black hole microstate counting and their macroscopic ...€¦ · black holes are classical...

Introduction Summary Microstate counting Macroscopic analysis

Black Hole Microstate Counting and TheirMacroscopic Counterpart

Ashoke Sen

Harish-Chandra Research Institute, Allahabad, India

CERN, January 2010


Plan1. Introduction

2. Summary

3. Microstate counting

4. Macroscopic analysis

Refs: arXiv:0708.1270, arXiv:0809.3304


Black Holes

Black holes are classical solutions of the equations ofmotion of general theory of relativity.

Their gravitational attraction is so large that even lightcannot escape a black hole.

Thus classically they behave as perfect black bodies atzero temperature.

Each black hole is surrounded by an event horizon thatacts as a one way membrane.


In quantum theory a black hole behaves as athermodynamic system with definite temperature, entropyetc.

T =κ

2π, SBH =

A4 GN

Bekenstein, Hawking

κ: acceleration due to gravity at the horizon of the blackhole.

A: Area of the event horizon

GN: Newton’s gravitational constant

Our units: ~ = c = kB = 1


For ordinary objects the entropy of a system has amicroscopic interpretation.

We fix the macroscopic parameters (e.g. total electriccharge, energy etc.) and count the number of quantumstates – known as microstates – each of which has thesame charge, energy etc.

dmicro: number of such microstates

Define microscopic (statistical) entropy:

Smicro = ln dmicro

Question: Does the entropy of a black hole have a similarstatistical interpretation?


Does the entropy of a black hole have a statisticalinterpretation?

Answering this question in the affirmative is essential forany consistent theory of quantum gravity. Hawking

Otherwise it leads to violation of the laws of quantummechanics.


In order to investigate the statistical origin of black holeentropy we need a quantum theory of gravity.

We shall carry out our investigation in string theory.

Even though there is a unique string (M)-theory, it can existin many different stable and metastable phases.

Without knowing precisely which phase of string theorydescribes the part of the universe we live in, we cannotdirectly compare string theory to experiments.

However there are some issues, like the issues involvingblack hole thermodynamics, which are universal, andhence can be addressed in any phase of string theory.


Convenient phases: supersymmetric phases

– have a symmetry that relates bosonic states to fermionicstates.

Many aspects of black hole thermodynamics have beenstudied in such supersymmetric phases of the theory, butwe shall focus our attention on one particular aspect.

– entropy of the black hole in the zero temperature limit(supersymmetric, extremal black holes).

Advantage: Such a black hole is a stable state of thetheory.


Strategy:

1. Identify a supersymmetric black hole carrying a certainset of electric charges {Qi} and magnetic charges {Pi} andcalculate its entropy SBH(Q,P) using theBekenstein-Hawking formula.

Note: since we are considering a generic phase of stringtheory, it may have more that one Maxwell field and hencemultiple charges.


2. Identify the supersymmetric quantum states in stringtheory carrying the same set of charges.

These will include the fundamental strings but also otherobjects in string theory which are required for consistencyof string theory.

Calculate the number dmicro(Q,P) of these states.

3. Compare Smicro ≡ ln dmicro(Q,P) with SBH(Q,P).


For a class of supersymmetric extremal black holes instring theory one indeed finds a match:

A/4GN = ln dmicro

Strominger, Vafa

This formula is quite remarkable since it relates ageometric quantity in black hole space-time to a countingproblem that does not make any direct reference to blackholes.


The Bekenstein-Hawking formula is an approximateformula that holds in classical general theory of relativity.

While string theory gives a theory of gravity that reducesto Einstein’s theory when gravity is weak, there arecorrections.

Thus the Bekenstein-Hawking formula for the entropyworks well only when gravity at the horizon is weak.

Typically this requires the charges to be large.


The calculation on the microscopic side also simplifieswhen the charges are large.

Instead of doing exact counting of quantum states, we canuse approximations which give the result for large charges.


For ordinary systems, thermodynamics provides anapproximate description that becomes exact in the limit oflarge volume.

