Localization & Exact Quantum Entropy · 2012. 10. 9. · Wald entropy can incorporate the...

Motivation Strategy Supergravity String Theory Conclusions

Localization&

Exact Quantum Entropy

Atish Dabholkar

CERN, Geneva

Centre National de la Recherche ScientifiqueUniversite Pierre et Marie Curie Paris VI

Progress in QFT and String Theory6 April 2012

Atish Dabholkar (Paris/CERN) Localization & Quantum Entropy April 2012, Osaka 1 / 46

1 MotivationQuantum EntropyAdS2/CFT1 Holography

2 StrategyMicroscopic CountingLocalization

3 SupergravityOff-shell supergravity for LocalizationLocalizing Instanton and its Renormalized action

4 String TheoryReducing the problem to supergravityPutting it all together

5 ConclusionsSummary and Outlook

Based on

A.D. Joao Gomes, Sameer Murthy, “Localization and ExactHolography,” arXiv:1111.1163

A.D. Joao Gomes, Sameer Murthy, “Quantum Black Holes,Localization, and the Topological String,” arXiv:1012.0265

A.D., Sameer Murthy, Don Zagier; “Quantum Black Holes,Wall-crossing, and Mock Modular Forms,” arXiv:12mm.nnnn

Quantum Entropy

Two Related Motivations

Entropy of black holes remains one of the most important and preciseclues about the microstructure of quantum gravity.

Can we compute exact quantum entropy of black holes including allcorrections both microscopically and macroscopically?

Holography has emerged as one of the central concepts regarding thedegrees of freedom of quantum gravity.

Can we find simple example of AdS/CFT holgraphy where we might beable to ‘prove’ it exactly?

Quantum Entropy

Black Hole Entropy Bekenstein [72]; Hawking[75]

For a BPS black hole with charge vector (q, p), for large charges, theleading Bekenestein- Hawking entropy precisely matches the logarithm ofthe degeneracy of the corresponding quantum microstates

A(q, p)

4= log(d(q, p)) + O(1/Q)

Strominger & Vafa [96]

This beautiful approximate agreement raises two important questions:

What exact formula is this an approximation to?

Can we systematically compute corrections to both sides of thisformula, perturbatively and nonperturbatively in 1/Q and may beeven exactly for arbitrary finite values of the charges?

Quantum Entropy

Black Hole Entropy Bekenstein [72]; Hawking[75]

For a BPS black hole with charge vector (q, p), for large charges, theleading Bekenestein- Hawking entropy precisely matches the logarithm ofthe degeneracy of the corresponding quantum microstates

A(q, p)

4= log(d(q, p)) + O(1/Q)

Strominger & Vafa [96]

This beautiful approximate agreement raises two important questions:

What exact formula is this an approximation to?

Can we systematically compute corrections to both sides of thisformula, perturbatively and nonperturbatively in 1/Q and may beeven exactly for arbitrary finite values of the charges?

Quantum Entropy

Finite size effects

We do not know which phase of string theory might correspond tothe real world. For such a theory under construction, a useful strategyis to focus on universal properties that must hold in all phases.One universal requirement for a quantum theory of gravity is that inany phase of the theory that admits a black hole, it must be possibleto interpret black hole as a statistical ensemble of quantum states.

Finite size corrections to the entropy, unlike the leading area formula,depend on the details of the phase, and provide a sensitive probe ofshort distance degrees of freedom.

This is an extremely stringent constraint on the consistency of the theorysince it must hold in all phases of the theory.

Quantum Entropy

Finite size effects

Quantum Entropy

Finite size effects

Quantum Entropy

Finite size effects

Quantum Entropy

Nonperturbative Spectrum

Conversely, we can use the knowledge of exact black hole results fromthe bulk to learn about the exact nonperturbative spectrum of thetheory.

One can compare any putative counting fromula against the blackhole entropy. This has already proved to be a useful guide inunderstanding for example the exact spectrum of quarter-BPS dyonsin N=4 theories. In particular to fix subtle issues about dependence ofthe counting formula on wall-crossing, arithmetic duality invariants.

Quantum Entropy

Nonperturbative Spectrum

Conversely, we can use the knowledge of exact black hole results fromthe bulk to learn about the exact nonperturbative spectrum of thetheory.

