120
SK
F
R
SKN
S
2
:::S
6
G
R
S
2
:::S
6
N
+
1
18
H
(3)
RSK
F
RSK
N
+
1
4320
20H
(3)
S
1
S
2
S
3
C
S
4
S
5
S
6
+H
(6)
S
1
:::S
6
G
S
1
:::S
6
N
: (3.18)
As in [41] the (?F^F) terms cancel out, and it is important to note that this
happens independently of the choice of the parameter, .
As we will describe below, the 1-formanalogue of the presentH
(3)
was omitted
in the analysis of [41] for the assumption of simple-connectedness of the four-
dimensional slices, .
24
3.3 Komarintegralsineleven-dimensionalsuper-
gravity
In the following we will set up the eleven-dimensional Komar integral. In order
to properly normalize the asymptotic mass, we will compactify spacetime on a
6-torus according to M
11
! M
5
T
6
. This way we can easily relate the sought
eleven-dimensional normalization factor with the known ve-dimensional one from
[41].
If K is a Killing vector, then the Komar integral [63, 64, 65, 66, 41] is
Z
@
?dK =
Z
@
(@
M
K
N
@
N
K
M
)d
MN
=2
Z
R
MN
K
M
d
N
: (3.19)
If K is timelike at in nity, we can use a coordinate with K
@
@t
, so near in nity
the 1-form is then
Kg
00
dt: (3.20)
Now we assume,
M =A
11
Z
10
d(?
11
dK) =A
11
Z
S
9
?
11
dK; (3.21)
where A
11
is the eleven-dimensional normalization factor and S
9
the 9-sphere
bounding
10
.
Compactifyingspacetimeasoutlinedabove,thelastexpressioncanberewritten
25
to
M =A
11
Z
10
d(?
5
dK^dvol
6
j
r=1
) =A
11
Z
4
T
6
d(?
5
dK)^dvol
6
j
r=1
=A
11
vol
6
Z
4
d(?
5
dK) =A
11
vol
6
Z
S
3
?
5
dK; (3.22)
wherevol
6
is the volume of theT
6
at spaces in nity.
The relation of the normalization factors can be directly concluded; in partic-
ular, with the known ve-dimensional factor,
A
5
=
3
32G
5
; (3.23)
we obtain the eleven-dimensional one:
A
11
=
A
5
vol
6
=
3
32G
5
vol
6
: (3.24)
Hence, the eleven-dimensional Komar integral is:
M =
3
32G
5
vol
6
Z
S
9
?
11
dK =
3
16G
5
vol
6
Z
10
R
MN
K
M
d
N
: (3.25)
Using (3:18) in (3:25) and assuming that the boundary terms fall o¤ su¢ ciently
fast at in nity, the conserved mass is given by
M =
3
16G
5
vol
6
Z
10
h
1
18
H
(3)
S
1
S
2
S
3
F
S
1
S
2
S
3
N
+
12
C
S
4
S
5
S
6
G
S
1
:::S
6
N
+
1
4320
H
(6)
S
1
:::S
6
G
S
1
:::S
6
N
i
d
N
: (3.26)
26
Rewritten in terms of di¤erential forms, this becomes
M =
1
32G
5
vol
6
Z
10
H
(3)
^(2GC^F)H
(6)
^F
: (3.27)
Note that the di¤erential form,
2GC^F;
is closed by virtue of (3:8). Since F, H
(3)
and H
(6)
are closed by de nition, the
integral is purely topological for all values of . This means that the contribution
of explicit Chern-Simons terms in the Komar mass formula is precisely to turn the
latter into an integral over cohomology.
It should be remarked that due to compacti cation, the harmonic uxes, es-
pecially H
(6)
, can vary their number of legsin the internal space and external
spacetime. So,theeleven-dimensionalcaseallowsforstudyingthephysicalimplica-
tions of uxes of di¤erent degrees withinvarious compacti cations. The particular
choice leading to the ve-dimensional case studied in [41], is worked out in the
following subchapter.
3.4 Recap of the ve-dimensional case
In the ve-dimensional case of [41], cohomology is of degree two the analogue of
the above H
(6)
. There is also a 1-form analogue of H
(3)
, but it was left away to
ensure simple-connectedness of space.
We are going to repeat the calculations done in [41] and in addition allow for
non-trivial1-formsandanarbitraryconstantcoe¢ cientoftheChern-Simonsterm,
27
. Finally, we will compare this to the eleven-dimensional case.
The action [32] is
S =
Z
?
5
RQ
IJ
dX
I
^?
5
dX
J
Q
IJ
F
I
^?
5
F
J
6
C
IJK
F
I
^F
J
^A
K
;
(3.28)
whereC
IJK
=j
IJK
j andX
I
,I = 1;2;3, are scalar elds arising fromreducing the
eleven-dimensional metric,
ds
2
11
=ds
2
5
+
Z
2
Z
3
Z
2
1
1
3
dx
2
5
+dx
2
6
+
Z
1
Z
3
Z
2
2
1
3
dx
2
7
+dx
2
8
+
Z
1
Z
2
Z
2
3
1
3
dx
2
9
+dx
2
10
;
(3.29)
with the reparametrization,
X
1
=
Z
2
Z
3
Z
2
1
1
3
; X
2
=
Z
1
Z
3
Z
2
2
1
3
; X
3
=
Z
1
Z
2
Z
2
3
1
3
; (3.30)
to ful ll the constraintX
1
X
2
X
3
= 1:
Moreover, there is a metric for the kinetic terms,
Q
IJ
=
1
2
diag
1
X
1
2
;
1
X
2
2
;
1
X
3
2
; (3.31)
necessary also for the dualization of the eld strength
4
,F
I
=dA
I
,
G
I
=Q
IJ
?
5
F
J
: (3.32)
The analysis done for equation (5.8) in [41] leads with an arbitrary Chern-Simons
4
We use calligraphic script at some places to avoid confusion between the ve- and eleven-
dimensional objects.
28
coe¢ cient to
dG
I
=
4
C
IJK
F
J
^F
K
: (3.33)
3.4.1 Incorporating 1-forms
Besides the argument of simple-connectedness, non-trivial 1-forms have not been
considered in [41] because their contribution was assumed to not provide new in-
teresting physics. Here we incorporate them for completeness of the further below
stated dictionary of the elds. In eleven dimensions, such 1-forms can arise from a
3-form, and so might generate new interesting terms.
Equation (5.13) of [41] can be extended to
d
i
K
F
I
= 0,K
F
I
=@
I
+H
(1)I
: (3.34)
As a consequence we get
d(i
K
G
I
) =i
K
dG
I
=
4
C
ILM
i
K
F
L
^F
M
=
2
C
ILM
d
L
+H
(1)L
^F
M
=
2
C
ILM
d
L
F
M
H
(1)L
^A
M
; (3.35)
so
K
G
I
= 2@
[
I]
2
C
ILM
L
F
M
2H
(1)L
[
A
M
]
+H
(2)
I
: (3.36)
Note, that also here we included an arbitrary constant coe¢ cient, , in front of
theChern-Simonstermforwhich,likeintheeleven-dimensionalcase, = 1means
supersymmetry.
29
It follows
K
Q
IJ
F
I
F
J
=r
Q
IJ
I
F
J
+
16
C
IJK
I
F
J
F
K
+Q
IJ
H
(1)I
F
J
(3.37)
K
Q
IJ
G
I
G
J
= 2r
Q
IJ
I
G
J
4
C
ILM
I
F
L
F
M
+Q
IJ
C
ILM
H
(1)L
A
M
+H
(2)
I
G
J
; (3.38)
and hence
K
R
=
1
3
r
2Q
IJ
I
F
J
+Q
IJ
I
G
J
+
2
3
Q
IJ
H
(1)I
F
J
+
1
6
Q
IJ
C
ILM
H
(1)L
A
M
+H
(2)
I
G
J
: (3.39)
Excluding inner boundaries, the Komar mass formula becomes
16G
5
3
M =
Z
4
R
K
d
(3.40)
=
Z
4
h
2
3
H
(1)
I
F
I
+
1
6
Q
IJ
C
ILM
H
(1)L
A
M
+H
(2)
I
G
J
i
d
:
The generalized version of Smarr s formula given in [41] is now
M =
1
32G
5
Z
4
h
H
(1)
I
^
4G
I
C
IJK
A
J
^F
K
2H
(2)
I
^F
I
i
: (3.41)
Also here note that the di¤erential form,
4G
I
C
IJK
A
J
^F
K
;
is closed in virtue of (3:33). As in the eleven-dimensional generalization of (3:41),
30
eq. (3:27), the Chern-Simons contributions do only support pure topology for all
values of .
3.4.2 Dimensional reduction
The ve-dimensionalmassformulain[41]andtheoneabovecanbereproducedby
dimensional reduction of the eleven-dimensional expression (3:27).
The ve-dimensional elds embed into the eleven-dimensional ones via
C =A
1
^dx
5
^dx
6
+A
2
^dx
7
^dx
8
+A
3
^dx
9
^dx
10
(3.42)
F =F
1
^dx
5
^dx
6
+F
2
^dx
7
^dx
8
+F
3
^dx
9
^dx
10
: (3.43)
In order to express G = ?
11
F in terms of the G
I
= Q
IJ
?
5
F
J
, we go to a
representation in frames:
e
0
=Z
1
(dt+k) e
i
=
p
ii
dx
i
e
5
=
Z
2
Z
3
Z
1
1
6
dx
5
e
6
=
Z
2
Z
3
Z
1
1
6
dx
6
e
7
=
Z
1
Z
3
Z
2
1
6
dx
7
e
8
=
Z
1
Z
3
Z
2
1
6
dx
8
e
9
=
Z
1
Z
2
Z
3
1
6
dx
9
e
10
=
Z
1
Z
2
Z
3
1
6
dx
10
(3.44)
We compute explicitely the rst term of (3:43) and then nd the other two by
analogy. It holds:
?
11
e
^e
^e
5
^e
6
=?
5
(e
^e
)^e
7
^:::^e
10
; (3.45)
which can be rewritten with (3:44) to
Z
2
Z
3
Z
1
1
3
?
11
e
^e
^dx
5
^dx
6
=
Z
1
Z
3
Z
2
1
3
Z
1
Z
2
Z
3
1
3
?
5
(e
^e
)^dx
7
^:::^dx
10
:
(3.46)
31
and thus becomes with (3:30)(3:32)
?
11
F
1
^dx
5
^dx
6
=
1
X
2
1
?
5
F
1
^dx
7
^:::^dx
10
= 2Q
11
?
