Introduction AdS/CFT correspondence Quantum entropy function Kerr/CFT correspondence Final comments Appendix

Black Holes and AdS/CFT

Ashoke Sen

Harish-Chandra Research Institute, Allahabad, India

Hamburg, July 2009

Introduction AdS/CFT correspondence Quantum entropy function Kerr/CFT correspondence Final comments Appendix

AdS/CFT has played an important role in developing ourunderstanding of string theory.

A large part of the studies in AdS/CFT correspondenceinvolves black holes in some way.

In particular in this conference there have already beentalks on the application of AdS/CFT to the study ofhydrodynamics of strongly coupled gauge theories.

These studies involve black holes in an essential way.

In view of this I shall focus this talk on a different aspect ofthe relation between black holes and AdS/CFTcorrespondence.

Introduction AdS/CFT correspondence Quantum entropy function Kerr/CFT correspondence Final comments Appendix

Extremal Black Holes and AdS2/CFT1

This is perhaps the least understood case of AdS/CFTcorrespondence.

We shall study two aspects:

1. Quantum entropy functionA.S., arXiv:0809.3304 + · · ·

2. Kerr/CFT correspondence.Guica, Hartman, Song, Strominger, arXiv:0809.4266

+ · · ·

Introduction AdS/CFT correspondence Quantum entropy function Kerr/CFT correspondence Final comments Appendix

IntroductionOne of the successes of string theory has been anexplanation of the Bekenstein-Hawking entropy of a classof supersymmetric black holes in terms of microscopicquantum states.

SBH(~Q) = ln dmicro(~Q)

Strominger, Vafa

dmicro(~Q): degeneracy of microstates carrying a given setof charges ~Q

SBH(~Q) = A/4GN

Introduction AdS/CFT correspondence Quantum entropy function Kerr/CFT correspondence Final comments Appendix

This formula is quite remarkable since it relates ageometric quantity in space-time to a counting problem.

One would clearly like to have a better understanding ofthis corresponence / find generalizations.

We shall study this question in the context of extremal, ı.e.zero temperature black holes.

Since they do not Hawking radiate, the notion ofdegeneracy is better defined for these black holes.

Often, but not always, they preserve part of thesupersymmetry, and hence are stable.

Introduction AdS/CFT correspondence Quantum entropy function Kerr/CFT correspondence Final comments Appendix

What kind of qusetions would we like to address?

1. The Bekenstein-Hawking formula needs generalizationin string theory where the action contains higher derivativecorrections and quantum corrections.

The effect of higher derivative corrections are taken intoaccount in a more general formula due to Wald, but whatabout quantum corrections?

– Goal of the quantum entropy function formalism.

This is necessary if we want to make a precisioncomparison between the microscopic and themacroscopic entropies of the black hole.

Introduction AdS/CFT correspondence Quantum entropy function Kerr/CFT correspondence Final comments Appendix

An example: Spectrum of a class of supersymmetric statesin heterotic string theory compactified on a sixdimensional torus.

charge2 degeneracy dmicro ln dmicro SBH

2 50064 10.82 6.28

4 32861184 17.31 12.57

6 16193130552 23.51 18.85

8 7999169992704 29.71 25.13

10 4074192429737760 35.943 31.42

Introduction AdS/CFT correspondence Quantum entropy function Kerr/CFT correspondence Final comments Appendix

2. Understand the entropy of extremal non-supersymetricblack holes e.g. extreme Kerr black hole from amicroscopic viewpoint.

– Goal of Kerr/CFT correspondence.

Eventually we would like to use these as the starting pointfor studying the thermodynamics of non-extremal blackholes.

Introduction AdS/CFT correspondence Quantum entropy function Kerr/CFT correspondence Final comments Appendix

In both studies AdS2/CFT1 correspondence will play animportant role.

What is AdS2?