Is the situation with black holes similar, ı.e. they onlycapture the information about the system in the limit oflarge charge and mass?

Or, could it be that the relation A/4GN = ln dmicro is anapproximation to an exact result?

Our goal will be to search for an exact formula to which theabove is an approximation


In order to address this issue we have to work on twofronts.

1. Count the number of microstates to greater accuracy.

2. Calculate black hole entropy to greater accuracy.

We can then compare the two to see if they agree beyondthe large charge limit.

In these lectures I shall describe the progress on bothfronts.


Progress in microscopic countingIn a wide class of phases of string theory with 16 or moreunbroken supercharges one now has a completeunderstanding of the microscopic ‘degeneracies’ ofsupersymmetric black holes.

Typically such theories have multiple Maxwell fields.

⇒ the black hole is characterized by multiple electric andmagnetic charges, collectively denoted by (Q,P).

The degeneracy is expressed as a function dmicro(Q,P) ofthe charges.

Dijkgraaf, Verlinde, Verlinde; Shih, Strominger, Yin;David, Jatkar, A.S.; Dabholkar, Gaiotto, Nampuri;Banerjee, Srivastava, A.S.; Dabholkar, Gomes, Murthy; · · ·


In these theories dmicro(Q,P) is expressed as Fourierexpansion coefficients of some well-known (and not sowell-known) functions, e.g. Jacobi theta functions, Igusacusp forms etc.

⇒ ‘experimental data’ to be explained by a ‘theory of blackholes’.

In the large charge limit these degeneracies agree with theexponential of the Bekenstein-Hawking entropy of blackholes carrying the same set of charges

In these lectures I shall explain the counting for oneparticular example and then describe the results in variousother examples.


Progress in black hole entropy computation

Can we find an exact formula for the black hole entropythat can be compared with ln dmicro(Q,P)?

This will require us to take into account

1. Stringy (α′) corrections.

2. Quantum (gs) corrections.

We shall describe an approach to finding such a generalformula for black hole entropy using AdS2/CFT1correspondence, and then compare the results to themicroscopic results.


The role of index

The counting of microstates is always done in a regionwhere gravity is weak and hence the states do not form ablack hole.

In order to be able to compare it with the black holeentropy we must focus on quantities which do not changeas we change the coupling from small to large value.

– needs appropriate supersymmetric index.

The appropriate index in D=4 is the helicity trace index.Gregori, Kiritsis, Kounnas, Obers, Petropoulos, Pioline


Suppose we have a BPS state that breaks 4nsupersymmetries.

→ there will be 4n fermion zero modes (goldstino) on theworld-line of the state.

Quantization of these zero modes will produce Bose-Fermidegenerate states.

Thus Tr(−1)F vanishes.

Define: B2n = 1(2n)!Tr(−1)F(2h)2n = Tr(−1)2h(2h)2n

h: third component of angular momentum in rest frame.

For every pair of fermion zero modes Tr(−1)F(2h) gives anon-vanishing result, leading to a non-zero B2n.


Note: Since on the microscopic side we compute an index,we must ensure that on the black hole side also wecompute an index.

Otherwise we cannot compare the two results.


An example

We consider type IIB string theory on K 3× S1 × S1 with thefollowing configuration:

1) One Kaluza-Klein monopole along S1

2) One D5-brane wrapped on K 3× S1

3) (Q1 + 1) D1-branes wrapped on S1

4) −n units of momentum along S1

5) J units of momentum along S1

– BMPV black hole at the center of Taub-NUT

Shih, Strominger, Yin


Although this appears to be a very special system, a largefraction of the dyons in this theory can be related to this bya duality transformation.

Thus by knowing the index associated with this specialsystem, we can calculate the index associated with a finitefraction of all the states in the theory.


This system breaks 12 of the 16 supersymmetries.

Thus the relevant index is B6.

We shall denote the B6 associated with this system by−d(Q1,n,J).

