Supersymmetric Mechanics - Vol. 3: Attractors and Black Holes in Supersymmetric Gravity

374 Pages·2008·3.96 MB·english

by  Iosif Bena

Lecture Notes in Physics
FoundingEditors:W.Beiglbo¨ck,J.Ehlers,K.Hepp,H.Weidenmu¨ller
EditorialBoard
R.Beig,Vienna,Austria
W.Beiglbo¨ck,Heidelberg,Germany
W.Domcke,Garching,Germany
B.-G.Englert,Singapore
U.Frisch,Nice,France
F.Guinea,Madrid,Spain
P.Ha¨nggi,Augsburg,Germany
G.Hasinger,Garching,Germany
W.Hillebrandt,Garching,Germany
R.L.Jaffe,Cambridge,MA,USA
W.Janke,Leipzig,Germany
H.v.Lo¨hneysen,Karlsruhe,Germany
M.Mangano,Geneva,Switzerland
J.-M.Raimond,Paris,France
D.Sornette,Zurich,Switzerland
S.Theisen,Potsdam,Germany
D.Vollhardt,Augsburg,Germany
W.Weise,Garching,Germany
J.Zittartz,Ko¨ln,Germany
TheLectureNotesinPhysics
TheseriesLectureNotesinPhysics(LNP),foundedin1969,reportsnewdevelopments
in physics research and teaching – quickly and informally, but with a high quality and
theexplicitaimtosummarizeandcommunicatecurrentknowledgeinanaccessibleway.
Bookspublishedinthisseriesareconceivedasbridgingmaterialbetweenadvancedgrad-
uatetextbooksandtheforefrontofresearchandtoservethreepurposes:
• tobeacompactandmodernup-to-datesourceofreferenceonawell-definedtopic
• to serve as an accessible introduction to the field to postgraduate students and
nonspecialistresearchersfromrelatedareas
• tobeasourceofadvancedteachingmaterialforspecializedseminars,coursesand
schools
Both monographs and multi-author volumes will be considered for publication. Edited
volumes should, however, consist of a very limited number of contributions only. Pro-
ceedingswillnotbeconsideredforLNP.
VolumespublishedinLNParedisseminatedbothinprintandinelectronicformats,the
electronicarchivebeingavailableatspringerlink.com.Theseriescontentisindexed,ab-
stractedandreferencedbymanyabstractingandinformationservices,bibliographicnet-
works,subscriptionagencies,librarynetworks,andconsortia.
ProposalsshouldbesenttoamemberoftheEditorialBoard,ordirectlytothemanaging
editoratSpringer:
ChristianCaron
SpringerHeidelberg
PhysicsEditorialDepartmentI
Tiergartenstrasse17
69121Heidelberg/Germany
[email protected]
S. Bellucci (Ed.)
Supersymmetric
Mechanics - Vol. 3
Attractors and Black Holes in Supersymmetric Gravity
123
StefanoBellucci
IstitutoNazionalediFisica
Nucleare
LaboratoriNazionalidiFrascati
viaE.Fermi,40
00044FrascatiRM
Italy
[email protected]
Bellucci,S.(Ed.),SupersymmetricMechanics-Vol.3,Lect.NotesPhys.755(Springer,
BerlinHeidelberg2008),DOI10.1007/978-3-540-79523-0
ISBN:978-3-540-79522-3 e-ISBN:978-3-540-79523-0
DOI10.1007/978-3-540-79523-0
LectureNotesinPhysicsISSN:0075-8450
LibraryofCongressControlNumber:2006926534
(cid:2)c 2008Springer-VerlagBerlinHeidelberg
Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis
concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,
reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication
orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,
1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare
liabletoprosecutionundertheGermanCopyrightLaw.
Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply,
evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws
andregulationsandthereforefreeforgeneraluse.
Coverdesign:eStudioCalamarS.L.,F.Steinen-Broo,Pau/Girona,Spain
Printedonacid-freepaper
9 8 7 6 5 4 3 2 1
springer.com
Preface
This is the third volume in a series of books on the general topics of Supersym-
metric Mechanics, with the first and second volumes being published as Lecture
Notes in Physics Vol. 698, Supersymmetric Mechanics – Vol. 1: Supersymmetry,
NoncommutativityandMatrixModels(ISBN:3-540-33313-4),andLectureNotes
inPhysicsVol.701,SupersymmetricMechanics–Vol.2:TheAttractorMechanism
andSpaceTimeSingularities(ISBN:3-540-34156-0).
The aim of this ongoing collection is to provide a reference corpus of suitable,
introductorymaterialtothefield,bygatheringthesignificantlyexpandedandedited
versionsofalltutoriallectures,givenovertheyearsatthewell-establishedannual
INFN-LaboratoriNazionalidiFrascatiWinterSchoolontheAttractorMechanism,
directedbymyself.