AdS2 may be regarded as a two dimensional Lorentzianspace embedded in a 3-dimensional space of signature(+,-,-) via the relation:

x2 − y2 − z2 = −a2

a: some constant giving the radius of AdS2.

This space has an SO(2,1) isometry.

Introduction AdS/CFT correspondence Quantum entropy function Kerr/CFT correspondence Final comments Appendix

Why AdS2?

All known extremal black holes have an AdS2 factor in theirnear horizon geometry.

– time translation symmetry gets enhanced to SO(2, 1) inthe near horizon limit.

Introduction AdS/CFT correspondence Quantum entropy function Kerr/CFT correspondence Final comments Appendix

Reissner-Nordstrom solution in D = 4:

ds2 = −(1− ρ+/ρ)(1− ρ−/ρ)dτ2

+dρ2

(1− ρ+/ρ)(1− ρ−/ρ)

+ρ2(dθ2 + sin2 θdφ2)

Define

2λ = ρ+ − ρ−, t =λ τ

ρ2+

, r =2ρ− ρ+ − ρ−

2λ

and take λ → 0 limit keeping r, t fixed.

ds2 = ρ2+

[−(r2 − 1)dt2 +

dr2

r2 − 1

]+ ρ2

+(dθ2 + sin2 θdφ2)

AdS2 × S2

Introduction AdS/CFT correspondence Quantum entropy function Kerr/CFT correspondence Final comments Appendix

Postulate: Any extremal black hole has an AdS2 factor /SO(2, 1) isometry in the near horizon geometry.

– partially proved

Kunduri, Lucietti, Reall; Figueras, Kunduri, Lucietti, Rangamani

Introduction AdS/CFT correspondence Quantum entropy function Kerr/CFT correspondence Final comments Appendix

This simple postulate leads to non-trivial consequences.

1. It gives a proof of attractor mechanism in any classicaltheory of gravity with or without higher derivative terms.

The entropy of an extremal black hole depends only on itscharges and not on any other asymptotic data, e.g. thevacuum expectation value of various moduli scalar fields.

2. For spherically symmetric extremal black holes thisgives a simple method for computing the entropy withoutsolving any differential equation.

Introduction AdS/CFT correspondence Quantum entropy function Kerr/CFT correspondence Final comments Appendix

Review of AdS/CFT Correspondence

We begin by describing the basic observations / postulatesleading to the AdS/CFT correspondence.

1. The boundary of euclidean AdSd+1 is a d-dimensionalsphere Sd.

2. Given a string theory in euclidean AdSd+1 there is anassociated d dimensional euclidean CFT on Sd such that

Zgravity = ZCFT

Zgravity: result of path integral of the string theory onAdSd+1.

ZCFT: result of the path integral over the CFT fields on Sd.

Introduction AdS/CFT correspondence Quantum entropy function Kerr/CFT correspondence Final comments Appendix

3. Often the AdSd+1 background arises as the near horizongeometry of an extremal black brane.

In this case there is a simple way to identify the dual CFT.

It is the theory obtained by taking the low energy limit ofthe quantum theory on the brane system that produces theblack hole geometry.

We shall now apply this to AdS2/CFT1 correspondence.

Introduction AdS/CFT correspondence Quantum entropy function Kerr/CFT correspondence Final comments Appendix

Quantum entropy functionObservations:

1. In all known examples in string theory the microscopicsystem that produces the black hole has an energy gapthat separates the ground states from the first excitedstate.

For fixed values of charges, only the ground states survivein the low energy limit .

2. For d = 1, Sd is a circle S1 of some lenth L.

⇒ ZCFT = Tr(e−LH) = dmicro

dmicro: ground state degeneracy

Introduction AdS/CFT correspondence Quantum entropy function Kerr/CFT correspondence Final comments Appendix

AdS2/CFT1 correspondence now implies:

Zgravity = ZCFT = dmicro

We declare ln Zgravity to be the quantum generalization ofthe black hole entropy.

– Quantum entropy function.