Define

Z (ρ, σ, v) =∑

Q1,n,J

(−1)J+1 d(Q1,n, J) e2πi(Q1ρ+nσ+Jv)


In the weakly coupled type IIB description the low energydynamics of the system is described by three weaklyinteracting pieces:

1) The closed string excitations around the Kaluza-Kleinmonopole

2) The dynamics of the D1-D5 center of mass coordinate inthe Kaluza-Klein monopole background

3) The motion of the D1 branes along K3.


The dyon partition function is obtained as the product ofthe partition function of these three subsystems.

We calculate the partition function of each subsystem as afunction of (ρ, σ, v) and then take their product. David,A.S.

Taking the size of S1 to be large compared to otherdimensions we can regard each subsystem as a 1+1dimensional CFT.

BPS condition forces the modes carrying positivemomentum along S1 (right-moving modes) to be frozeninto their ground state.

Only left-moving modes can be excited.


Dynamics of Kaluza-Klein monopole

1. 24 left-moving and 8 right-moving bosons, arising from

– the 3 transverse motion (3+3)

– component of 2-form fields along the harmonic 2-form inTaub-NUT (2+2)

– component of the 4-form field along the wedge productof the harmonic 2-form on Taub-NUT and a harmonic2-form on K3 (19+3)

2. 8 right-moving fermions arising as goldstinos.

ZKK = e−2πiσ∞∏

n=1

{(1− e2πinσ)−24

}


Dynamics of the D1-D5 center of mass motion in the KKmonopole background

This is described by a supersymmetric sigma model withTaub-NUT space as the target space.

By taking the size of the Taub-NUT space to be large wecan take the oscillator modes to be those of a free fieldtheory but the zero mode dynamics is described by asupersymmetric quantum mechanics problem.

ZCM =∞∏

n=1

{(1− e2πinσ)4 (1− e2πinσ+2πiv )−2 (1− e2πinσ−2πiv )−2}

×e−2πiv (1− e−2πiv )−2


D1-brane motion along K3

1. First consider a single D1-brane, wrapped w times alongS1 and carrying fixed momenta along S1 and S1.

– the dynamics is described by a supersymmetric sigmamodel with target space K3

– the number of states of this system can be counted bystandard method, by going to the orbifold limit.

2. A generic state contains multiple D1-branes of this type,carrying different amounts of winding along S1 anddifferent momenta along S1 and S1.

Total number of states can be determined from the resultof step 1 by simple combinatorics.


Result:

ZD1 = e−2πiρ∏

l,b,k∈Zk≥0,l>0

{1− exp(2πi(kσ + lρ+ bv))

}−c(4lk−b2)

Dijkgraaf, Moore, Verlinde, Verlinde

Definition of c(n):

F (τ, z) ≡ 8[ϑ2(τ, z)2

ϑ2(τ,0)2 +ϑ3(τ, z)2

ϑ3(τ,0)2 +ϑ4(τ, z)2

ϑ4(τ,0)2

]F (τ, z) =

∑b∈zz,n

c(4n − b2) qn e2πizb

Physically c(4n − b2) counts the number of BPS states inthe supersymmetric sigma model with target space K3.


After taking the product we get

Z = e−2πiρ∏

l,b,k∈Zk≥0,l≥0,b<0 for k=l=0

{1− exp(2πi(kσ + lρ+ bv))

}−c(4lk−b2)

Z (ρ, σ, v) = 1/Φ10(ρ, σ, v)

Φ10: weight 10 Igusa cusp form of Sp(2,ZZ).Dijkgraaf, Verlinde, Verlinde


Z (ρ, σ, v) =∑

Q1,n,J

(−1)J+1 d(Q1,n, J) e2πi(Q1ρ+nσ+Jv)

↓

d(Q1,n, J) = (−1)J+1∫

dρdσdv e−2πi(Q1ρ+nσ+Jv) Z (ρ, σ, v)

We shall express this in a more duality invariant notationby defining

P2 = 2Q1, Q2 = 2n, Q.P = J

This is motivated by going to the heterotic on T 6 frame.