5
F
1
^dx
7
^:::^dx
10
(3.47)
= 2G
1
^dx
7
^:::^dx
10
:
Analogously proceeded forG
2
andG
3
, we nally achieve
G = 2G
1
^dx
7
^dx
8
^dx
9
^dx
10
+2G
2
^dx
5
^dx
6
^dx
9
^dx
10
+2G
3
^dx
5
^dx
6
^dx
7
^dx
8
: (3.48)
With(3:13),(3:15),(3:34),(3:36),(3:42),(3:43),and(3:48)we ndtherelations
between the non-trivial forms,
H
(3)
=H
(1)1
^dx
5
^dx
6
+H
(1)2
^dx
7
^dx
8
+H
(1)3
^dx
9
^dx
10
(3.49)
H
(6)
= 2H
(2)
1
^dx
7
^dx
8
^dx
9
^dx
10
+2H
(2)
2
^dx
5
^dx
6
^dx
9
^dx
10
+2H
(2)
3
^dx
5
^dx
6
^dx
7
^dx
8
; (3.50)
and the remaining forms,
(2)
=
1
dx
5
^dx
6
+
2
dx
7
^dx
8
+
3
dx
9
^dx
10
(3.51)
(5)
= 2
(1)
1
^dx
7
^dx
8
^dx
9
^dx
10
+2
(1)
2
^dx
5
^dx
6
^dx
9
^dx
10
+2
(1)
3
^dx
5
^dx
6
^dx
7
^dx
8
: (3.52)
Compactifying on a 6-torus, with dvol
6
= dx
5
^:::^dx
10
according to (3:29),
and using the above stated dictionary, the eleven-dimensional integral ammounts
32
to
Z
10
H
(3)
^(2GC^F)H
(6)
^F
=
Z
4
T
6
h
H
(1)
I
^
4G
I
C
IJK
A
J
^F
K
2H
(2)
I
^F
I
i
^dvol
6
(3.53)
=vol
6
Z
4
h
H
(1)
I
^
4G
I
C
IJK
A
J
^F
K
2H
(2)
I
^F
I
i
:
From this result we see with (3:27) and (3:41) that if = the Komar masses of
both dimensional cases are related like
M
(11)
=M
(5)
: (3.54)
3.5 Summing-up
In this chapter, a mass formula for solitonic solutions has been derived for eleven-
dimensional supergravity compactifying on a simply connected and Ricci-at 6-
manifold. Furthermore, the Chern-Simons term was given an arbitrary constant
coe¢ cient in the action to see in how far the results are inuenced by this para-
meter. Finally, a generalized version of Smarrs formula has been obtained and
the arrival back at the ve-dimensional theory explicitly shown by performing the
eleven-dimensional Komar integral overT
6
.
The most intriguingaspect of the calculations done bothin[41] andhere is the
proofofthepossibilityofcontructingmassivesolitonsolutionswithouttheneedof
horizons and so showing that the techniques used in microstate geometries are the
only methods that can support solitons. Moreover, it could be shown that making
the Chern-Simons term arbitrary does not change this fact; the incorporation of
33
Chern-Simons interactions does not yield extra pieces in the mass formula in addi-
tion to cohomology, but is only signi cant for the purely topological nature of the
soliton mass.
34
Chapter 4
An almost-BPS spacetime with a
magnetic running bolt
4.1 Motivation
It has been shown that horizonless solitonic solutions of supergravity can indeed
be constructed purely by means of nontrivial topology. Aside from the previous
chapter, the Smarr formula has been derived in mutliple works by means of the
Komar integral formalism over cohomology [41, 44, 45, 46], one important result
beingtheroleofChern-Simonstermstoonlysupportthetopologicalnatureofthe
integral.
In this chapter we examine a non-BPS solution of supergravity for spacetime
withamagneticallychargedrunningboltandcomputeallitscontributionsfrom
topology and boundary that are owing into the total mass formula, to see which
pieces precisely cause the breaking of supersymmetry by rendering M 6=
3
I=1
Q
I
;
and which, in particular, make up M =M
3
I=1
Q
I
.
35
For this purpose, we recall the ve-dimensional results of [41], but impose a
di¤erentchoiceofboundaryconditions. Weuseaspacetimewithamagneticbolt
in its center and that is asymptoticallyR
1;3
S
1
. To construct it, we de ne in the
fashion of [67] a four-dimensional Ricci-at base space which carries a Euclidean
Schwarzschild metric and magnetic ux froma oating braneansatz [72] for the
Maxwell elds.
Topologically, the spacetime can be described by entirely two homological cy-
cles: The bolt 2-sphere and a non-compact cycle extending from the center to
in nity. In this spirit, the main analysis will be done also in the framework of
intersection homology.
The supersymmetry conditions require that the curvature tensor be either self-
dual or anti-self-dual and tell how this duality has to be correlated with the one
of the magnetic parts of the Maxwell- elds. Since the rotation group in four-
dimensional space decomposes likeSO(4) =SU (2)
self-dual
SU (2)
anti-self-dual
, only
one half of the Killing-spinors would feelspaces holonomy and the other half
at space. In simple examples, this half-atness, and the preservation or breaking
of supersymmetry can be easily arranged by just changing a sign in the duality of
the elds [69, 70, 71].
However, the curvature tensor for the Euclidean Schwarzschild bolt is neither
self-dual nor anti-self-dual, so one essential BPS-condition is not ful lled; but,
because the Schwarzschild-geometry is Ricci-at, the (almost-)BPS equations of
motionarestillsatis ed[67,72],andhenceonespeaksofanalmost-BPS-solution.
This provides a ground for more general solutions.
Before computing the uxes and the Komar integral, we will briey consider
a vastly simpli ed scenario in which the angular momentum of the running bolt
36
is set to zero and the ve-dimensional warp-factor set to one. The reason for this
is to demonstrate in a very clear and quick way that in a spacetime which is not
asymptotically behaving like R
1;d
, the theorem M
Komar
= M
ADM
does in general
not hold anymore. In particular, the Komar mass vanishes under the simpli ed
conditions, while the ADM mass equals the mass parameter of the Schwarzschild
solution. The latter is responsible for the violation of the BPS-bound according to
M =Q
1
+Q
2
+Q
3
+m.
This raises a very important question: Is the extra-mass term violating the
BPS-boundalwaysthedi¤erencebetweentheADMmassandtheKomarmass,or,
are there situations in which Komar also reects terms breaking supersymmetry?
The answer will be illuminated in this chapter along with the derivation of
the Smarr formula based on the Komar-integral formalism in the sense of [41] for
the present boundary conditions and for non-BPS solutions. Further analysis is
done within intersection homology in particular, it will be examined how the
BPS-bound breaking extra-mass is composed by the period-integrals of the elds
and uxes in virtue of the homological cycles. This is to see which intersecting
components exist in the present topology and how they contribute to the breaking
of supersymmetry.
4.2 ADM versus Komar
BeforewemoveontothecomputationoftheKomarmass,itisessentialtoremark
its relation to the ADM mass and illustrate this by an easy and quick example.
On a rst note, in case of space being asymptoticallyR
1;d
, that is, no compact-
i ed dimensions, it is known that the two masses are equal in value. The presence
of the S
1
-direction in the spacetime considered here though, does in fact create a
37
di¤erence, as can be seen by looking at a strongly simpli ed version of the current
Ricci-at metric:
ds
2
5
=dt
2
+
1
2m
r
d
2
+
1
2m
r
1
dr
2
+r
2
d
2
+sin
2
d
2
: (4.1)
This metric is Ricci-at.
At rst, it is important to outline that the ADM mass is in general the more
authoritativemeasureforthegravitationalmassofasystem,sinceKomar,asstated
earlier, requires stationarity.
For a time-like Killing vector,K =
@
@t
, with dual 1-formK =g
00
dt, the Komar
integral becomes
M
Z
X
3
?
5
dK
Z
X
3
@
@r
(g
00
)?
5
(dr^dt); (4.2)
and fromg
00
=1 we can see directly that the Komar mass vanishes:
M
Komar
= 0: (4.3)
To elaborate on the mass in more detail, we consider orbits in this simpli ed
metric.
The geodesic equations at =
2
are:
dt
d
=E;
1
2m
r
d
d
=L
1
; r
2d
d
=L
2
;
(4.4)
whereE, L
1
, andL
2
are conserved quantities.
Theradialequationcanbeobtainedthroughthenormalizationconditionofthe
38
four-velocity, u
=
dx
d
,
1 =g
dx
d
dx
d
=E
2
+
L
2
2
r
2
+
1
2m
r
1
h
L
2
1
+
dr
d
2
i
; (4.5)
and so, keeping only terms of up to rst order in
1
r
at in nity,
dr
d
2
=
1
2m
r
E
2
1
L
2
1
: (4.6)
The radial acceleration at in nity is now:
a
r
=
d
2
r
d
2
=
E
2
1
q
(1
2m
r
)(E
2
1)L
2
1
m
r
2
!
E
2
1
p
E
2
L
2
1
1
m
r
2
: (4.7)
Setting o¤any rotation, L
1
=L
2
= 0, we read o¤the Keplarian mass seems to be
M
Kepler
=
p
E
2
1 m; (4.8)
whichcarriesafactorof
p
E
2
1. Inparticular,onehasa
r
= 0forE = 1;thusthis
simplecalculationdoesnotdirectlyseetheintrinsicmassofthebackground. To
resolvethisissue,wegotoa3+1descriptionintermsofgravityin3+1dimensions
to look at the ADM mass.
Dimensionally reducing the metric along the-direction, means to introduce a
conformal scale factor,
:
ds
2
5
=
1
2m
r
d
2
+
ds
2
4
: (4.9)
The goal is that ds
2
4
will be the metric apparent to observers in 3+1 dimensions.
As mentioned above, Komar and ADM mass are equal in value if asymptotics are
39
R
1;3
.
The scale factor is necessary to ensure that
1
G
5
Z
d
5
x
p
g
(5)
R
(5)
=
1
G
4
Z
d
4
x
p
g
(4)
R
(4)
+derivatives of scale factors.
(4.10)
With these scaling factors one has
g
(5)
!
1
2m
r
4
g
(4)
andR
(5)
!
1
R
(4)
; (4.11)
and hence
p
g
(5)
R
(5)
!
q
1
2m
r
p
g
(4)
R
(4)
:
Thus one must take
=
1
2m
r
1
2
: (4.12)
Without the scale factor
, the four-dimensional Newton constant would gain
radial dependence through multiplication by a power of
1
2m
r
.
Rewriting (4:1) in this sense,
ds
2
5
=
1
2m
r
d
2
+
1
2m
r
1
2
n
1
2m
r
1
2
dt
2
(4.13)
+
1
2m
r
1
2
dr
2
+
1
2m
r
r
2
d
2
+sin
2
d
2
o
;
leads to the reduced four-dimensional metric:
ds
2
4
=
1
2m
r
1
2
dt
2
+
1
2m
r
1
2
dr
2
+
1
2m
r
r
2
d
2
+sin
2
d
2
; (4.14)
40
andreadingo¤g
(4)
00
=
1
2m
r
1
2
,yieldsthestatedexpressionfortheADM-mass:
M
ADM
=m: (4.15)
ThisresultisthemassparameteroftheSchwarzschildsolution,whichprecisely
accounts for the BPS-bound breaking extra-mass term of the solution within the
current simpli cations of zero charge.
OneconcludesthattheKomarmassdoesnotdetectthebreakingofBPS/super-
symmetry while the ADM mass does.
After deriving the Komar mass and Maxwell-charges under the more general
conditions in the following, we will examine the obvious question whether this is
generally true for spacetimes asymptotically behaving likeR
1;3
S
1
, or, possibly
M
Komar
=Q
1
+Q
1
+Q
3
+M for some non-vanishing M.
4.3 Preliminaries
Sinceweareworkingin vedimensions,theactionisthesameas(3:28)with = 1.
The conditions for the scalar elds are also the same as (3:29)(3:31).
However, as opposed to the ve-dimensional recap of the last chapter, we do
not allow for cohomology of degree one, so the equations of motion and symmetry
equations correspond to the those in subchapter 3.4 for H
(1)
I
= 0 and = 1.
Hence, we only give a brief summary of the mathematical background emerging
the ve-dimensional Komar integral including all the boundary terms this time.
41
FromvaryingtheactionwereceivetheEinsteinandtheMaxwellequations[41],
R
=Q
IJ
F
I
F
J
1
6
g
F
I
F
J
+@
X
I
@
X
J
(4.16)
J
CS
I
=r
Q
IJ
F
J
; (4.17)
with the ve-dimensional Chern-Simons 1-form current,
J
CS
I
=
1
16
C
IJK
F
J
F
K
: (4.18)
In di¤erential form language, the Maxwell equations give the known identity for
the dual eld strengths,
dG
I
=
1
4
C
IJK
F
J
^F
K
: (4.19)
Eq. (4:16) can be rewritten such that the RHS is free of any trace terms,
R
=Q
IJ
2
3
F
I
F
J
+@
X
I
@
X
J
+
1
6
Q
IJ
G
I
G
J
; (4.20)
especially since this form is much more helpful for the derivation of the Komar
mass formula.