The equality between black hole entropy and ln dmicro nowbecomes a consequence of AdS2/CFT1 correspondence.

Introduction AdS/CFT correspondence Quantum entropy function Kerr/CFT correspondence Final comments Appendix

Consistency check: In the classical limit this reduces tothe exponential of the Wald entropy.

This exact formula serves as a starting point for computingsystematic corrections to the black hole degeneracy andcomparing it with the microscopic result where the resultis known.

There has been several checks including comparison ofsome non-perturbative corrections on both sides.

However lot more work is still left to be done to check thecomplete equality between Zgravity and dmicro.

Introduction AdS/CFT correspondence Quantum entropy function Kerr/CFT correspondence Final comments Appendix

Kerr/CFT correspondenceLike all extremal black holes, extremal rotating black holesalso have AdS2 factor in their near horizon geometry.

Thus we might expect to apply AdS2/CFT1correspondence for studying these black holes.

However here we proceed somewhat differently.

Introduction AdS/CFT correspondence Quantum entropy function Kerr/CFT correspondence Final comments Appendix

In conventional approach to defining the quantum theoryon AdSd+1 we choose the asymptotic boundary conditionon different fields in such a way that asymptotic structureof space-time remains unmodified.

In AdS2 this essentially forces the system to be in itsground state, carrying a fixed set of charges.

In the study of Kerr/CFT correspondence we relax this andallow for a more general class of field configurations.

Introduction AdS/CFT correspondence Quantum entropy function Kerr/CFT correspondence Final comments Appendix

For this more general class of boundary conditions we canfind a set of general coordinate transformations whichpreserve the boundary condition and generate a Virasoroalgebra.

The central charge of the algebra is

c = 12J/~

J: classical angular momentum.

Introduction AdS/CFT correspondence Quantum entropy function Kerr/CFT correspondence Final comments Appendix

Extremal black holes have zero temperature.

However Kerr black holes have two isometries, – timetranslation and rotation – and hence one can associate two‘temperatures’ to this black hole.

The temperature associated with one linear combination ofthe two isometries must vanish due to the extremalitycondition.

However the ‘temperature’ associated with the other linearcombination is non-zero and takes the value 1/2π.

Using the information about the central charge and thetemperature one can calculate the entropy.

Introduction AdS/CFT correspondence Quantum entropy function Kerr/CFT correspondence Final comments Appendix

Entropy =π2

3c T =

π2

312J~

12π

=2πJ~

This agrees with the Bekenstein-Hawking entropy.

This provides a possible understanding of the entropy of aKerr black hole as the thermal entropy of a dual CFT.

– Kerr/CFT correspondence.

This analysis has been generalized to theories other thanpure Einstein gravity.

Introduction AdS/CFT correspondence Quantum entropy function Kerr/CFT correspondence Final comments Appendix

Final commentsAdS2/CFT1 is the least understood of all the dualities.

However this is the duality that allows us to focus directlyon extremal black holes for which the microscopic side ofthe story is much better understood.

Thus a deeper understanding of the AdS2/CFT1correspondence is likely to shed further light on thecorrespondence between the microscopic entropy and theblack hole entropy.

Conversely, tests of equality between black hole entropyand ln dmicro can be used to provide precision tests ofAdS2/CFT1 correspondence.

Introduction AdS/CFT correspondence Quantum entropy function Kerr/CFT correspondence Final comments Appendix

A (not completely systematic) one loop calculation givesthe following result for the entropy of black holes inheterotic string theory on a six-torus:

charge2 degeneracy dmicro ln dmicro SBH S1−loop

2 50064 10.82 6.28 10.62

4 32861184 17.31 12.57 16.90

6 16193130552 23.51 18.85 23.19

8 7999169992704 29.71 25.13 29.47

10 4074192429737760 35.943 31.42 35.75

Cardoso, de Wit, Kapelli, Mohaupt; David, Jatkar, A.S.