In this frame Q and P refers to electric and magneticcharges and Q2, P2 and Q.P are T-duality invariant innerproducts.


Asymptotic expansion

d(Q,P) = (−1)Q.P+1∫

dρdσdv e−πi(P2ρ+Q2σ+2Q.Pv) Z(ρ, σ,v)

In order to compare this with the black hole entropy weneed to find its behaviour for large Q2, P2, Q.P.

Z has poles at the zeroes of Φ10.

We perform one of the three integrals using residuetheorem, picking up contributions from various poles.


The leading contribution comes from the pole at

(ρσ − v2) + v = 0

After picking up the residue at the pole we are left with atwo dimensional integral:

d(Q,P) '∫

d2τ e−F(Q2,P2,Q.P,τ1,τ2)

We evaluate this integral by saddle point method

Expand F around its extremum and carry out the integralusing perturbation theory.

Perturbation parameter: inverse of Q2, P2, Q.P.


Result for ln d(Q,P) :

π√

Q2P2 − (Q.P)2

−φ

(Q.PP2 ,

√Q2P2 − (Q.P)2

P2

)+O

(1

Q2,P2,Q.P

)φ(τ1, τ2) ≡ 12 ln τ2 + 24 ln η(τ1 + iτ2) + 24 ln η(−τ1 + iτ2)

Cardoso, de Wit, Kappeli, Mohaupt


Walls of marginal stability

Our result for the D1-D5-KK monopole system was derivedfor weakly coupled type IIB string theory.

As we move around in the moduli space we may hit wallsof marginal stability at which the quarter BPS dyon underconsideration become unstable against decay into a pair ofhalf-BPS dyons.

At these walls the degeneracy jumps and hence we cannottrust our formula on the other side of the wall.

Can we compute the degeneracy on the other side?


It turns out that with the help of S-duality we can alwaysbring the moduli to a domain where the type IIB theory is inthe weakly coupled domain and we can trust our originalformula.

The net outcome of this analysis is that in differentdomains the index is given by the formula:

d(Q,P) = (−1)Q.P+1∫

Cdρdσdv e−πi(P2ρ+Q2σ+2Q.Pv) Z (ρ, σ, v)

C: A choice of ‘contour’ that picks a 3 real dimensionalsubspace of integration in the 3 complex dimensionalspace.

The choice of C depends on the domain in the modulispace where we compute the index.

A.S.; Dabholkar, Gaiotto, Nampuri; Cheng, Verlinde


d(Q,P) = (−1)Q.P+1∫

Cdρdσdv e−πi(P2ρ+Q2σ+2Q.Pv) Z (ρ, σ, v)

The jumps in the index across walls of marginal stabilityare encoded in the residues at the poles in Z that weencounter while deforming the contour corresponding toone domain to the contour corresponding to the otherdomain.

On the black hole side these jumps correspond to(dis-)appearance of two-centered black holes as we crosswalls of marginal stability.


Other duality orbits:

We have said that the results are valid for a subset of dyonswhich can be related via duality transformation to the D1-D5-psystem analyzed here.

What about dyons which are outside these duality orbits?

The inequivalent duality orbits are characterized by an integer rand can be related to a system of IIB on K3× S1 × S1 with:

1. 1 KK monopole along S1.

2. r D5-branes wrapped on K3× S1

3. (Q1 + 1)r D1-branes wrapped on S1

4. −n units of momentum along S1 Dabholkar, Gaiotto, Nampuri

5. rJ units of momentum along S1. Banerjee, A.S.


Can we derive the −B6 index for these dyons?

This has not yet been done from first principles.

But a guess has been made.

In the domain of the moduli space where 2-centered blackholes are absent ∑

s|r

d(

Q1rs,n,J

rs

)d(Q1,n,J) is the same function that appeared earlier.

Banerjee, Srivastava, A.S.; Dabholkar, Gomes, Murthy

During the rest of the lectures we shall focus on the r = 1case.