One can compare any putative counting fromula against the blackhole entropy. This has already proved to be a useful guide inunderstanding for example the exact spectrum of quarter-BPS dyonsin N=4 theories. In particular to fix subtle issues about dependence ofthe counting formula on wall-crossing, arithmetic duality invariants.

Quantum Entropy

Wald entropy and beyond

Wald[94]; Iyer & Wald [94]; Jacobson, Kang, & Myers [93]

Wald entropy can incorporate the corrections to Bekenstein-Hawkingentropy from all higher-derivative local terms in the effective action.

This does not include quantum effects from integrating out masslessfields. These are in general essential for example for duality invariance.

For a systematic comparison one needs a manifestly duality covariantformalism that generalizes Wald entropy. Such a formalism has beenproposed recently by Sen using AdS2/CFT1 holography.

Quantum Entropy

AdS2/CFT1 Holography

Quantum Entropy and AdS2/CFT1 Sen [08]

Near horizon geometry of a BPS black hole is AdS2 × S2. Quantumentropy can be defined as as partition function of the AdS2.

W (q, p) =

⟨exp

[− i qi

∫ 2π

0Ai dθ

]⟩finite

Functional integral over all string fields. The Wilson line insertion isnecessary so that classical variational problem is well defined.

The black hole is made up of a complicated brane configuration. Theworldvolume theory typically has a gap. Focusing on low energy statesgives CFT1 with a degenerate, finite dimensional Hilbert space. Partitionfunction d(q, p) is simply the dimension of this Hilbert space.

Functional Integral Boundary Conditions

For a theory with some vector fields Ai and scalar fields φa, we havethe fall-off conditions

ds20 = v

[(r 2 + O(1)

)dθ2 +

r 2 + O(1)

φa = ua + O(1/r) , Ai = −i e i (r − O(1))dθ , (1)

Magnetic charges are fluxes on the S2. Constants v , e i , ua fixed toattractor values v∗, e

i∗, u

a∗ determined purely in terms of the charges,

and set the boundary condition for the functional integral.

Quantum entropy is purely a function of the charges (q, p).

The functional integral is infrared divergent due to infinite volume of theAdS2. Holographic renormalization to define the finite part.

Renormalized functional integral

Put a cutoff at a large r = r0.

Lagrangian Lbulk is a local functional, hence the action has the form

Sbulk = C0r0 + C1 + O(r−10 ) ,

with C0,C1 independent of r0.

C0 can be removed by a boundary counter-term (boundarycosmological constant). C1 is field dependent to be integrated over.

Quantum Entropy gives a proper generalization of Wald entropy to includenot only higher-derivative local terms but also the effect of integratingover massless fields. This is essential for duality invariance and for asystematic comparison since nonlocal effects can contribute to same order.

Sbulk = C0r0 + C1 + O(r−10 ) ,

To summarize, our most ambitious goal will be twofold.

1 Compute d(q, p) from bound state dynamics of branes.

2 Compute W (q, p) from the bulk for arbitrary finite charges byevaluating the functional integral over string fields in AdS2.Check if the two agree.

The first problem has now been solved in some cases. We now know theexact spectrum of both half and quarter-BPS dyonic black holes for allcharges at all points in the moduli space for certain N = 4 theories. Andfor a large class of states in N = 8 theories.Maldacena, Moore, Strominger [98]

Dijkgraaf, Verlinde, Verlinde; [96] Gaiotto, Strominger, Yin; David, Sen [05];

David, Jatkar, Sen; Dabholkar, Nampuri; Dabholkar, Gaiotto [06];

Sen; Dabholkar, Gaiotto, Nampuri; Cheng, Verlinde [07];

Banerjee, Sen, Srivastava; Dabholkar, Gomes, Murthy [08]

To summarize, our most ambitious goal will be twofold.

1 Compute d(q, p) from bound state dynamics of branes.

2 Compute W (q, p) from the bulk for arbitrary finite charges byevaluating the functional integral over string fields in AdS2.Check if the two agree.