Asbefore,weassumethemetrictohaveatime-likeKillingvector,K
,andcan
hence write the ve-dimensional mass formula in terms of a Komar integral in ve
dimensions,
M =
3
32G
5
Z
X
3
?
5
dK; (4.21)
where X
3
is the 3-boundary of the ve-dimensional spacetime. Smoothness of
spatial sections,
4
, allows in virtue of properties of the Killing vector to rewrite
42
this formula as an integral over such byX
3
bound space-like hypersurfaces:
M =
3
32G
5
Z
X
3
?
5
dK =
3
16G
5
Z
4
K
R
d
: (4.22)
Assumingagainthatthematter eldshavethesymmetriesofthemetric,means
them to be invariant under the Lie-derivative along the Killing vector, K,
L
K
F = 0 =L
K
G; (4.23)
so we get, similar to (3:34)(3:36), the equations,
0 =d(i
K
F),i
K
F
I
=d
I
andi
K
G
I
=d
I
1
2
C
IJK
J
F
K
+H
(2)
I
; (4.24)
where
I
are magnetostatic potentials of theG
I
and electrostatic potentials of the
F
I
, respectively;
I
are globally de ned 1-forms and H
(2)
I
2 H
2
(M
5
) closed but
not exact 2-forms.
With (4:24) the Einstein equations (4:20) become
K
R
=
1
3
r
2Q
IJ
I
F
J
+Q
IJ
I
G
J
+
1
6
Q
IJ
H
(2)
I
G
J
: (4.25)
From this follows the Komar mass integral [63, 64, 65, 66, 41] over the spatial
hypersurface,
4
, including the boundary terms overX
3
:
1
M =
1
16G
5
Z
4
H
(2)
I
^F
I
Z
X
3
2
I
G
I
I
^F
I
: (4.26)
Sincein[41]thespacetimewasassumedtobeasymptotictoR
1;4
,theboundary
1
The here used convention divX = X ) dZ
I
=
^
r
2
Z
I
, where is the to d adjoint
exterior derivative, means for here: r
2
K
=R
K
, so the opposite sign as in [41].
43
integral was takenoverX
3
=S
3
. Forthe remainderof this chapterthe spacetimes
will be asymptotic toR
1;3
S
1
and so we haveX
3
=S
2
1
S
1
.
As we will see later, there is a gauge choice for which
I
can be made zero at
in nity. Thistogetherwiththefactthatthe
I
areexactandvanishingatin nity,
will prove the Komar mass to be an integral purely over cohomology, given by the
rst term in (4:26):
M =
1
16G
5
Z
4
H
(2)
I
^F
I
: (4.27)
The Maxwell-charge is computed like:
Q
I
=
1
vol(X
3
)
Z
X
3
G
I
=
1
32
2
m
Z
4
dG
I
=
1
128
2
m
C
IJK
Z
4
F
J
^F
K
: (4.28)
4.4 Metric and equations of motion
The ve-dimensional metric, called the running bolt[67] is a time bration over
Euclidian Schwarzschild:
ds
2
5
=Z
2
(dt+k)
2
+Zds
2
4
=Z
2
(dt+k)
2
+Z
h
1
2m
r
d
2
+
1
2m
r
1
dr
2
+r
2
d
2
+sin
2
d
2
i
;
(4.29)
wherekistheangular-momentum1-formoftherunningboltandZ thewarpfactor
linking the ve- and four-dimensional metrics. The coordinate,, results from the
Wick-rotationofthetime-coordinateintheEuclidianSchwarzschildbasemanifold;
it parameterizes theS
1
with periodicity +8m.
44
The Maxwell elds are set up by the oating braneansatz [72],
A
I
="Z
1
I
(dt+k)+B
(I)
; (4.30)
where " is set by the (anti-)self-duality of the elds. The magnetic eld strengths
are
(I)
=dB
(I)
: (4.31)
The three forms,Z
I
,
(I)
andk, are determined through the equations [74, 75, 69,
67]:
(I)
="?
4
(I)
; (4.32)
r
2
Z
I
=
1
2
"C
IJK
?
4
(J)
^
(K)
; (4.33)
dk+"?
4
dk ="Z
I
(I)
: (4.34)
Note, that (4:32)(4:34) are purely represented on the base manifold.
Following the choice of solution for the eld strength made in [67],
(I)
=q
I
1
r
2
d^dr+"d
2
; (4.35)
we have also
Z
I
= 1
1
2m
1
r
C
IJK
q
J
q
K
(4.36)
k =(r)d ="
1
r
1
2m
3
I=1
q
I
3
2m
q
1
q
2
q
3
1
r
+
1
2m
d; (4.37)
wheretheq
I
areM5-chargesassociatedwiththemagnetic eldstrengthcomponent
in (4:35).
45
It is important to note that exact terms proportional to d have been chosen
such that k vanishes on the bolt, which is essential for regularity and to remove
closed timelike curves. With this choice, the asymptotic limit of the angular mo-
mentum does not vanish but has a nite value:
r!1
!
=
"
2m
3
I=1
q
I
3
4m
2
q
1
q
2
q
3
: (4.38)
It is this nite limit which led to the name running bolt.
Transforming (4:38) leads to a formula for the magnetic charges:
3
I=1
q
I
=2"m
+
3
4m
2
q
1
q
2
q
3
: (4.39)
4.5 The Komar integrals
The results for the mass and charges can also be achieved by directly calculating
the asymptotic integrals.
It is very helpful to use the frames of the ve-dimensional metric,
e
0
=Z
1
(dt+d) e
1
=Z
1
2
1
2m
r
1
2
d
e
2
=Z
1
2
1
2m
r
1
2
dr e
3
^e
4
=Zr
2
d
2
: (4.40)
We can write the timelike Killing vector near in nity:
K =
@
@t
; (4.41)
46
and so nd for the mass:
M =
3
32G
5
Z
S
1
S
2
1
?
5
dK
=
3
32G
5
Z
S
1
S
2
1
?
5
[@
r
g
00
dr^dt+@
r
g
01
dr^d]
=
3
32G
5
Z
S
1
S
2
1
2
1
@
r
g
00
@
r
g
01
d^r
2
d
2
=
3
32G
5
Z
S
1
S
2
1
1
3m
I
C
IJK
q
J
q
K
"
q
I
d^d
2
=
G
5
3
I=1
(C
IJK
q
J
q
K
+3"m
q
I
): (4.42)
Also, for the charges it is
Q
I
=
1
32
2
m
Z
X
3
G
I
=
1
32
2
m
Z
X
3
1
2
"r
2
1
2m
r
Z
0
I
+"r
2
Z
I
Z
3
0
+q
I
Z
2
I
Z
3
d^d
2
=
"
4m
C
IJK
q
J
q
K
2
(q
I
q
J
): (4.43)
The goal now is to see how these results are produced by cohomology.
47
4.6 Topological data
Figure 1: Schematic of the homological 2-cycles (from left): bolt and non-compact
cycle
The spacetime at hand has entirely two cycles the bolt and the non-compact
cycle extending from r = 2m to in nity. Each of these carries an independent
cohomological ux: d
2
, which is carried by the bolt cycle (B), and its dual,
1
r
2
dr^d, which is carried by the non-compact cycle ().
The 2-form, d
2
, is manifestly harmonic. Although its dual can be written as
a total derivative,
1
r
2
dr^d =d
1
2m
1
r
d
, it has a nonvanishing value,
1
2m
d,
at in nity and thus is not exact.
From this, a basis for the cohomology can be directly inferred:
v
B
=
1
4
d
2
(4.44)
v
=
1
4r
2
dr^d: (4.45)
It is immediately clear thatv
B
is cohomology. Forv
, on the other hand, it is not
so obvious; but as explained above, its potential,
1
4
1
r
0
1
r
d, is either singular
48
attheboltornon-vanishingatin nity, dependingonr
0
, andhencecohomological.
Fromtheknownresultsofthe eldsandthesymmetryconditions,wewillderive
the cohomological 2-form uxes. These and the elds will be used to compute the
period-integrals the topological building blocksto particularly analyse the
mass, charges, and BPS-bound breaking extra-mass within intersection homology.
4.6.1 Deriving the uxes
In the following we derive expressions for the elds and uxes from the RHS of
(4:26) to understand their contributations to the mass formula.
From (4:30);(4:31) and (4:35) follows the Maxwell- eld strength, F
I
= dA
I
,
which decomposes into an exact and a (on the base manifold) harmonic part,
F
I
=d
^
A
I
+"q
I
d
2
(2m"
+q
I
)
1
r
2
dr^d; (4.46)
where
^
A
I
="
Z
1
I
+
1
2m
r
d +dt: (4.47)
Sincethetime-coordinate,t,isnotpartofthebasespace,its1-form,dt,isirrelevant
for the cohomology.
Notethatthe
^
A
I
vanishattheboltandatin nityandarethusgloballysmooth.
Choosing the Killing vector like (4:41), we get from (4:24a):
I
="Z
1
I
I
; (4.48)
where
I
are constants.
49
It is directly obvious that the choice
I
="; (4.49)
causes the
I
to vanish at in nity, whereZ
I
! 1. This way the term 2
I
G
I
drops
fromtheboundaryintegral of (4:26). Also, thebelowcomputedexact1-forms,
I
,
from (4:25) have to fall o¤at in nity; and so the Komar mass is rendered purely
topological.
With (4:30) and (3:32) we have
G
I
=
1
2
"r
2
1
2m
r
Z
0
I
+"r
2
Z
I
Z
3
0
+q
I
Z
2
I
Z
3
d^d
2
+
1
2
Z
I
Z
3
"r
2
0
+q
I
Z
I
dt^d
2
+
"
2r
2
q
I
Z
2
I
Z
3
dt^d^dr; (4.50)
and nd from this together with (4:48),
i
K
G
I
+
1
2
C
IJK
J
F
K
=
1
2
C
IJK
J
F
K
1
2
d
Z
I
Z
3
(dt+d)
; (4.51)
which is a manifestly closed expression.
Using (4:24b), we put all exact pieces of (4:51) intod
I
so that we have:
I
=
1
2
Z
I
Z
3
1
2m
r
"C
IJK
J
Z
1
K
1
2m
r
d +dt;
(4.52)
which is smooth at the bolt and vanishes at in nity.
The remaining terms in (4:51) sum up to the 2-form harmonic:
H
(2)
I
=
"
2
C
IJK
q
J
K
d
2
m
1
2
C
IJK
(q
J
+2m"
)
K
1
r
2
dr^d: (4.53)
50
As mentioned earlier, the 2-form,
1
r
2
dr^ d, has nonvanishing potential at
in nity and thus is cohomological.
4.6.2 Intersection homology
Theintersectiontechniquerelatesanintegraloftwowedged2-formsoverthewhole
four-dimensional base space to integrals of the single 2-forms over the homological
2-cycles.
Applying the gauge choice (4:49),
I
= ", rendering the Komar mass purely
topological, the period-integrals, forming the topological building blocks, am-
mount to:
C
B
=S
2
r=2m
C
=S
1
[2m;1[
R
F
I
4"q
I
4(2m"
+q
I
)
R
H
(2)
I
2(
3
J=1
q
J
q
I
) 2["(
3
J=1
q
J
q
I
)+2m
]
Table 1: Integrals of the 2-forms over the 2-cycles
It is instructive to introduce a canonical integer basis for the cohomology,
Z
C
A
v
A
0
=
A
0
A
and
Z
4
v
A
^v
A
0
=I
AA
0
; (4.54)
with
F
I
=
I
A
v
A
andH
(2)
I
= ~
I;A
v
A
; (4.55)
where
I
A
and ~
I;A
(A = B;) are precisely the entries of the above tables (the
building blocks), andI
AA
0
=I
A
0
A
is the inverse intersection matrix.