The present set of notes results again from the participation and dedication of
prestigiouslecturers,suchasIosifBena,SergioFerrara,RenataKallosh,PerKraus,
FinnLarsen,andBorisPioline.Asusual,thelecturesweresubsequentlycarefully
edited and reworked, taking into account the extensive follow-up discussions. The
presentvolumeemphasizestopicsofgreatrecentinterest,namelygeneralconcepts
ofattractorsinsupersymmetricgravityandblackholes.
A two-parameter family of spherically symmetric, static, asymptotically flat,
electricallychargedsingularmetricsind=4isdescribedbytheso-calledReissner-
No¨rdstromsolution.Itmayberigorouslyshownthatthesphericallysymmetricso-
lution of N =2, d =4 Maxwell-Einstein supergravity represented by an extremal
Reissner-No¨rdstrom black hole preserves one-half of the supersymmetry isome-
tries out of the eight related to the asymptotical limit given by the N =2, d =4
Minkowskibackground.
Whenapproachingtheeventhorizonoftheblackhole,onegetsarestorationof
thepreviouslylostfouradditionalsupersymmetries,hencereobtainingamaximally
symmetric N =2 metric background, namely the 4-d Bertotti-Robinson AdS×S2
blackholemetric.
In the earlier book Supersymmetric Mechanics – Vol. 2, a general dynamical
principle was considered, namely the “attractor mechanism”, which governs the
dynamics inside the moduli space, with supersymmetry being related to dynami-
cal systems with fixed points describing the corresponding equilibrium state and
thestabilityproperties.Ifthismechanismholds,inapproachingsomefixedvalues,
v
vi Preface
whichdependsolelyupontheelectricandmagneticchargesofthetheory,theorbits
of the dynamical evolution lose all memory of their initial conditions, and yet the
overalldynamicsremainsfullydeterministic.Historically,thefirstattractorexample
insupersymmetricsystemsemergedfromtheconsiderationofextremeblackholes
inN=2, d=4,5Maxwell-Einsteinsupergravitiescoupledwithmattermultiplets.
In the present volume, some of the founders of the research in this field, interact-
ing among themselves, as well as with younger collaborators, yield a pedagogical
introductiontothesubject.
In his lectures, Iosif Bena (co-authored by Nick Warner) gives an introduc-
tion to the construction and analysis of three-charge configurations in string the-
ory and supergravity and describes the corresponding implications for the physics
of black holes in string theory. Sergio Ferrara (co-authored by Mike Duff) re-
views some recently established connections between the mathematics of black
hole entropy in string theory and that of multipartite entanglement in quantum in-
formation theory, a topic that could be of great interest also for experimental test-
ing and perhaps even for potential applications. The lectures by Renata Kallosh
(co-authored by Stefano Bellucci, Sergio Ferrara, and Alessio Marrani) provides
a pedagogical, introductory review of the Attractor Mechanism (at work in two
different 4-dimensional frameworks: extremal black holes in N =2 supergravity
and N =1 flux compactifications. AdS black holes and their connection to two-
3
dimensionalconformalfieldtheoriesviatheAdS/CFTcorrespondencearethesub-
ject of the lectures by Per Kraus, including background material on gravity in
AdS , in the context of the holographic renormalization. Also Finn Larsen in his
3
lectures yields a pedagogical introduction to the attractor mechanism, in particu-
lar in five dimensions, concentrating chiefly on supersymmetry-preserving black
holes in five dimensions, both with and without spherical symmetry, being mo-
tivated essentially by the consideration of black rings, as well as rotating black
holes. Pioline in his contribution “Black Holes, Topological Strings and Quantum
Attractors” reviews recent developments on the relation between the macroscopic
entropy of four-dimensional BPS black holes and the microscopic counting of
states.
I wish to thank all lecturers and participants of the School for contributing to
createanalmostmagicalatmospheretoprogressinthelearningandthefurtherre-
searching in this absolutely fascinating topic. I wish to thank most warmly Mrs.
SilviaColasantiforhergenerouseffortsinthesecretarialworkandinvariousorga-
nizationalaspects.MygratitudegoestoINFNandinparticulartoMarioCalvettifor
supportingtheSchool.InwelcomingourbrandnewdaughterErica,mythoughtsgo
tomywifeGloriaandourbelovedCostanza,Eleonora,andAnnalisaforproviding
meeverydayjoy,withoutwhichIcouldneverhaveaccomplishedthiseffort.