The first problem has now been solved in some cases. We now know theexact spectrum of both half and quarter-BPS dyonic black holes for allcharges at all points in the moduli space for certain N = 4 theories. Andfor a large class of states in N = 8 theories.Maldacena, Moore, Strominger [98]

Dijkgraaf, Verlinde, Verlinde; [96] Gaiotto, Strominger, Yin; David, Sen [05];

David, Jatkar, Sen; Dabholkar, Nampuri; Dabholkar, Gaiotto [06];

Sen; Dabholkar, Gaiotto, Nampuri; Cheng, Verlinde [07];

Banerjee, Sen, Srivastava; Dabholkar, Gomes, Murthy [08]

Microscopic Counting

Type II string theory on a T 6

Exact microscopic degeneracy of one-eighth BPS dyonic black holes withcharge vector (Q,P) are given in terms of the Fourier coefficients of thefollowing counting function.

F (τ, z) =ϑ2

1(τ, z)

η6(τ).

where ϑ1 is the Jacobi theta function and η is the Dedekind function.With q := e2πiτ and y := e2πiz , they have the product representations

ϑ1(τ, z) = q18 (y

12 − y−

12 )∞∏n=1

(1− qn)(1− yqn)(1− y−1qn) ,

η(τ) = q1

∞∏n=1

(1− qn) .

Rademacher expansion

The degeneracy d(∆) depends only the U-duality invariant∆ = Q2P2 − (Q · P)2 and admits an exact expansion

d(∆) =∞∑c=1

dc(∆)

dc(∆) = (−1)∆+1 2π I 72

(π√∆

c9/2Kc(∆) .

Iρ(z) =1

∫ ε+i∞

ε−i∞

σρ+1eσ+ z2

4σ dσ,

is a modifield Bessel function and Kc(∆) is the Kloosterman sum.

Kloosterman Sum

The Kloosterman sum has intricate number theoretic structure defined by

Kc(∆) := e5πi/4∑

−c≤d<0;(d,c)=1

e2πi dc

(∆/4) M(γc,d)−1`1 e2πi a

c(−1/4)

with ` = ∆ mod 2 and ad = 1 mod c ,

γc,d =

(a (ad − 1)/cc d

)and M is a particular representation of SL(2,Z).

Our goal now will be to evaluate the formal expression for W(q,p) bydoing the functional integral over string fields in AdS2.

This is of course highly nontrivial and may seem foolishly ambitious.

Surprisingly, one can go quite far using localization techniques toreduce the functional integral to finite number of ordinary integrals.

With enough supersymmetry, it seems possible to even evaluate theseordinary integrals all the way under certain assumptions.

Bulk of my talk will be about some recent progress in the evaluations ofW (q, p) functional integral using localization.

Localization

Consider a supermanifold M with an odd (fermionic) vector field Qbe such that Q2 = H for some compact bosonic vector field H.

To evaluate an integral of a Q-invariant function h with Q-invariantmeasure we first deform it

dµ h e−S → I (λ) :=

dµ h e−S−λQV ,

where V is a fermionic, H-invariant function.

Easy to check I ′(λ) = 0, so I (λ) is independent of λ. Hence, insteadof at λ = 0, we can evaluate it at λ =∞ where semiclassicalapproximation is exact.

Witten[88, 91], Duistermaat-Heckmann [82], Schwarz & Zaboronsky [95]

Localization

dµ h e−S → I (λ) :=

Localization

dµ h e−S → I (λ) :=

Localization

Choice of the Supercharge

In this limit, the functional integral localizes onto the critical points ofthe functional SQ := QV . This reduces the functional integral overfield space to a this localizing submanifold.

To apply to our problem, we pick Q which squares to H = 4(L− J).Here L generates rotation of the Euclidean AdS2 which is a disk andJ generates a rotation of S2, so H is compact.

Given this choice of Q we choose the localizing action functional to be

SQ = QV ; V = (QΨ,Ψ)

where Ψ denotes schematically all fermions of the theory.

To apply this directly in string theory is not feasible given the state ofstring field theory. So we will first solve a simpler problem in supergravity.

Localization

A simpler problem in supergravity

Consider W (q, p) which is the same functional integral but insupergravity coupled to only nv + 1 vector multiplets.

This is still a complicated functional integral over spacetime fields.