The choice of the cohomological basis (4:44) (4:45) manifestly ful lls the
51
above-stated orthonormality condition.
The integrals for the mass and charge formulae become then
Z
4
H
(2)
I
^F
I
= 16
2
3
I=1
(C
IJK
q
J
q
K
+3"m
q
I
) = ~
I;A
I
A
0I
AA
0
(4.56)
C
IJK
Z
4
F
J
^F
K
=32
2
"C
IJK
q
J
(2m"
+q
K
) =C
IJK
J
A
K
A
0I
AA
0
: (4.57)
In particular, this means that the integrals need to be reproduced by composing
theproductsoftheperiod-integrals,
I
A
and ~
I;A
, bytheintegercoe¢ cientsofI
AA
0
in the sense of the last expressions, which can only be achieved by
I
AA
0
=I
AA
0 =
0
B
@
0 1
1 0
1
C
A
: (4.58)
Thisismerelyatrivialone-timeintersectionbetweentheboltandthenon-compact
cycle, proving that there is no self-intersecting homology at hand.
Note: If we take the building blocks from table 1 and compute
I
A
v
A
, then we
get
F
I
harmonic
="q
I
d
2
(2m"
+q
I
)
1
r
2
dr^d; (4.59)
but it is
F
I
F
I
harmonic
=d
^
A; (4.60)
what is cohomologous to the total Maxwell- eld strength, where
^
A is a global
1-form falling o¤at in nity.
In this spirit, it is now easy to compute the total Komar mass and Maxwell-
charge.
52
4.7 Mass, charges, and the breaking of super-
symmetry
In this section, we will evaluate the expressions for the Komar mass and the
Maxwell-charges by means of the just introduced intersection homology method.
The fact that
I
! 0 at in nity and the gauge choice,
I
="; (4.61)
forwhichthefunction
I
goeszeroaccordingto(4:48),maketheboundaryintegral
drop out of (4:26), rendering the Komar mass an integral purely over cohomology
(4:27):
M =
1
16G
5
Z
4
H
(2)
I
^F
I
: (4.62)
From (4:62) and (4:28) we get with (4:56)(4:57) the mass and charges:
M =
G
5
3
I=1
(3m"
q
I
+C
IJK
q
J
q
K
) (4.63)
Q
I
=
"
4m
C
IJK
q
J
(2m"
+q
K
); (4.64)
which match (4:42)(4:43).
From that follows
M =
"m
G
5
4
3
I=1
Q
I
3
I=1
q
I
; (4.65)
53
and hence the BPS-bound violating extra-mass
M =M +
4"m
G
5
3
I=1
Q
I
=
"m
G
5
3
I=1
q
I
(4.66)
=
1
16G
5
Z
4
H
(2)
I
^F
I
+
"
2
3
I=1
C
IJK
Z
4
F
J
^F
K
(4.67)
=
1
16G
5
K
A
0
~
K;A
+
"
2
3
I=1
C
IJK
J
A
I
AA
0
: (4.68)
Interestingly, the breaking of supersymmetry is caused by the total M5-charges,
q
I
.
Very important note: The factored term,
K;A
= ~
K;A
+
"
2
3
I=1
C
IJK
J
A
; (4.69)
considered at each cycle separately,
K;B
= 0 (4.70)
K;
=4m
; (4.71)
vanishes identically for the bolt-cycle, A = B, so all contribution to the breaking
of supersymmetry comes from the non-compact cycle, A =:
M =
1
16G
5
K
B
~
K;
+
"
2
3
I=1
C
IJK
J
I
B
(4.72)
=
1
16G
5
3
K=1
4"q
K
(4m
) (4.73)
=
"m
G
5
3
K=1
q
K
: (4.74)
54
4.8 Summing-up
The Komarmass foranalmost-BPSsolutionof supergravityhas beenderivedina
ve-dimensionalstationaryspacetime,wherespacewasgivenaboltatthecenter
and made asymptoticallyS
1
R
3
.
Oneessentialquestionaddressedinthischapteris,whethertheextra-massterm
violatingtheBPS-boundinaspacetimeasymptoticallybehavinglikeR
1;3
S
1
,can
be detected by the Komar mass. It was shown that for a vastly simpli ed scenario
in which the magnetic charges got turned o¤, this is not given; but in the more
general situation considered in this chapter, the Komar mass also contains terms
breaking supersymmetry.
The very goal was to determine explicitely if and how the mass and charges
follow from topology within the framework of intersection homology, and espe-
ciallywhichhomologicalcyclesaccountparticularlyfortheextra-masscausingthe
violation of the BPS-bound.
Although the topology of the base space does not inhabit any self-intersecting
homology,asopposedtotheGibbons-Hawkingbase,themassandchargeintegrals
indeed turned out to exhibit explicit contributions from topology.
It could also be shown that one can use a choice of gauge, so that the integrals
become completely topological.
In any case, the nal result is fairly clear supersymmetry is broken solely in
virtue of the non-compact cycle and hence the boundary at in nity. This brings
up the obvious question whether a non-vanishing extra-mass termgenerally comes
from the non-compact cohomology only.
55
Chapter 5
Topological contributions to the
BPS-bound violation in a 2-center
solution
5.1 Motivation
In the previous chapter, it was shown that BPS-bound violating mass terms are,
in the presence of compact dimensions, not solely seenby the ADM mass but
indeed also arise in the Komar mass from spaces topology. This was due to the
cohomology dual to the non-compact cycle; in particular, the latters intersection
with the compact bolt cycle.
This raises the obvious question how supersymmetry-breaking emerges from
the cohomological structure of more complex non-BPS solutions.
In this chapter, we consider a more general non-supersymmetric solution of
supergravityina ve-dimensionalspacetime,thistimewithtwointersectingpieces
56
ofcompacthomology: Anon-extremalcenterinformofachargedbolt,constructed
in a similar fashion as in [67, 47], and an extremal Gibbons-Hawking center. From
previous works, there are known extremal results for BPS [68] and almost-BPS
systems[69,70,71]thesituationconsideredhereisanon-extremalgeneralization
for a non-BPS system.
Inparticular,wehavearunning-bolthomology2-spherelinkedwithaGibbons-
Hawking nut by a bubble carrying additional uxes. This solution was derived in
[73], rst for an arbitrary number of extremal Gibbons-Hawking centers and then
specializedtoonlyonesointotaltwoindependentpiecesofintersectingcompact
homology and a non-compact cycle, much as in the last chapter. The interesting
newfeatureistolookattheinteractionbetweentheboltandtheGibbons-Hawking
nut.
1
The main part of the calculations in this chapter is the explicit examination
of the cohomological uxes coming from the homological 2-cycles that result from
the bolt and the center-linking bubble. The focus then is on the analysis of the
topologicalintegralsovertheharmonicsintermsofintersectionhomology,inorder
to see how each cycle contributes to the mass, the charges, and the BPS-bound
violating extra-mass. Although the latter got contribution from the non-compact
cycleinthepreviouschapter,theintersectionmatrixturnedouttoberathertrivial
noself-intersectinghomology. Here, ontheotherhand, amorecomplicatedform
is to be expected, because every cycle has self-intersection. Special emphasis is on
the question if and how the supersymmetry-breaking extra-mass term results in
part from spaces topology aside from pure boundary e¤ects.
1
Another interesting non-supersymmetric multi-center solution with more than one homolog-
ical 2-cycle can be found in [76]
57
5.2 The 2-center solution
Figure 1: Schematic of the homological 2-cycles (from left): bolt, bubble, and
non-compact cycle
Inthefollowingweoutlinethemainresultsofthenon-supersymmetric2-center
solution given in [73].
The geometry of the spacetime and the three U (1)-gauge elds are a solution
of the Einstein-Maxwell equations in the oating brane ansatz [72] within ve-
dimensional ungauged supergravity. The stationary spacetime carries a Killing
vector, K =
@
@t
, and has the metric:
ds
2
5
=
1
2
LL
a
L
a
2
3
dt+
^
k
2
+
1
2
LL
a
L
a
1
3
ds
2
4
; (5.1)
where the functionsL andL
a
andthe angularmomentum1-form
^
k will be de ned
below.
The parametera in (5:1) counts vectormultiplets; likein[67, 41, 47] we choose
to have two (a = 2;3) to have a total of three Maxwell-charges, A
I
(I = 1;2;3).
This enables non-trivial Chern-Simons interactions.
However, like [73] we use the STU truncation in which the elds and uxes
with index I = 1 are treated in a di¤erent way than I =a. In detail, the raise of
58
the latter index,K
a
=
ab
K
b
, happens with anSO(1;1) metric following from the
non-zero Chern-Simons couplingC
1ab
=
ab
=
0
B
@
0 1
1 0
1
C
A
.
As for the running bolt solution from the last chapter and Gibbons-Hawking
metrics,thefour-dimensionalbasespaceisaU (1)- brationovera3-manifoldwhich
isasymptoticallyR
3
atin nity,renderingtheasymptoticsofthewholespatialbase
S
1
R
3
at in nity.
In particular, the 3-manifold is parameterized by (r;;) and the ber by .
As in the last chapter, the latter de nes a compact spatial dimension the S
1
,
this time with periodicity +4k, wherek is the scale parameter of theS
1
.
Thespecial topologyconsideredhere, is constitutedbyanon-extremal charged
bolt at r = c and an extremal Gibbons-Hawking center at r = R > c and = 0.
Thisleadstoanon-extremalandnon-BPSgeneralizationofknownextremalresults
for BPS [68] and almost-BPS systems [69, 70, 71].
The most crucial aspect of this topology is that, the - ber pinches o¤at two
locationstheboltandtheGibbons-Hawkingcenter. This - beralonganinterval
between r =c and r =R at = 0 de nes the new non-trivial compact homology
cycle.
The metric of the four-dimensional base manifold is
ds
2
4
=V
1
d +!
0
2
+V
dr
2
+
r
2
c
2
d
2
+sin
2
d
2
; (5.2)
where V has poles at r = c and (r;) = (R;0). This causes the - ber to pinch
o¤, but leaves the metric smooth.
This leads to the main di¤erence to the running-bolt spacetime of the last
chapter, which had only one pinch-o¤point for the circle- bration.
59
As we will see in detail, the fact that pinches o¤at two centers generates a
new homological cycle de ned by the ber and the radial interval between the two
pinch-o¤points.
In particular, if a periodic coordinate pinches o¤ at only one existing center,
like in the running-bolt solution, it can be xed at the bolt at the cost of creating
ux that does not vanish at in nity and hence giving rise to a non-compact cycle.
Two pinch-o¤ points, as considered here, give rise to both a further compact as
well as the non-compact cycle.
Entirely, there are three independent homological 2-cycles ( g. 1): The bolt-
sphere atr =c; the bubble-cycle being the - bered radial line along the positive
z-axis,connectingtheboltsnorthpoleandtheGibbons-Hawkingcenterat(r;) =
(R;0); the non-compact cycle, also running radially along the positivez-axis from
the Gibbons-Hawking center, z =R, towards in nity, bred by the -circles.
The appendix gives more information on the functions used here
2
along with
their asymptotics they are very complicated, which reinforces the interest in
cohomology , but we will need some details here
3
:
V (r;) =
r+m
2(r
2
c
2
)
r+m
+
2k
R+m
Rrc
2
cos
p
r
2
1
c
2
sin
2
(5.3)
^
k =!