Frascati,December2007 StefanoBellucci
Contents
BlackHoles,BlackRings,andtheirMicrostates...................... 1
IosifBenaandNicholasP.Warner
1 Introduction.............................................. 1
2 Three-ChargeMicroscopicConfigurations .................... 6
3 BlackRingsandSupertubes ................................ 16
4 Geometric Interlude: Four-Dimensional Black Holes
andFive-DimensionalFoam ................................ 26
5 SolutionsonaGibbons-HawkingBase ....................... 31
6 BubbledGeometries....................................... 45
7 MicrostatesforBlackHolesandBlackRings.................. 57
8 MergersandDeepMicrostates .............................. 63
9 ImplicationsforBlack-HolePhysics ......................... 78
References..................................................... 87
BlackHoleEntropyandQuantumInformation ...................... 93
M.J.DuffandS.Ferrara
1 BlackHolesandQubits .................................... 93
2 TheN=2STUModel...................................... 95
3 Cayley’sHyperdeterminant ................................. 98
4 BlackHoleEntropy .......................................100
5 TheN=8Generalization ..................................101
6 DecompositionofE ....................................102
7(7)
7 TripartiteEntanglementof7Qubits ..........................103
8 Cartan’sE Invariant ....................................105
7(7)
9 TheBlackHoleAnalogy ...................................108
10 Subsectors ...............................................111
11 Conclusions..............................................112
References.....................................................113
vii
viii Contents
ExtremalBlackHoleandFluxVacuaAttractors ..................... 115
S.Bellucci,S.Ferrara,R.KalloshandA.Marrani
1 Introduction..............................................116
2 SpecialKa¨hlerGeometry...................................123
3 Extremal Black Hole Attractor Equations inN = 2, d = 4
(ungauged)Supergravity ...................................129
4 FluxVacuaAttractorEquationsinN =1,d =4Supergravity
fromTypeIIBon CY3×T2 ...................................160
Z
2
5 SomeRecentDevelopmentsonExtremalBlackHoleAttractors ..176
References.....................................................186
LecturesonBlackHolesandtheAdS /CFT Correspondence.......... 193
3 2
P.Kraus
1 Introduction..............................................193
2 GravityinAsymptoticallyAdS Spacetimes...................197
3
3 ChargedBlackHolesandChern-SimonsTerms ................210
4 StringTheoryConstructions ................................215
5 PartitionFunctionsandEllipticGenera .......................225
6 ComputationofPartitionFunctionsinGravity:Warmup
Examples................................................231
7 ComputationofPartitionFunctionsinSupergravity.............235
8 ComputationofBPSSpectra................................240
9 Conclusion...............................................243
References.....................................................244
TheAttractorMechanisminFiveDimensions ....................... 249
F.Larsen
1 Introduction..............................................249
2 TheBasicsoftheAttractorMechanism.......................250
3 BlackRingAttractors......................................262
4 ExtremizationPrinciples ...................................275
References.....................................................280
Lectures on Black Holes, Topological Strings, and Quantum
Attractors(2.0) ................................................. 283
B.Pioline
1 Introduction..............................................283
2 ExtremalBlackHolesinStringTheory .......................285
3 SpecialGeometryandBlackHoleAttractors ..................293
4 TopologicalStringPrimer ..................................311
5 HigherDerivativeCorrectionsandTopologicalStrings ..........323
6 PrecisionCountingofSmallBlackHoles .....................329
7 QuantumAttractorsandAutomorphicPartitionFunctions .......336
8 Conclusion...............................................365
References.....................................................366
Black Holes, Black Rings, and their Microstates
IosifBenaandNicholasP.Warner
Abstract In this review article, we describe some of the recent progress towards
the construction and analysis of three-charge configurations in string theory and
supergravity. We begin by describing the Born-Infeld construction of three-charge
supertubes with two dipole charges and then discuss the general method of con-
structingthree-chargesolutionsinfivedimensions.Weexplainindetailtheuseof
these methods to construct black rings, black holes, as well as smooth microstate
geometrieswithblackholeandblackringcharges,butwithnohorizon.Wepresent
argumentsthatmanyofthesemicrostategeometriesaredualtoboundarystatesthat
belongtothesamesectoroftheD1-D5-PCFTasthetypicalstates.Weendwithan
extendeddiscussionoftheimplicationsofthisworkforthephysicsofblackholes
instringtheory.
1 Introduction
Blackholesareveryinterestingobjects,whosephysicsbringsquantummechanics
and general relativity into sharp contrast. Perhaps the best known, and sharpest,
exampleofsuchcontrastisHawking’sinformationparadox[1,2].Thishasprovided
averyvaluableguideandtestinggroundinformulatingaquantumtheoryofgravity.
Indeed, it is one of the relatively few issues that we know must be explained by a
viabletheoryofquantumgravity.
String theory is a quantum theory of gravity and has had several astounding
successes in describing properties of black holes. In particular, Strominger and
Vafa have shown [3] that one can count microscopic configurations of branes and
strings at zero gravitational coupling and exactly match their statistical entropy to
IosifBena
Service de Physique The´orique, CEA/Saclay, F-91191 Gif-sur-Yvette Cedex, France,
[email protected]
Nicholas.P.Warner
Department of Physics and Astronomy, University of Southern California, Los Angeles, CA
90089-0484,USA,[email protected]
Bena,I., Warner, N.P.: BlackHoles, BlackRings and theirMicrostates. Lect.Notes Phys. 755,
1–92(2008)
DOI10.1007/978-3-540-79523-0 1 (cid:2)c Springer-VerlagBerlinHeidelberg2008