We will show using localization that this functional integral reduces to anordinary integral over nv + 1 real parameters. Huge simplification.

To apply localization inside the functional integral, one requires anoff-shell formulation. In general, off-shell supergravity is notoriouslycomplicated but for N = 2 vector multiplets an elegant formalism existsthat gauges the full superconformal group.

Localization

Off-shell supergravity for Localization

Multiplets and Off-shell supersymmetry transformations

Gravity multiplet: Vielbein, spin connection, auxiliary fields andfermions

Vector multiplet: vector field AIµ, complex scalar X I and an SU(2)

triplet of auxiliary fields Y Iij and fermions. Here i is SU(2) doublet.

X I ,ΩIi ,A

δΩi = 2 /DX εi +1

2εijFµνγ

µνεj + Yijεj + 2Xηi ,

where ε, η are the (superconformal) supersymmetry parameters.de Wit, Lauwers, van Holten, Van Proeyen [1980]

X I ,ΩIi ,A

2εijFµνγ

X I ,ΩIi ,A

2εijFµνγ

X I ,ΩIi ,A

2εijFµνγ

Importance of being off-shell

The beauty of off-shell supergravity is that the supersymmetrytransformations are specified once and for all and do not depend onthe choice of the action. This is crucial for localization.

Auxiliary fields which are normally eliminated from the physical actionacquire nontrivial position dependence for the localizing instantonsolutions.

With this setup, the bosonic part (QΩ,QΩ) of the QV action is a sum ofperfect squares. Setting each of these terms to zero gives a set offirst-order differential equations. Happily, it turns out they can be solvedexactly to obtain an analytic solution.

The bosonic part of the localizing action (QΨ,QΨ)

cosh(η)[K − 2sech(η)H)

]2+ 4 cosh(η)

[H1 + H tanh(η)

]2+ 4 cosh(η)[H2

0 + H22 + H2

[f −01 − J − 1

A(sin(ψ)J3 − sinh(η)J1)

[f +01 + J − 1

B(sin(ψ)J3 + sinh(η)J1)

[f −03 +

A(sin(ψ)J1 + sinh(η)J3)

[f +03 +

B(sin(ψ)J1 − sinh(η)J3)

[f −02 +

A(sin(ψ)J0 + sinh(η)J2)

[f +02 −

B(sin(ψ)J0 + sinh(η)J2)

+4 cosh(η)

AB[sinh(η)J0 − sin(ψ)J2]2

+4 cosh(η) sinh2(η)

1 + J23 ] ,

whereH Ia := eµa ∂µH I , J I

a := eµa ∂µJ I ,

A := cosh(η) + cos(ψ) , B := cosh(η)− cos(ψ) .

It is understood that all squares are summed over the index I .

Localizing Instanton and its Renormalized action

Localizing instanton Solution

X I = X I∗ +

r, X I = X I

∗ +C I

Y I11 = −Y I2

2 =2C I

r 2, f I

µν = 0 .

Solves a major piece of the problem by identifying the off-shell fieldconfigurations onto which the functional integral localizes. Thus, afunctional integral is reduced to a finite dimensional ordinary integral. Thisinstanton is universal and does not depend on the physical action.

Scalar fields move away from the attractor values X I∗ inside the AdS2

‘climbing up’ the entropy function potential. Q supersymmetry is stillmaintained because auxiliary fields get nontrivial position dependence.

Localizing instanton Solution

X I = X I∗ +

r, X I = X I

∗ +C I

Y I11 = −Y I2

2 =2C I

r 2, f I

µν = 0 .

Solves a major piece of the problem by identifying the off-shell fieldconfigurations onto which the functional integral localizes. Thus, afunctional integral is reduced to a finite dimensional ordinary integral. Thisinstanton is universal and does not depend on the physical action.

Scalar fields move away from the attractor values X I∗ inside the AdS2

‘climbing up’ the entropy function potential. Q supersymmetry is stillmaintained because auxiliary fields get nontrivial position dependence.