M
V
d +!
0
(5.4)
!
0
(r;) =
1
2
(m
+
m
)cos+
2k
R+m
R
2
m
rR(rm
)cosc
2
sin
2
p
r
2
1
c
2
sin
2
d (5.5)
!(r;) =
e
R
2(R+m
)
2
u
a
u
a
h
1
r+R
r
1
(1cos)+
c
2
Rr
1
sin
2
i
d (5.6)
2
The functions used here are taken from eqs. (2:25), (2:26), (3:55), (4:2)(4:6) of [73].
3
In [73] the functions were equipped with an extra-parameter, n
A
, that can be set equal to 1,
which we do throughout this chapter.
60
L
a
(r;) =
(r+m
)(c
2
+m
r)
2m
(r
2
c
2
)
la
V
+u
a
(5.7)
L(r;) =
e
2
2m
2
1
V
l
a
l
a
e
2
c
2
(c+m
)
2
f
1
r+f
2
(m
+r)(m
+R)
u
a
u
a
(5.8)
M (r;) =
e
2m
l
a
L
a
+
e
2(m
+R)
Rr
m
+r
V +
(c
2
+m
r)(f
1
r+f
2
)
2c
2
(c+m
)
2
(r
2
c
2
)
u
a
u
a
; (5.9)
where r
1
=
p
r
2
+R
2
2Rrcos is the distance measured from the Gibbons-
Hawking center; m
, m
+
, e
, l
a
, u
a
, f
1
, and f
2
are parameters of the solution
which are non-trivially interrelated (see appendix A).
The functionV has two singularities one at the bolt, going like
k
rc
, and one
at the Gibbons-Hawking point, going like
k
jrRj
at = 0; in the metric (5:2) it
poses a coordinate singularity and so does not harm regularity at the centers.
4
Like the geometry, the Maxwell- elds are solutions to the Einstein-Maxwell
equations; their potentials in the oating brane ansatz are:
A
1
=
1
L
dt+
^
k
(5.10)
1
2e
(r+m
)(c
2
+m
r)
V(r
2
c
2
)
d +!
0
+
c
2
m
2
cosd
A
a
=
1
La
dt+
^
k
e
m
l
a
V
d +!
0
(5.11)
+e
u
a
2
r+m
d +!
0
cos+
2k
R+m
rRcos
p
r
2
1
c
2
sin
2
d
;
where in both cases the rst line represents the terms which are globally de ned
yet not exact, for they do not fall o¤at in nity.
4
For a much more detailed outline of the regularity analysis, see [73].
61
From this follow their eld-strengths, F
I
=dA
I
,
F
1
=d
1
L
dt+
^
k
+Z
1
V
r
2
c
2
d
2
dr^
d +!
0
(5.12)
+Z
2
d^
d +!
0
+V sindr^d
F
a
=d
1
La
dt+
^
k
e
m
l
a
V
d +!
0
(5.13)
+
2e
u
a
(r+m
)
2
V
r
2
c
2
d
2
dr^
d +!
0
;
where the terms have been sorted according to the ones of the potentials.
Extracting the topological bits out of the globally de ned terms, to make them
exact, can be done with help of the cohomological basis which we derive later.
Adding them to the other terms, however, would render them not harmonic any-
more, so the present topology has both self-dual and anti-self-dual parts.
The functions Z
1;2
, have the form:
Z
1
=
(c
2
+m
r)(r+m
)
2e
V(r
2
c
2
)
h
r+m
2V(r
2
c
2
)
m
c
2
+m
r
+
1
(r+m
)(r+m
+
)
2V(r
2
c
2
)
R
Rrc
2
cos
+
Rcosr
r
2
1
c
2
sin
2
i
(5.14)
Z
2
=
(c
2
+m
r)(r+m
)
2e
V
1
(r+m
)(r+m
+
)
2V(r
2
c
2
)
R
Rrc
2
cos
+
Rcosr
r
2
1
c
2
sin
2
sin
Rrcos
:
(5.15)
The dual eld strengths have to be computed by
G
1
=
1
2
L
4
3
1
2
L
c
L
c
2
3
?
5
F
1
(5.16)
G
a
=
1
2
L
2
3
1
2
L
c
L
c
4
3
(L
a
)
2
?
5
F
a
; (5.17)
where the prefactors correspond to the Q
IJ
from (3:32), composed in the same
62
manner as in (3:31) of the scalars of the solution:
X
1
=L
2
3
1
2
L
a
L
a
1
3
andX
a
=L
1
3
1
2
L
b
L
b
2
3
L
a
:
ThederivationoftheKomarmassformulawasdoneinthelastchapterin(4:26);
the three U (1)-charges relevant for this subchapter, are given by the properly
normalized formula:
Q
I
=
1
vol(X
3
)
Z
X
3
G
I
=
1
16
2
k
Z
4
dG
I
=
1
64
2
k
C
IJK
Z
4
F
J
^F
K
; (5.18)
where the last step follows from the equations of motion for the Maxwell- elds
(4:19).
5.3 The Komar integrals
Likeinthepreviouschapter,themassandchargescanalsobecomputedbymeans
of the asymptotic integral.
We nd the frames very helpful,
e
0
=
1
2
LL
a
L
a
1
3
dt+
^
k
e
1
=
1
2
LL
a
L
a
1
6
V
1
2
(d +!
0
)
e
2
=
1
2
LL
a
L
a
1
6
V
1
2
dr e
3
^e
4
=
1
2
LL
a
L
a
1
3
V (r
2
c
2
)d
2
;
(5.19)
since we need to work with dual forms.
63
FromtheKomarmassintegral(4:21)wegetwiththeKillingvectorK =g
0
dx
,
M =
3
32G
5
Z
X
3
?
5
dK
=
3
32G
5
Z
X
3
?
5
(@
r
g
00
dr^dt+@
r
g
01
dr^d +@
r
g
04
dr^d)
=
3
32G
5
Z
X
3
1
2
LL
a
L
a
2
3
@
r
g
00
+
2
@rg
00
@rg
01
(
1
2
LLaL
a
)
1
3
!
d ^
r
2
c
2
d
2
=
k
G
5
lim
r!1
h
1
2
2
LLcL
c
L
0
L
+L
0
a
1
La
+
3
2LLcL
c
M
V
0
i
r
2
c
2
=
k
G
5
lim
r!1
n
L
0
^
L
+
1
la+ua
h
L
0
a
1
2
^
L
2
l
a
+u
a
L
0
^
L
+L
0
a
1
la+ua
+
3
8
^
L
1
l
a
+u
a
M
V
0
io
r
2
=
k
2G
5
n
4ke
2
^
L
h
1
(R+m
)
2
+
4c+n(c+m
)
(c+m
)
3
i
u
a
u
a
(5.20)
1
la+ua
h
3
4
a
e
2m
l
a
e
^
L
R+2m
R+m
u
b
u
b
c
2
e
+
e
2
^
L
(u
a
l
a
)
io
;
where we de ned for brevity
^
L = lim
r!1
L and
= lim
r!1
M
V
; (5.21)
and similarly,
a
=
l
a
+u
a
2e
m
l
a
(5.22)
~
a
=
l
a
+u
a
4e
m
l
a
: (5.23)
We have also used the asymptotics from appendix B to obtain
^
L =
2e
m
e
2
m
2
l
a
(l
a
+2u
a
)+
e
2
m
1
R+m
u
a
u
a
(5.24)
V
0
!
(n+1)km
r
2
(5.25)
L
0
!
e
2
r
2
h
m
c2(n+1)k
m
2
l
a
l
a
4k(Rc)
(R+m)(c+m)
2
u
a
u
a
+
c+m
e
2
^
L
i
(5.26)
64
L
0
a
!
m
2
c
2
2(n+1)km
m
r
2
l
a
(5.27)
M
V
0
!
2e
r
2
h
c
2
m
2
m
2
l
a
l
a
l
a
u
a
1
2
Rm
R+m
u
a
u
a
+
(n+1)k
e
i
m
e
L
0
: (5.28)
For the rst charge holds, using (5:18) and (5:16),
Q
1
=
1
16
2
k
Z
X
3
G
1
=
1
32
2
k
2
^
L
2
LaL
a
2
3
Z
X
3
?
5
F
1
=
1
32
2
k
Z
X
3
n
L
0
+
1
(la+ua)(l
a
+u
a
)
h
M
V
0
+
^
LZ
1
i
o
r
2
d ^d
2
=ke
2
h
1
(R+m
)
2
+
4c+n(c+m
)
(c+m
)
3
i
u
a
u
a
+
1
4
a
(m
a
2e
u
a
); (5.29)
where we used the further asymptotic,
lim
r!1
Z
1
r
2
c
2
=
c
2
+m
2
e
: (5.30)
For the remaining charges it is,
Q
a
=
1
16
2
k
Z
X
3
G
a
=
1
32
2
k
1
L
2
(L
a
)
2
1
2
LL
c
L
c
4
3
Z
X
3
?
5
F
a
=
1
32
2
k
Z
X
3
n
L
0
a
+
2LL
a
h
M
V
0
2e
m
L
a
2l
a
V
0
m
(r+m
)
2
u
a
io
r
2
d ^d
2
=
1
4
h
2m
^
L
e
m
u
a
a
+
m
2
c
2
e
a
i
; (5.31)
where by the bar index, a, we denote the abscence of the sum convention to avoid
confusion.
The results for the mass and the charges will in the following be derived in
terms of cohomology, which will turn out to be essentially less complicated.
65
5.4 Deriving the uxes
In the following, we derive expressions for the elds and uxes to understand their
contributions to the mass formula. As above, we write the timelike Killing vector
near in nity likeK =
@
@t
.
From (5:12)(5:13) we get
i
K
F
1
=d
1
L
(5.32)
i
K
F
a
=d
1
La
; (5.33)
and so with (4:24a):
1
=
1
1
L
(5.34)
a
=
a
1
La
; (5.35)
wherethe
I
arefreely-choosableconstants. Wewilllater xthemthough, sothat
I
! 0 at in nity.
From (5:12)(5:17)and (5:34)(5:35)we nd,decomposing (4:24b)according
to the STU truncation:
1
=
e
(r+m
)
2
(u
a
a
2)
V
r
2
c
2
d
2
dr^
d +!
0
+d
h
1
LcL
c
(
a
L
a
1)
dt+
^
k
e
2m
V
l
a
a
d +!
0
i
(5.36)
a
=
e
(r+m
)
2
u
a
1
+
1
2
ab
b
Z
1
V
r
2
c
2
d
2
dr^
d +!
0
+
1
2
ab
b
Z
2
V sindr^d+d^
d +!
0
(5.37)
+d
h
1
2
1
L
a
1
+
1
L
ab
b
1
LL
a
dt+
^
k
e
2m
V
l
a
1
d +!
0
i
;
66
where we wrote for short,
1
=i
K
G
1
+
1
2
ab
a
F
b
(5.38)
a
=i
K
G
a
+
1
2
ab
1
F
b
+
b
F
1
: (5.39)
These formulae suggest obvious choices for the analogues of (4:52) and (4:53):
~
H
(2)
1
=
e
(r+m
)
2
(u
a
a
2)
V
r
2
c
2
d
2
dr^
d +!
0
(5.40)
~
H
(2)
a
=
h
e
(r+m
)
2
u
a
1
+
1
2
ab
b
Z
1
i
V
r
2
c
2
d
2
dr^
d +!
0
(5.41)
+
1
2
ab
b
Z
2
V sindr^d+d^
d +!
0
~
1
=
1
LcL
c
(
a
L
a
1)
dt+
^
k
e
2m
V
l
a
a
d +!