Generalization I: Twisted index

Take type IIB theory on K3× S1 × S1.

On subspaces of the moduli space of K3, we encounterenhanced discrete symmetries which preserve theholomorphic (2,0)-form on K3.

Thus these symmetries commute with supersymmetry.

Let us work on such a subspace of the moduli space with aZZN discrete symmetry generated by g.

Consider the index

Bg6 =

16!

Tr((−1)F(2h)6g

)


Bg6 can be calculated using the same method described

earlier by keeping track of the g quantum numbers ofvarious modes which contribute to the partition function.

Result:

Bg6(Q,P) = (−1)Q.P

∫C

dρdσdv e−πi(P2ρ+Q2σ+2Q.Pv) Zg(ρ, σ,v)

Zg: a different function, also with nice modular properties,and poles in the complex (ρ, σ, v) space.

We can find the behaviour of this for large charges by thesame method described earlier.

Result:ln |B6

g(Q,P)| ' π√

Q2P2 − (Q.P)2/N


Generalization 2: CHL models

Begin with type IIB string theory on K3× S1 × S1 with a ZZNsymmetry as described in the previous case.

Take an orbifold of this theory by g accompanied by 2π/Nshift along S1.

Can we calculate B6 in this theory?

Again it can be calculated in the same way as beforekeeping track of the g quantum number of various modesand also including contribution from twisted sector states.


Result:

Bg6(Q,P) == (−1)Q.P

∫C


Zg(ρ, σ,v) is yet another new function, also with nicemodular properties and poles in the (ρ, σ, v) space.

We can calculate its behaviour for large charges as before.


Result for ln |B6(Q,P)| for large charges:

π√

Q2P2 − (Q.P)2

−φ

(Q.PP2 ,

√Q2P2 − (Q.P)2

P2

)+O

(1

Q2,P2,Q.P

)φ(τ1, τ2) ≡ (k + 2) ln τ2 + ln g(τ1 + iτ2) + ln g(−τ1 + iτ2)

k are known numbers and g(τ) are known functions,depending on the choice of N.

Note: In the original type IIB string theory on K3× S1 × S1,

k = 10, g(τ) = η(τ)24


Generalization III: Twisted index in CHL models

Consider type IIB string theory on K3× S1 × S1 with aZZM × ZZN discrete symmetry that commutes withsupersymmetry.

gM: generator of ZZM

gN: generator of ZZN.

Now consider an orbifold of this theory by a ZZM symmetrygenerated by gM together with 1/M unit of shift along S1.

gN still generates a symmetry of the theory.

Define:Bg

6 =16!

Tr((−1)F(2h)6gN

)Can we calculate this?


Result

Bg6(Q,P) == (−1)Q.P

∫C


Zg: yet another set of functions, also with nice modularproperties, and poles in the complex (ρ, σ, v) space.

We can find the behaviour of this for large charges by thesame method described earlier.

Result:ln |Bg

6(Q,P)| ' π√

Q2P2 − (Q.P)2/N


Macroscopic analysisGoal:

1. Develop tools for computing the entropy of extremalblack holes including stringy and quantum corrections.

2. Apply it to black holes carrying the same charges forwhich we have computed the microscopic index.

3. Compare the macroscopic results with the microscopicresults.

We shall begin by describing a general formula forcomputing the black hole degeneracy on the macroscopicside


In general the macroscopic degeneracy, denoted by dmacrocan have two kinds of contributions:

1. From degrees of freedom living outside the horizon(hair)

Example: The fermion zero modes associated with thebroken supersymmetry generators.

2. From the horizon.


Horizon

Horizon

HorizonHair

Q1

Q

Q

2

n

Qhair


We shall denote the degeneracy associated with thehorizon degrees of freedom by dhor and those associatedwith the hair degrees of freedom by dhair.

The proposed formula for dmacro:

∑n

∑{~Qi},~QhairPn

i=1~Qi +

~Qhair =~Q

{n∏

i=1

dhor (~Qi)

}dhair (~Qhair ; {~Qi})

dhair can be calculated by explicitly identifying andquantizing the hair modes.