We now need to evaluate the physical action on the localizinginstantons after proper renormalization to compute Sren(C , q, p)

(−i(X I FI − FI XI )) · (−1

[i∇µFI∇µX I

4iFIJ(F−Iab −

4X IT ij

ab εij)(F−abJ − 1

4X JT ij

ab εij)

8iFI (F +I

ab −1

4X ITabij ε

ij)T ijab εij −

8iFIJY I

ijYJij − i

32F (Tabij ε

AC − 1

AA(εikεjl Bij Bkl − 2F−abF−ab)

2i F−abF

AI(F−Iab −

4X IT ij

ab εij)−1

4i BijFAI

Y Iij + h.c.]

− i(X I FI − FI XI ) · (∇aVa −

2V aVa −

4|Mij |2 + DaΦi

αDaΦαi ) .

We now need to evaluate the physical action on the localizinginstantons after proper renormalization to compute Sren(C , q, p)

(−i(X I FI − FI XI )) · (−1

[i∇µFI∇µX I

4iFIJ(F−Iab −

4X IT ij

ab εij)(F−abJ − 1

4X JT ij

ab εij)

8iFI (F +I

ab −1

4X ITabij ε

ij)T ijab εij −

8iFIJY I

ijYJij − i

32F (Tabij ε

AC − 1

AA(εikεjl Bij Bkl − 2F−abF−ab)

2i F−abF

AI(F−Iab −

4X IT ij

ab εij)−1

4i BijFAI

Y Iij + h.c.]

− i(X I FI − FI XI ) · (∇aVa −

2V aVa −

4|Mij |2 + DaΦi

αDaΦαi ) .

Renormalized action

Substituting our localizing instanton solution in the above action wecan extract the finite piece after removing the leading divergent piecelinear in r0 by holographic renormalization.

After a tedious algebra, one obtains a remarkably simple form for therenormalized action Sren as a function of C I.

Sren(φ, q, p) = −πqIφI + F(φ, p) (2)

with φI := e I∗ + 2iC I and F given by

F(φ, p) = −2πi

[F(φI + ipI

)− F

(φI − ipI

where e I∗ are the attractor values of the electric field.

Renormalized action

Substituting our localizing instanton solution in the above action wecan extract the finite piece after removing the leading divergent piecelinear in r0 by holographic renormalization.

After a tedious algebra, one obtains a remarkably simple form for therenormalized action Sren as a function of C I.

Sren(φ, q, p) = −πqIφI + F(φ, p) (2)

with φI := e I∗ + 2iC I and F given by

F(φ, p) = −2πi

[F(φI + ipI

)− F

(φI − ipI

where e I∗ are the attractor values of the electric field.

Note that Sren(φ, q, p) equals precisely the classical entropy functionE(e, q, p). In particular, exp(Sren) is the topological string partitionfunction. The physics is however completely different.

E(e, q, p) depends on the the attractor values X∗ of the scalar fields.Sren(φ, q, p) depends on the value of the scalar fields at the center ofAdS2. This difference is very important.

Even though scalar fields are fixed at the boundary, their value at theorigin can fluctuate and we can integrate over them for large values.

It is something of a surprise that the renormalized action for these veryoff-shell field configurations takes the same form as the on-shell classicalentropy function.

Reducing the problem to supergravity

We at present lack a useful definition of functional integral over stringfields. To apply localization, we proceed in three steps.

Three Steps:

1 Integrate out massive string and Kaluza-Klein modes to obtain a localWilsonian effective action for the massless supergravity fields.

2 Solve a supergravity problem to evaluate W (q, p).

3 Use the results in Step II to evaluate W (q, p) There are Zc orbifoldsof AdS2 that have the same boundary conditions and hencecontribute to the functional integral. Hence, W (q, p) has the form

W (q, p) =∞∑c=1

Wc(q, p) .

Sen[09, 10], Pioline & Murthy [10]

Three Steps:

W (q, p) =∞∑c=1

Wc(q, p) .

Three Steps:

W (q, p) =∞∑c=1

Wc(q, p) .

Evaluation Wc(q, p) is related to the problem of evaluation of W (q, p) ina simple way, under certain assumptions.

General form of the answer for c = 1

W1(n,w) =

eSren(n,w ,φ) |Zinst(φ, n,w)|2 Zdet(φ) [dφ]µ .