0
(5.42)
~
a
=
1
2
1
L
a
1
+
1
L
ab
b
1
LL
a
dt+
^
k
e
2m
V
l
a
1
d +!
0
: (5.43)
Note: Thetildeindicatesthatthed
~
I
arenotexact,sincethe
~
I
donotvanish
at in nity and are thus singular there; exactness can be achieved, however, by
extracting the cohomological bits from the d
~
I
by means of the cohomological
basis, which we will derive later, and shift them into the
~
H
(2)
I
.
The
~
H
(2)
I
are manifestly self-dual, and one can check that d
~
H
(2)
I
= 0. They
are thus harmonic and can locally be written as
~
H
(2)
I
=d
~
B
I
, where the
~
B
I
are not
globallyde nedsincetheydonotvanishatthepinch-o¤pointsofthe -coordinate:
~
B
1
=e
(u
a
a
2)
1
r+m
d +!
0
cos
2
+
k
R+m
rRcos
p
r
2
1
c
2
sin
2
d
(5.44)
~
B
a
=e
u
a
1
1
r+m
d +!
0
cos
2
+
k
R+m
rRcos
p
r
2
1
c
2
sin
2
d
1
4e
ab
b
(r+m
)(c
2
+m
r)
V(r
2
c
2
)
d +!
0
+
c
2
m
2
cosd
: (5.45)
67
Note that, becauseV has a singularity at each center,V
1
goes zero there and
ensures nite norms except for the terms with the additional factor (rc)
1
which cancels the zero at the bolt and hence the goodbehavior.
As one can clearly see from the analysis, the factors in the volume integral of
(4:26),
~
H
(2)
I
and F
I
, are each equipped with two dual ux terms, V (r
2
c
2
)d
2
and dr^(d +!
0
), wedge-multiplying to spaces volume form. Hence, homology
allows for the existence of purely topological terms in the mass formula, which
cannotbeconvertedintoboundaryterms,andevenhasself-intersection,aswewill
see in the following.
5.5 Topological data and intersection homology
In this subchapter, we explicitly derive the topological ingredients owing into the
formulaforthemassandcharges. Thegoalistoseeifandhoweachofthe2-cycles
contribution goes into either, so that we can precisely make out their sources of
topologyinvirtueofintersectionhomologywithspecialemphasisonthebreaking
of supersymmetry.
As mentioned earlier, the cohomological uxes (5:40) (5:41) are still miss-
ing cohomology from the d
~
I
. Since the
~
I
are only irregular at in nity, these
68
cohomological pieces just concern the non-compact cycle, C
:
H
(2)
1
=
~
H
(2)
1
+
Z
C
d
~
1
v
=
~
H
(2)
1
+k
h
2
a
1
la+ua
l
a
+u
a
4e
m
l
a
a
i
v
(5.46)
H
(2)
a
=
~
H
(2)
a
+
Z
C
d
~
a
v
=
~
H
(2)
a
+2k
h
l
a
+u
a
2e
m
l
a
1
+
^
L
ab
b
^
L(l
a
+u
a
)
i
v
; (5.47)
wherev
isthecohomologicalbasisvectorforthenon-compactcyclederivedbelow
in (5:62).
The Komar mass and charges are not dependent on any choice of gauge, but it
is very convenient to choose the
I
such that the boundary integral term 2
I
G
I
in
(5:50) vanishes; with (5:34) and (5:35) this is achieved by
1
=
1
^
L
=
h
e
2
m
2
l
a
l
a
+
e
2
c
2
(c+m
)
m
+
Rc
R+m
4c
2
k
(c+m
)
2
u
a
u
a
i
1
(5.48)
a
= lim
r!1
1
La
=
1
la+ua
: (5.49)
With this choice the equations (4:26) and (5:18) then become completely topo-
logical expressions:
M =
1
16G
5
Z
4
H
(2)
I
^F
I
(5.50)
Q
I
=
1
64
2
k
C
IJK
Z
4
F
J
^F
K
: (5.51)
We now compute the topological building blocksof the elds and uxes by
which(5:50)(5:51)shallberepresentedintheframeworkofintersectionhomology.
Likeinthelastchapter,wederivetheperiod-integralsbydoingtheintegralsofthe
69
2-form uxes over all topologically relevant 2-cycles (see g. 2):
C
B
=S
2
r=c
(Bolt) C
=S
1
[c;R] (Bubble)
R
F
1
2(c
2
m
2
)+(c+m
)
2
n
2e
(c+m
)
2
e
R
F
a
16cke
(c+m
)
2
u
a
8ke
(cR)
(c+m
)(R+m
)
u
a
R
H
(2)
1
8cke
(c+m
)
2
l
a
1
la+ua
4ke
(cR)
(c+m
)(R+m
)
l
a
1
la+ua
R
H
(2)
a
8cke
(c+m
)
2
1
^
L
u
a
+
2(c
2
m
2
)+(c+m
)
2
n
2e
1
l
a
+u
a
4ke
(cR)
(c+m
)(R+m
)
1
^
L
u
a
+
(c+m)
2
2e
1
l
a
+u
a
Table 2: Integrals of the 2-forms over bolt and bubble cycle
C
=S
1
[R;1[ (Non-compact cycle)
R
F
1
4k
^
L
2m
e
R
F
a
4k
h
la+ua
2e
m
l
a
+
m
R+m
u
a
i
R
H
(2)
1
k
l
a
+u
a
e
m
4R
R+m
l
a
1
la+ua
R
H
(2)
a
2k
h
^
L
2m
e
1
l
a
+u
a
2e
m
l
a
+
m
R+m
u
a
1
^
L
i
Table 3: Integrals of the 2-forms over the non-compact cycle
The parametern hereby represents the bolts NUT-charge.
Itisinstructiveagaintointroduceacanonicalintegerbasisforthecohomology,
Z
C
A
v
A
0
=
A
0
A
and
Z
4
v
A
^v
A
0
=I
AA
0
; (5.52)
with
F
I
=
I
A
v
A
andH
(2)
I
= ~
I;A
v
A
; (5.53)
where
I
A
and ~
I;A
(A = B;;) represent the entries of the above tables, and
I
AA
0
=I
A
0
A
is the inverse intersection matrix.
70
In this spirit, the bulk integral (5:50) becomes
Z
4
H
(2)
I
^F
I
= ~
I;A
I
A
0I
AA
0
; (5.54)
and analogously for (5:51). Reproducing the integrals by composing the products
of the period-integrals,
I
A
and ~
I;A
, by the integer coe¢ cients ofI
AA
0
in the sense
of (5:54), can only be achieved by
I
AA
0
=
0
B
B
B
B
@
0 1 1
1 n n
1 n n1
1
C
C
C
C
A
,I
AA
0 =
0
B
B
B
B
@
n 1 0
1 1 1
0 1 1
1
C
C
C
C
A
: (5.55)
It is obvious that in the present spacetime the homological structure is signi -
cantly more complex than in the one of the last chapter.
The o¤-diagonal 1s in I
AA
0 are directly clear: The bolt intersects the bubble
at the former s north pole; the bubble intersects the non-compact cycle in the
Gibbons-Hawking point. Less intuitive are the self-intersections of each cycle it
is n-fold for the bolt for NUT-charge of n; the self-intersections of the bubble and
thenon-compactcyclearise, liketheintersectionbetweenthem, fromthetopology
of the nut linking these cycles.
Sincewewanttostudytheexplicittopologicalcontributionofeachintersecting
part later, it is important to see in how far the elements of I
AA
0, representing the
intersections, are reected in I
AA
0
, representing the composition of the building
blocks. To shed light on this, we leave the intersection numbers in I
AA
0 more
71
general:
I
AA
0 =
0
B
B
B
B
@
I
BB
I
B
0
I
B
I
I
0 I
I
1
C
C
C
C
A
: (5.56)
After inversion, we receive
I
AA
0
=
1
det(I
AA
0)
0
B
B
B
B
@
I
I
I
2
I
B
I
I
B
I
I
B
I
I
BB
I
I
BB
I
I
B
I
I
BB
I
I
BB
I
I
2
B
1
C
C
C
C
A
: (5.57)
Note, that this form is merely a schematic tracking deviceto qualitatively
illustrate how the contributions of intersection from I
AA
0 distribute among the
non-zero entries ofI
AA
0
; hence, the determinant, det(I
AA
0) = 1, and the upper left
entry,I
I
I
2
= 0 (not contributing to topology), are not of interest in this
regard. So, the matrix can be written as:
I
AA
0
=
0
B
B
B
B
@
0 I
B
I
I
B
I
I
B
I
I
BB
I
I
BB
I
I
B
I
I
BB
I
I
BB
I
I
2
B
1
C
C
C
C
A
(5.58)
=I
B
0
B
B
B
B
@
0 I
I
I
0 0
I
0 I
B
1
C
C
C
C
A
+I
BB
0
B
B
B
B
@
0 0 0
0 I
I
0 I
I
1
C
C
C
C
A
: (5.59)
The last expression indeed gives a clue about how to disentangle homology.
First, I
AA
0
decomposes with respect to the bolt-intersections (I
BB
and I
B
).
The resulting constituents are predominantly de ned by the homology of the sub-
system of the bubble and the non-compact cycle, that is, the topology of the nut;
72
and if this was turned o¤, then there would still be a contributionI
2
B
= 1
from the bolt-bubble intersection.
This gives rise to the conclusion that there might be a possible topological
hierarchy between the bolt and the nut.
In any case, these insights will prove to be very helpful in spectralizing the
topological contributions later.
From (5:12)(5:15) and (5:53)(5:55) follows the cohomological basis:
v
B
=
1
4
d
2
(5.60)
v
=
1
Z
1
+
2
(r+m)
2
V
r
2
c
2
d
2
dr^
d +!
0
+
1
Z
2
d^
d +!
0
+V sindr^d
+
3
d
2
(5.61)
v
=
1
Z
1
+
2
(r+m)
2
V
r
2
c
2
d
2
dr^
d +!
0
+
1
Z
2
d^
d +!
0
+V sindr^d
+
3
d
2
; (5.62)
with the constant coe¢ cients,
1
=
e
c+m
(c+m
)
3
4km
(cR)
1
=
e
cR
(c+m
)
3
4km
(cR)
2
=
m
(R+m
)(c+m
)
(c+m
)
3
4km
(cR)
2
=
1
4k
(R+m
)(c+m
)
3
(c+m
)
3
4km
(cR)
3
=
8ckm
(R+m
)(c+m
)
3
[2(cm
)+(c+m
)n]
4(c+m
)[(c+m
)
3
4km
(cR)]
3
=
c+m
4
(n2)(Rc)(c+m
)+2c(3R2c+m
)
(c+m
)
3
4km
(cR)
(5.63)
Note: v
andv
are each self-dual and hence harmonic, up to ad
2
-term.
The orthonormality condition (5:52) is obviously ful lled by v
B
. However, to
show the same for v
and v
, one has to evaluate the integrals of the coe¢ cent
73
functions over the cycles with help of
Z
S
2
r=c
V
r
2
c
2
Z
1
d
2
=
2(c
2
m
2
)
e
+
(c+m
)
2
e
n (5.64)
Z
R
c
Z
1
j
=0
dr =
(c+m
)
2
4ke
(5.65)
Z
1
R
Z
1
j
=0
dr =
m
e
: (5.66)
Now, we can easily write the uxes in terms of the cohomology basis:
H
(2)
1
= ~
1;A
v
A
=
h
8ce
(c+m
)
2
l
a
v
B
+
4e
(cR)
(c+m
)(R+m
)
l
a
v
l
a
+u
a
e
m
4R
R+m
l
a
v
i
k
la+ua
(5.67)
H
(2)
a
= ~
a;A
v
A
=
8cke
(c+m
)
2
1
^
L
u
a
+
(c
2
m
2
)
e
+
(c+m
)
2
n
2e
1
l
a
+u
a
v
B
+
h
4ke
(cR)
(c+m
)(R+m
)
1
^
L
u
a
+
(c+m)
2
2e
1
l
a
+u
a
i
v
(5.68)
2k
h
2e
m
l
a
+
m
R+m
u
a
1
^
L
^
L
2m
e
1
l
a
+u
a
i
v
1
=
~
1
Z
C
d
~
1
!