To leading order dhor ∼ exp[SBH].

Our main goal is to calculate corrections to dhor.


Degeneracy to indexOn the microscopic side we usually compute an index

On the other hand dhor computes degeneracy.

How do we compare the two?

Strategy: Use dhor to compute the index on themacroscopic side.

We shall illustrate this for the helicity trace Bn for a fourdimensional single centered black hole.


For a black hole that breaks 2n supercharges we define

Bn =1n!

Tr(−1)2h(2h)n

h: 3rd component of angular momentum in rest frame

For each pair of broken supersymmetries we have a pair offermion zero modes.

→ a bose-fermi pair of degenerate states after quantization.

Tr(−1)2h on such a pair of states will vanish unless weprevent it by inserting a factor of 2h.

Thus for 2n broken supersymmetries we need n insertionsof 2h ı.e. Bn.


Bn =1n!

Tr(−1)2hhor+2hhair(2hhor + 2hhair)n

Since typically all the fermion zero modes associated withbroken supersymmetries live on the hair, we need aninsertion of (2hhair)

n into the trace.

Bn =1n!

Tr(−1)2hhor+2hhair(2hhair)n =

∑B0;horBn;hair

Often for single centered black holes the only hair degreesof freedom are the fermion zero modes associated withbroken supersymmetry, leading to Bn;hair = (−1)n/2.

In that case Bn = (−1)n/2B0;hor = (−1)n/2Trhor(−1)2hhor .


Bn = (−1)n/2B0;hor = (−1)n/2Trhor(−1)2hhor

In 4D only hhor = 0 black holes are supersymmetric

Trhor(−1)2hhor = Trhor(1) = dhor

→Bn = (−1)n/2dhor

This explains how we can compare the helicity trace indexcomputed in the microscopic theory with dhor computed inthe macroscopic theory.


We now focus on the computation of dhor.

AdS2 space plays a special role in this analysis.

We shall begin by reviewing why this is so.


What is AdS2?

Take a three dimensional space labelled by coordinates(x,y, z) and metric

ds2 = dx2 − dy2 − dz2

AdS2 may be regarded as a two dimensional Lorentzianspace embedded in this 3-dimensional space via therelation:

x2 − y2 − z2 = −a2

a: some constant giving the radius of AdS2.

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


x2 − y2 − z2 = −a2

Introduce independent coordinates (η, t):

x = a sinh η cosh t, y = a cosh η, z = a sinh η sinh t

dx2 − dy2 − dz2 = a2(dη2 − sinh2 η dt2)

Define: r = cosh η

ds2 = a2[

dr2

r2 − 1− (r2 − 1)dt2

], r ≥ 1


Why AdS2?

All known black holes develop an AdS2 factor in their nearhorizon geometry in the extremal limit.

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


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


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

The full near horizon geometry takes the form AdS2 × K

K: some compact space.


Wald’s general formula for computing higher deivativecorrections to classical black hole entropy takes a verysimple prescriptions for black holes with an AdS2 factor inthe near horizon geometry.

We shall illustrate this in the context of sphericallysymmetric black holes in four dimensional theories.

In this case the near horizon geometry has an AdS2 × S2

factor.


Consider an arbitrary general coordinate invariant theoryof gravity coupled to a set of gauge fields A(i)

µ and neutralscalar fields {φs}.