[dφ]µ is the induced measure on the localizing manifold.

|Zinst |2 is contribution of instantons localized at north pole of S2 andanti-instantons localized at the south pole.

Zdet(φ) is one loop determinant.

W1(n,w) =

No off-shell formalism with N = 4 with finite number of auxiliary fields. Sowe will proceed in the N = 2 language.

Assumptions

Drop two gravitini multiplets. These contain four vector fields but noscalar fields. If the black hole does not couple to these vector fields, itshould be reasonable to drop this.

Drop hyper multiplets. The off-shell supersymmetry transformationsof the vector multiplets do not change by adding hypers. So ourlocalizing instantons will continue to exist.

Drop D-terms. A large class of D-terms are known not contribute toWald entropy. de Wit, Katmadas, Van Zalk [10]

Assumptions

Prepotential

For our example with this N = 2 restriction, the prepotential is

F (X , A) = −1

7∑a,b=2

CabX aX b,

in the Type-IIA frame. Here Cab is the intersection matrix for the 62-cycles of T 6. Choose a simple charge configuration with five charges q0

of D0-branes, q2 D2-branes and p1, p2, p3 D4-branes wrapping threeindependent 4-cycles in T 6.

Renormalized Action

We would like to evaluate the renormalized action Sren(φ, q, p) as afuncton of the collective coordinates φI and the charges.

Renormalized action

Since the prepotential corresponds to two-derivative action which hasE7,7(R) duality symmetry we expect to be able express the action in termsof the duality invariant ∆.

Duality-invariant variables

It is useful to define new variables

σ = −πp1p2p3/φ0 , τ a = − 1√φ0

(φa +

φ1 pa

)α ∼ φ1 .

Renormalized Action

Sren =

)− π τ2

2+π α2

withz2 = π2∆ .

Putting it all together

Evaluation of W1

One-loop determinants

The localizing action is purely quadratic in the fields. Hence the quadtraticfluctuation operator does not depend on the collective coordinates or thecharges. Hence the determinant factor is an overall numerical constant.

Instantons

Various branes can contribute to the effective action. The instantoncontribution to the prepotential is computed (partially) by the topologicalstring. However, for the N = 8 theory the classical prepotential isquantum exact and there are no corrections.

Induced Measure

Given the measure on space of fields X , the measure on the localizingsubmanifold parametrized by φ can be deduced from the induced metric.

Integration Measure

The line element on φ-space is

dΣ2 = MIJ δφI δφJ

with the metric

MIJ = KIJ −1

∂φI∂K

∂φJ

given in terms of the Kahler potential

e−K := −i(X I FI − X IFI )

Measure ∫ 7∏I=0

dφI√

det(M) .

Final Answer for W1(∆)

Conformal compensator

The φ0 variable can be thought of as the conformal compensating field.Conformal factor of the metric has wrong sign kinetic term in Euclideangravity and hence its contour of integration must be analytically continuedto make the functional integral well-defined.

The Gaussian integrals can be readily done to obtain

∫ +∞

−i∞

σ9/2exp[σ +

4σ] with z = π

√∆ .

Up to an overall constant, this is precisely the integral representation ofthe first term in the Rademacher expansion — I7/2(π

√∆) !

Comparison of d(∆) with W1(∆) and exponential of Wald Entropy

∆ 3 4 7 8 11 12

d(∆) 8 12 39 56 152 208

W1(∆) 7.97 12.20 38.99 55.72 152.04 208.45

exp(π√

∆) 230.7 535.4 4071.9 7228.3 33506 53252.3

The area of the horizon goes as 4π√

∆ in Planck units. Already for∆ = 12 this area would be much larger than one, and one might expectthat the Wald entropy would be a good approximation. Not true! Thediscrepancy between the degeneracy and the exponential of the Waldentropy arises entirely from integration over massless fields.

Index, Degeneracy and Fermions

Table: Some Fourier coefficients

∆ -1 0 3 4 7 8 11 12 15

C (∆) 1 −2 8 −12 39 −56 152 −208 513

It is striking the the sign of the Fourier coefficients is alternating. Notethat

d(∆) = (−1)∆+1C (∆)

Since F is the partition function of indexed degeneracy there is no a priorireason why they should they be positive. Surprising from the point of viewof the microscopic theory. Holography gives a simple physical explanationof this fact.