=
1
LcL
c
1
la+ua
L
a
1
dt+
^
k
e
2m
V
l
a
1
la+ua
d +!
0
k
la+ua
~
a
!
(5.69)
a
=
~
a
Z
C
d
~
a
!
=
1
2
1
L
a
1
+
1
L
ab
b
1
LL
a
dt+
^
k
e
2m
V
l
a
1
d +!
0
2k
^
L
a
!
;
(5.70)
74
where!
is the potential forv
=d!
,
!
=
1
r+m
2
(r+m
)(c
2
+m
r)
2e
V(r
2
c
2
)
1
d +!
0
(5.71)
cos
2
c
2
m
2
e
1
+
2
+2
3
+
k
R+m
rRcos
p
r
2
1
c
2
sin
2
2
d: (5.72)
Analogously it holds for the elds,
F
1
=
1
A
v
A
(5.73)
=
2(c
2
m
2
)
e
+
(c+m
)
2
n
e
v
B
+
(c+m
)
2
e
v
+4k
^
L
2m
e
v
(5.74)
F
a
=
a
A
v
A
(5.75)
=
8ke
c+m
u
a
2c
c+m
v
B
+
cR
R+m
v
+4k
h
la+ua
2e
m
l
a
+
m
R+m
u
a
i
v
;
(5.76)
and the potentials of theH
(2)
I
,
B
1
=
~
B
1
+
Z
C
d
~
1
!
=
~
B
1
+
k
la+ua
~
a
!
(5.77)
B
a
=
~
B
a
+
Z
C
d
~
a
!
=
~
B
a
+
2k
^
L
a
!
: (5.78)
5.6 Mass,charges,andBPS-boundbreakingfrom
cohomology
We now have the data we need to compute the total mass and charges.
75
From (5:50)(5:55) follow the expressions:
M =
1
16G
5
Z
4
H
(2)
I
^F
I
=
1
16G
5
~
I;A
I
A
0I
AA
0
(5.79)
=
k
2G
5
n
4ke
2
^
L
h
1
(R+m
)
2
+
4c+n(c+m
)
(c+m
)
3
i
u
a
u
a
1
la+ua
h
3
4
a
e
2m
l
a
e
^
L
R+2m
R+m
u
b
u
b
c
2
e
+
e
2
^
L
(u
a
l
a
)
io
(5.80)
Q
1
=
1
64
2
k
ab
Z
4
F
a
^F
b
=
1
64
2
k
ab
a
A
b
A
0I
AA
0
(5.81)
=ke
2
h
1
(R+m
)
2
+
4c+n(c+m
)
(c+m
)
3
i
u
a
u
a
+
1
4
a
(m
a
2e
u
a
) (5.82)
Q
a
=
1
32
2
k
ab
Z
4
F
1
^F
b
=
1
32
2
k
ab
1
A
b
A
0I
AA
0
(5.83)
=
1
4
h
2m
^
L
e
m
u
a
a
+
m
2
c
2
e
a
i
: (5.84)
These match (5:20), (5:29), and (5:31).
The core piece of this chapter is to examine the topological origin of the BPS-
boundbreakingextra-mass,forwhichpurposewewillsequencetheforegoingresults
with respect to intersecting homology.
Comparing (5:80) with (5:82)(5:84), the relation between the mass and the
total charge,
M =
2k
G
5
Q
1
+Q
2
+Q
3
+M; (5.85)
gives the sought extra-mass term,
M =M
2k
G
5
3
I=1
Q
I
(5.86)
=
1
16G
5
Z
4
H
(2)
I
^F
I
1
2
3
I=1
C
IJK
Z
4
F
J
^F
K
(5.87)
=
1
16G
5
K
A
0
~
K;A
1
2
3
I=1
C
IJK
J
A
I
AA
0
: (5.88)
76
With (5:80)(5:84) it ammounts to
M =
k
2G
5
3
a=2
n
4ke
2
1
^
L
1
h
1
(R+m
)
2
+
4c+n(c+m
)
(c+m
)
3
i
u
a
u
a
1
la+ua
3
4
a
e
2m
l
a
e
^
L
R+2m
R+m
u
b
u
b
c
2
e
e
^
L
h
u
a
1
la+ua
1
1
i
(5.89)
+
2
^
L
+
a
(2e
u
a
m
a
)+
m
2
c
2
e
3m
2
^
L
a
o
:
Like in the previous chapter, it is insightful to investigate the from (5:88) fac-
tored term,
K;A
= ~
K;A
1
2
3
I=1
C
IJK
J
A
; (5.90)
in more detail for every cycle:
1;B
=
8cke
(c+m
)
2
h
2
3
a=2
u
a
1
la+ua
1
i
(5.91)
a;B
=
8cke
(c+m
)
2
1
^
L
1
u
a
+
(c
2
m
2
)
e
+
(c+m
)
2
2e
n
1
l
a
+u
a
1
(5.92)
1;
=
4ke
(cR)
(c+m
)(R+m
)
h
2
3
a=2
u
a
1
la+ua
1
i
(5.93)
a;
=
4ke
(cR)
(c+m
)(R+m
)
1
^
L
1
u
a
+
(c+m)
2
2e
1
l
a
+u
a
1
(5.94)
1;
=k
3
a=2
nh
l
a
+u
a
4e
m
l
a
+
m
R+m
u
a
i
1
la+ua
1
l
a
+u
a
+
4e
R+m
o
(5.95)
a;
= 2k
h
^
L
2m
e
1
l
a
+u
a
1
2e
m
l
a
+
m
R+m
u
a
1
^
L
1
l
a
+u
a
i
:
(5.96)
In the previous chapter, all contribution from the bolt canceled out identically,
sothatonlythenon-compactcycleturnedouttoberesponsibleforthebreakingof
supersymmetry. Here we can clearly see that every cycle contributes, if one keeps
77
the parameters general.
In the following, we consider a special choice of parameters, in order to both
simplify the foregoing results and parallel the procedure even more with the one
from the previous chapter.
In chapter 4, the warp factor of the ve-dimensional metric, Z, goes to 1 at
in nity. For the warp factor,
1
2
LL
a
L
a
, of the present spacetime, however, this is
notsoobvious,sincetheasymptoticsofL
a
andLarecomposedoftheverystrictly
boundparameters(seeappendixB).Followingtheregularitydiscussionattheend
of [73], one condition outlined (eq. 4.17) is
VL> 0 andVL
a
> 0 everywhere, (5.97)
and so, with V !
1
2
at in nity, we learn that both L and L
a
must have positive
asymptotics. Hence, the choice
^
L = 1 andl
a
+u
a
=l
a
+u
a
=l+u = 1, (5.98)
is in agreement with that, and doing so we would have
1
2
LL
a
L
a
!
1
2
^
L(l
a
+u
a
)(l
a
+u
a
) =
^
L(l+u)
2
= 1; (5.99)
as desired.
This way (5:89) simpli es signi cantly to
M =
ke
2G
5
2l+5
3m
e
c
2
m
2
2e
2
; (5.100)
78
where rede ning the (restricted) degree of freedoml> 0 like 2l+5 =
2m
e
yields
M =
km
G
5
3
2
c
2
m
2
4m
e
: (5.101)
This result compares nicely to (4:66) from the running-bolt spacetime, especially
if one does
m
!m
Schwarzschild
andk
3
2
c
2
m
2
4m
e
!"
3
I=1
q
I
: (5.102)
The choice (5:98) applied to (5:91)(5:96), yields furthermore
1;B
=
16cke
(c+m
)
2
1;
=
8ke
(cR)
(c+m
)(R+m
)
1;
=2k
4e
R+m
a;B
= 0
a;
= 0
a;
=2k
; (5.103)
so clearly every cycle goes at least in part into the extra-mass and so contributes
to the violation of the BPS-bound, where, like in the previous chapter, the non-
compact cycle yields the strongest contribution.
To see the cyclescontribution more particularly in this light, we decompose
M /
K
A
K;A
0I
AA
0
into a part where the compact cycles only talkto one
another (c-c) (A;A
0
= B;), and one where the compact cycles correspond with
the non-compact cycle (c-n):
M
c-c
=
1
16G
5
1
1;B
+
1
B
n
1
1;
=
k
G
5
c
2
+Rm
R+m
(5.104)
M
c-n
=
1
16G
5
K
B
n
K
(n+1)
K
K;
+
1
(
1;B
n
1;
)
=
k
G
5
h
(23
)m
2
c
2
m
2
4e
c
2
+Rm
R+m
i
: (5.105)
79
So, there is a non-vanishing contribution from the compact cycles only.
Now, we take a closer look at how these results can be interpreted in the lan-
guage of intersection homology. The period-integrals compose the extra-mass in-
tegral (5:88) through the inverse matrix, I
AA
0
; from (5:59) we see which cycles
intersect in each component.
One nds from M /
K;A
K
A
0I
AA
0
and (5:59) that the vanishing of the com-
ponents,
a;B
and
a;
, weakens the contributions from the intersection terms:
I
B
I
, I
B
I
, I
BB
I
, and I
BB
I
; so, all intersections are a¤ected equally
many times except for the self-intersection of the bubble, I
. On the other
hand,theabove-mentioneddominanceofthenon-compactcyclemeanswith (5:59)
a stronger representation of the intersection terms: I
B
I
, I
2
B
, I
BB
I
, and
I
BB
I
; this time being all intersections except for the self-intersection of the
non-compact cycle, I
. So, under the bottom line one can make a qualitative
estimationof theorderof contributionstrengthfortheintersections(startingwith
the strongest):
I
B
! (I
BB
;I
)! (I
;I
): (5.106)
Since in the previous chapter the only contributing (and existing) intersection was
I
B
,onemayrisethequestionwhethertheintersectionoftheboltwithitsadjacent
cycle might be generally the dominant one.
In any case, it is striking that the breaking of supersymmetry has for this
solution of supergravity non-vanishing topological contributions from all cycles
intersections.
80
5.7 Summing-up
The cohomological uxes of a 2-center solution of ve-dimensional supergravity
have been derived, consisting of a non-extremal magnetic bolt and an extremal
Gibbons-Hawking center, both linked by a bubble 2-cycle.
A rst striking consequence from deriving the 2-form harmonics is that, com-
pared to the 1-center running boltsolution [67, 47], the 2-center situation at
hand exhibits additional cohomological ux as a direct implication of the bubble
pinching o¤at each of the two centers and thus generating a kind of interaction
uxwhich is dual to the bolt ux.
Thepresenthomologyisindeedself-intersectingandthuscreatingamorecom-
plex spectrum of purely topological contributions to the asymptotic mass and
Maxwell-charges.
From the derived harmonic uxes, the formulae for the mass and charges have
been computed and the topological pieces of the BPS bound-breaking termexplic-
itly analyzed in virtue of intersection homology. As a result, the extra-mass and
hence the breaking of supersymmetry are indeed supported by every homological
cycle in part even by the compact cycles alone and all existing intersections of
these evenbyn-foldself-intersectionof the bolt incase its NUT-charge is turned
on , where the intersection matrix got rewritten in a sensible manner to allow for
aninsightfuldi¤erentiationbetweenthevariouscontributionsfromhomology. The
bolt-bubble intersection turned out to be most pivotal.