The most general form of the near horizon geometry of anextremal black hole consistent with the symmetry ofAdS2 × S2:

ds2 ≡ gµνdxµdxν = v1

(−r2dt2 +

dr2

r2

)+v2

(dθ2 + sin2 θdφ2

)φs = us

F(i)rt = ei, F(i)

θφ =pi

4πsin θ ,


ds2 ≡ gµνdxµdxν = v1

(−r2dt2 +

dr2

r2

)+v2

(dθ2 + sin2 θdφ2

)φs = us F(i)

rt = ei, F(i)θφ =

pi

4πsin θ ,

For this background

Rαβγδ = −v1(gαγgβδ − gαδgβγ), α, β, γ, δ = r, t

Rmnpq = v2(gmpgnq − gmqgnp), m,n,p,q = θ, φ

Covariant derivatives of the Riemann tensor, scalar fieldsand gauge field strengths vanish.


ds2 = v1

(−r2dt2 +

dr2

r2

)+ v2

(dθ2 + sin2 θdφ2

)φs = us

F(i)rt = ei, F(i)

θφ =pi

4πsin θ ,

Let√−det gL be the Lagrangian density.

Define:f (~u, ~v , ~e, ~p) ≡

∫dθ dφ

√−det g L

E(~u, ~v , ~e, ~q, ~p) ≡ 2π(ei qi − f (~u, ~v , ~e, ~p))


Results:

For an extremal black hole of electric charge ~q andmagnetic charge ~p,

1 the values of {us}, {ei}, v1 and v2 are obtained byextremizing E(~u, ~v , ~e, ~q, ~p) with respect to thesevariables.

∂E∂us

= 0,∂E∂v1

= 0 ,∂E∂v2

= 0,∂E∂ei

= 0

2 SBH = E at the extremum.


These results provide us with

1. An algebraic method for computing the entropy ofextremal black holes without solving any differentialequation

2. A proof of the attractor mechanism – the black holeentropy is independent of the asymptotic moduli.

However this approach does not prove the existence of anextremal black hole carrying a given set of charges; itworks assuming that the solution exists.


What about quantum corrections?

Naive guess: apply Wald’s formula again, but replacing theclassical action by the 1PI action.

This will again give a simple algebraic method forcomputing the entropy.

This prescription is not complete since the 1PI actiontypically has non-local contribution due to massless statespropagating in the loops.

Nevertheless this has been used to compute corrections toblack hole entropy from local terms in the 1PI action withsignificant success.


Consider the CHL models obtained by ZZN orbifold of typeIIB on K 3× S1 × S1.

At tree level there are no corrections at the four derivativelevel, but at one loop these theories get correctionsproportional to the Gauss-Bonnet term in the 1PI action.√

−det g∆L

= ψ(τ, τ)√−det g

{RµνρσRµνρσ − 4RµνRµν + R2

}τ : modulus of the torus (S1 × S1).


Adding this correction to the supergravity action one cancalculate the correction to the entropy of a black hole inthe CHL model.

Result for the Wald entropy

π√

Q2P2 − (Q.P)2

−φ

(Q.PP2 ,

√Q2P2 − (Q.P)2

P2

)+O

(1

Q2,P2,Q.P

)φ(τ1, τ2) ≡ (k + 2) ln τ2 + ln g(τ1 + iτ2) + ln g(−τ1 + iτ2)

g(τ): a known function computed from the coefficient ofthe Gauss-Bonnet term.

– agrees exactly with the result for ln |B6(Q,P)| for largecharges.


Quantum corrections to dhor

Near horizon geometry of an extremal black hole alwayshas the form of AdS2 × K.

K: some compact space, possibly fibered over AdS2.

K includes the compact part of the space time as well asthe angular coordinates in the black hole background, e.g.S2 for a four dimensional black hole.

We shall now describe a proposal for computing quantumcorrected entropy in terms of a path integral of stringtheory in this near horizon geometry.


Steps for computing dhor

1. Go to the euclidean formalism and represent the AdS2factor by the metric:

ds2 = v(

(r2 − 1)dθ2 +dr2

r2 − 1

), 1 ≤ r <∞, θ ≡ θ + 2π

r = cosh η gives the metric on a disk

ds2 = v(

sinh2 η dθ2 + dη2)

2. Regularize the infinite volume of AdS2 by putting acut-off r ≤ r0f(θ) for some smooth periodic function f(θ).

This makes the AdS2 boundary have a finite length L.


3. Define:Z ≡

⟨exp[−iqk

∮dθA(k)

θ ]

⟩〈〉: Path integral over string fields in the euclidean nearhorizon background geometry.

{qk}: electric charges carried by the black hole,representing electric flux of the U(1) gauge field A(k)

through AdS2

4. dhor = Zfinite is defined by expressing Z as

Z = eCL+O(L−1) × Zfinite

C: A constant


Note: Near the boundary of AdS2, the θ indepndentsolutions to the Maxwell’s equation has the form:

Ar = 0, Aθ = C1 + C2r

C1 (chemical potential) represents normalizable mode

C2 (electric charge) represents non-normalizable mode

→ the path integral must be carried our keeping C2(charge) fixed and integrating over C1 (chemical potential).

Thus the AdS2 path integral computes the entropy in themicrocanonical ensemble.


Consistency checks:

1. dhor gives us back exp[Swald] in the classical limit.

2. By AdS2/CFT1 correspondence Z computes

Tr(e−LH) = d0 e−L E0

in the CFT1 living on the boundary of AdS2.

H: Hamiltonian of dual CFT1.

(d0,E0): (degeneracy, energy) of the states of CFT1.

Thus dhor = Zfinite counts the degeneracy d0 of the dualCFT1.


Application: Computation of twisted index

Suppose we want to compute the index

Bgn = (−1)n/2 1

n!Tr(−1)2h (2h)n g

g: some ZZN symmetry generator.

– gets contribution from states which break 2n or less ginvariant supersymmetries.

After separating out the contribution from the hair degreesof freedom, we see that the relevant quantity associatedwith the horizon is Trhor((−1)2hhorg) = Trhor(g)

What macroscopic computation should we carry out?


By following the logic of AdS/CFT correspondence we findthat we need to compute:

Zfiniteg =

⟨exp[−iqk

∮dθA(k)

θ ]

⟩finite

with a g twisted boundary condition on the fields underθ → θ + 2π.

Other than this the asymptotic boundary condition mustbe identical to that of the attractor geometry since thecharges have not changed


Recall AdS2 metric:

ds2 = v[(r2 − 1)dθ2 +

dr2

r2 − 1

]The circle at infinity, parametrized by θ, is contractible atthe origin r = 1.

Thus a g twist under θ → θ + 2π is not admissible.

→ the AdS2 × S2 geometry is not a valid saddle point of thepath integral.


Question: Are there other saddle points which couldcontribute to the path integral?

Constraints:

1. It must have the same asymptotic geometry as theAdS2 × S2 geometry.

2. It must have a g twist under θ → θ + 2π.

3. It must preserve sufficient amount of supersymmetriesso that integration over the fermion zero modes do notmake the integral vanish.

Beasley, Gaiotto, Guica, Huang, Strominger, Yin;Banerjee, Banerjee, Gupta, Mandal, A.S.


There are indeed such saddle points in the path integral,constructed as follows.

1. Take the original near horizon geometry of the blackhole.

2. Take a ZZ N orbifold of this background with ZZ Ngenerated by simultaneous action of

a) 2π/N rotation in AdS2

a) 2π/N rotation in S2 (needed for preserving SUSY)

c) g.

Banerjee, Jatkar, A.S.; A.S.; Pioline, Murthy


To see that this has the same asymptotic geometry as theattractor geometry we make a rescaling:

θ → θ/N, r → N r

The metric takes the form:

v(

(r2 − N−2)dθ2 +dr2

r2 − N−2

)Orbifold action: θ → θ + 2π, φ→ φ+ 2π/N, g

The g transformation provides us with the correctboundary condition.

The φ shift can be regarded as a Wilson line, and hence isan allowed fluctuation in AdS2.


The classical action associated with this saddle point, afterremoving the divergent part proportional to the length ofthe boundary, is Swald/N.

Thus the contribution to the twisted partition function Bgfrom this saddle point is

Zfiniteg = exp [Swald/N]

This is exactly what we have found in the microscopicanalysis of the twisted index.

black hole microstate counting and their macroscopic ...€¦ · black holes are classical...

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