81
Chapter 6
Conclusion
Inthelonglineofe¤ortsinderivingandapplyingtheSmarrformulainahugeand
manifold framework of physical situations, the so-called no-go theorems, which
prohibit the existence of massive supergravity solitons in the abscence of singu-
larities and horizons, were shown to be circumvented in recent works by allowing
non-trivial topology to spacetimes of dimensions ve of higher.
In this thesis, further accomplishments on exploring the scope of applicability
and implications of Smarrs formula have been demonstrated and discussed within
the cases of three geometrically and topologically distinct spacetimes.
Since the known non-BPS solutions are very specialized, the assumption stood
to reason that the breaking of supersymmetry might be in general a boundary
phenomenon rather than an intrinsic part of the core solution. In this thesis, it
couldbeshownthatthebreakingofsupersymmetrygetsindeedcontributionsfrom
the core topology in addition to the boundary.
At rstweconsideredthemaximumgeneralizationintermsofdimensions. The
equations of motion have been reformulated in the fashion of the ve-dimensional
82
case of [41] within eleven-dimensional supergravity, and from that a mass formula
of solitonic solutions via compactifying on a simply connected and Ricci-at 6-
manifold was inferred. Furthermore, we gave the Chern-Simons term an arbitrary
constant coe¢ cient in the action to see in how far the results are inuenced by
this parameter. We nally obtained a generalized version of Smarrs formula and
also showed how to arrive back at the ve-dimensional theory by performing the
eleven-dimensional Komar integral overT
6
.
The most intriguingaspect of the calculations done bothin[41] andhere is the
proofofthepossibilityofconstructingmassivesolitonsolutionswithouttheneedof
horizons and so showing that the techniques used in microstate geometries are the
onlymethodsthatcansupportsolitons. Moreover, wecouldshowthatmakingthe
Chern-Simonstermarbitrarydoesnotchangethisfact;theincorporationofChern-
Simons interactions does not yield extra pieces in the mass formula in addition to
the topological terms, but is only signi cant for the purely topological nature of
the soliton mass.
Inthesecondscenario,wehavederivedtheKomarmassforannon-BPSsolution
of supergravity in a ve-dimensional stationary spacetime where we gave space a
magnetically charged boltat the center and made it asymptoticallyS
1
R
3
.
The very goal was to determine explicitely how each mass component follows
from topology and especially how cohomology accounts particularly for the extra-
mass causing the violation of the BPS-bound.
One essential question addressed was, whether the extra-mass term violating
theBPS-boundinspacetimes asymptoticallybehavinglikeR
1;3
S
1
, onlyappears
in the ADM mass while the Komar mass preserves the BPS formula.
It was shown that for a vast simpli cation of the running bolt solution, in
83
which the magnetic charges got turned o¤, this still holds; but in the more general
situation, the Komar mass contains terms breaking supersymmetry as well.
Themassformulaisalltopology. Thisisduetothefactthatthepresentspace-
time allows fortwoharmonicuxes thevolumeformof thebolt andits dual ux
on the non-compact cycle. Nontheless, the topology of the Euclidian Schwarz-
schildbasespacedoesnotinhabitanyself-intersectinghomologyasopposedtothe
Gibbons-Hawking base and the spacetime in chapter 5.
Inanycase,itistheuxesonthenon-compactcyclethatrenderthesupersymmetry-
breaking extra-mass term non-zero, which raises the question whether the latter is
in general a boundary term.
On top of this, a whole di¤erent picture arose when the cohomological uxes
of a 2-center solution of ve-dimensional supergravity were derived, which consist
of a non-extremal magnetic bolt and an extremal Gibbons-Hawking center, both
linked by a bubble 2-cycle.
A rst striking consequence from the in this light computed 2-form harmonics
is that the 2-center situation exhibits additional topological ux through the bub-
bles pinching o¤at two centers and thus generating a kind of interaction ux,
whichisdualtothebolt-ux. Asopposedtothe1-centerrunning-boltsolution,the
homologyofthissolutionturnedouttobeself-intersectingandthepurelytopolog-
icalcontributionstotheasymptoticmassandMaxwell-chargesmuchmorevarious.
Fromthe running-bolt solutionit is alreadyknownthat Komardoes indeedreect
the breaking of supersymmetry in a spacetime with the given asymptotics; the
main question, however, addressed at this point is, in which explicit manner it is
caused by the various given features of spaces topology.
From the derived harmonic uxes, the formulae for the mass and charges were
84
computed and the topological pieces of the BPS-bound breaking extra-mass term
explicitly analyzed in virtue of intersection homology. As a striking result, the
extra-mass and hence the breaking of supersymmetry are indeed supported by all
existing intersections of the homological cycles dominated by the bolt-bubble
intersection , so in part even by the compact cycles and hence the core solution
alone.
All together, one can draw the conclusion that topological con gurations with
more than one center in the base manifold of a supergravity solution of spacetime,
givingrisetothenecessaryadditionalcohomologicaluxes,areresponsibleforthis
remarkable physics.
85
Appendix A
Functions and constants
In chapter 5, a more general solution of supergravity has been considered. It
contains several degrees of freedom, that are non-trivially interrelated, and some
fairly bulky functions. Although the latter were already written out in chapter 5,
they shall be briey listed here again for completeness.
The functions used here are taken from eqs. (2:25), (2:26), (3:55), (4:2)(4:6)
of [73]:
V (r;) =
r+m
2(r
2
c
2
)
r+m
+
2k
R+m
Rrc
2
cos
p
r
2
+R
2
2Rrcosc
2
sin
2
(A.1)
!
0
(r;) =
1
2
(m
+
m
)cos+
2k
R+m
R
2
m
rR(rm
)cosc
2
sin
2
p
r
2
+R
2
2Rrcosc
2
sin
2
d (A.2)
!(r;) =
e
R
2(R+m
)
2
u
a
u
a
h
1
r+R
p
r
2
2Rrcos+R
2
(1cos) (A.3)
+
c
2
R
p
r
2
2Rrcos+R
2
sin
2
i
d (A.4)
^
k(r;) =!
M
V
d +!
0
(A.5)
86
L
a
(r;) =
(r+m
)(c
2
+m
r)
2m
(r
2
c
2
)
la
V
+u
a
(A.6)
L(r;) =
e
2
2m
2
1
V
l
a
l
a
e
2
c
2
(c+m
)
2
f
1
r+f
2
(m
+r)(m
+R)
u
a
u
a
(A.7)
M (r;) =
e
2m
l
a
L
a
+
e
2(m
+R)
Rr
m
+r
V +
(c
2
+m
r)(f
1
r+f
2
)
2c
2
(c+m
)
2
(r
2
c
2
)
u
a
u
a
: (A.8)
The constants e
, m
, l
a
, u
a
, k, f
1
, f
2
, and the NUT-charge, n, are connected
by the relations:
c
2
=m
+
m
2e
+
e
(A.9)
m
+
=c
1+
4k
c+m
+
2k
R+m
=
2kc
c+m
2m
c+m
n
(A.10)
R =m
+
2k(c+m
)
2
(c+m
)
2
4ck2k(c+m
)n
(A.11)
(n+1)k =
m
+
m
2
kR
R+m
(A.12)
f
1
=
m
+
(c+m)
2
(cR)+4c
2
k(R+m)
c+m
(A.13)
f
2
=
cm
+
(c+m)
2
(Rc)+4c
2
k(c
2
2RcRm)
c+m
: (A.14)
87
Appendix B
Asymptotic limits
B.1 Functions
TheasymptoticlimitsofthefunctionsattheboltandtheGibbons-Hawkingcenter
are:
r!c = 0;r!R
V
k
rc
k
jRrj
!
0
h
(m
m
+
)cos
2
(Rccos)
2
+(Rcosc)(c+m
)
Rccos
k
R+m
i
d
m
m
+
2
k
d (r7R)
!
e
R(1cos)
2(R+m
)
2
h
1+
c
2
(1+cos)R(c+R)
R
p
R
2
+c
2
2Rccos
i
u
a
u
a
d 0
^
k
e
R(1cos)
2(R+m
)
2
h
1+
c
2
(1+cos)R(c+R)
R
p
R
2
+c
2
2Rccos
i
u
a
u
a
d 0
L
a
(c+m
)
2
4km
l
a
+u
a
u
a
L
4ke
2
(c+m
)
3
Rc
R+m
u
a
u
a
e
2
(Rc)
2
c
2
(R+m
)
2
m
+
(c+m
)
2
4c
2
k
(c+m
)
3
u
a
u
a
M
V
0 0
Table 5: Asymptotics at the centers
Note: Forsomeofthefunctions,thelimitstowardstheGibbons-Hawkingpoint
88
are direction dependent; we chose the approach along the positive z-axis ( = 0
andr!R) since that is how the adjacent cycles run.
At in nity we have the limits, to leading orders:
V !
1
2
(B.1)
!
0
!
h
(m
+
m
)cos
2
+k
Rcos+m
R+m
i
d (B.2)
!!
e
2
R
2
c
2
(R+m
)
2
u
a
u
asin
2
r
d (B.3)
^
k!
h
d
(m
+
m
)cos
2
k
Rcos+m
R+m
d
i
(B.4)
L
a
!l
a
+u
a
(B.5)
L!
e
2
m
2
l
a
l
a
+
e
2
c
2
(c+m
)
m
+
(Rc)
R+m
4c
2
k
(c+m
)
2
u
a
u
a
(B.6)
M
V
!
e
m
l
a
(l
a
+u
a
)
e
2
h
(c+m
)c
2
m
m
+
(Rc)
c
2
(R+m
)(c+m
)
+
4km
(c+m
)
3
i
u
a
u
a
=
(B.7)
B.2 Fields and uxes
The elds and uxes yield at in nity:
F
1
!
m
2
c
2
2e
m
^
L
d
2
(B.8)
F
a
!
h
e
(2l
a
+u
a
)
m
la+ua
i
d
2
(B.9)
H
(2)
1
!
e
2
(u
a
a
2)d
2
(B.10)
+
h
1
4
2
a
1
la+ua
l
a
+u
a
e
m
l
a
a
i
d +!
0
+dt
H
(2)
a
!
e
2
u
a
1
c
2
m
2
4e
ab
b
d
2
(B.11)
+
1
2
h
l
a
+u
a
2e
m
l
a
1
+
^
L
ab
b
^
L(l
a
+u
a
)
i
d +!
0
+dt.
89
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Haas, Patrick A. (author)
Core Title
Supergravity solitons: topological contributions to the mass and the breaking of supersymmetry
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black holes,fuzzballs,oai:digitallibrary.usc.edu:usctheses,OAI-PMH Harvest,relativity,Smarr formula,spacetime solitons,string theory,supergravity,topology
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Warner, Nicholas (
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), Bars, Itzhak (
committee member
), Bonahon, Francis (
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), Däppen, Werner (
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Abstract (if available)
Abstract
In this thesis, we provide new insights to how mass arises from cohomology of spacetime—solitons using Smarr's formula. We introduce the mechanism with a historical review of its evolution to more generalized forms and move on to later important applications in the framework of the Komar integral formalism. In particular, this means presenting in detail its resulting forms and implications for different solutions of supergravity in topologically distinct spacetimes. The first situation in which we outline that non-zero mass for smooth and horizonless solutions can only be provided by cohomology, is an eleven-dimensional spacetime compactified on a general six-dimensional, Ricci-flat manifold
Tags
black holes
fuzzballs
relativity
Smarr formula
spacetime solitons
string theory
supergravity
topology
Linked assets
University of Southern California Dissertations and Theses