Index = Degeneracy

The near-horizon AdS2 geometry has an SU(1, 1) symmetry. With foursupersymmetries, closure of the supersymmetry algebra requires that thenear horizon symmetry must contain the supergroup SU(1, 1|2). Thisimplies that that such a supersymmetric horizon must have SU(2)symmetry which can be identified with spatial rotations. The AdS2 pathintegral naturally fixes the charges and not the chemical potentials andhence J = 0. Together, this implies

Tr(−1J

)= Tr (1) , (3)

Fermionic boundary conditions

Because the constant mode of the gauge field fluctuates, and since thefermions couple to it, periodic and antiperidic boundary conditions areequivalent.

Nonpertrubative Corrections

There is a family of freely acting supersymmetric Zc orbifolds with twistson AdS2 × S2 on shifts along S1. The shift can be effected by modifyingthe gauge field at infinity. Hence the Wilson line gives a phase both forwinding and momentum. Since the orbifold action is freely acting, oneobtains the same localizing instanton solution but the renormalized actionis divided by c because of the reduced volume.

For each c we then obtain∫dσ

σ9/2exp[σ +

c2σ] .

Kloosterman Sum

The Kloosterman sum is a sum over phases. Such phases naturally enterour story from the Wilson lines. Our orbifold is a symmetric twist onAdS2 × S2 and a shift in the charge lattice. The shifts can in principleaccount for the Kloosterman sum.

Summary and Outlook

Summary

Localization of the functional integral in Sugra and String Theory

We have shown that full functional integral of supergravity coupled tovector fields on AdS2 localizes onto the submanifold MQ of criticalpoints of the functional SQ where Q is a specific supersymmetry.

We have obtained exact analytic expression for a family of nontrivialcomplex instantons as exact solutions which are completely universaland independent of the form of the physical action.

In string theory, there are nonperturbative contributions fromorbifolds as well as from brane instantons.

Summary and Outlook

Summary

Summary and Outlook

Summary

Summary and Outlook

Rademacher expansion from Gravity

The gravitational theory has all the ingredients to exactly reproduce theentire Rademacher expansion.

W (∆) =∞∑c=1

Kc(∆)

I7/2(π√

I7/2(z) =1

∫ ε+i∞

ε−i∞

σ9/2eσ+ z2

4σ dt,

Kloosterman sum can possibly follow from the analysis of phases for theWilson lines but this needs to be done yet.

Some of the assumptions need to be better justified. Requires off-shellrealization of at least the charge Q with vector, hypers, D-terms or N = 4field content. Interesting problem in supergravity.

Summary and Outlook

Comparison with Earlier Work

It seems possible to compute finite size corrections to the entropy of BPSblack holes in a systematic way going well beyond Bekenstein-Hawking.

The leading Bessel function was partially derived in Dabholkar [04] andDabholkar, Denef, Pioline, Moore [05]. However,

it relied on the OSV conjecture Ooguri, Strominger, Vafa [04];

it was at best an asymptotic expansion since there was no systematicway of determining the measure or the range of integration since theOSV conjecture applied to a mixed ensemble.

it was not clear how to include the nonperturbative corrections toobtain for example the subleading Bessel functions.

it was not clear how to incorporate holomorphic anomalysystematically.

Summary and Outlook

Our derivation follows from first principles using standard rules offunctional integration and localization.

Functional integral localizes onto a nontrivial localizing instantonsolution of the off-shell theory . Auxiliary fields play an important role.

The measure or the range of integration can be determined followingusual collective coordinate quantization. Gives the exact Besselfunction and not just the asymptotic expansion.

Nonperturbative corrections from orbifolds give subleading Besselfunctions. Their sum is natural in microcanonical ensemble. It isjustified to keep subleading saddles because of localization.

Includes nonholomorphic contribution from integrating over masslessfields.

Summary and Outlook

From Gravity to Number Theory

The Rademacher expansion is an exact expansion. It is rapidlyconvergent but at no finite order can one assert integrality of thesum. It is a nontrivial fact that all these terms add up to an integerand that we can see the entire expansion from the bulk. To reproducethis completely we need to understand the Kloosterman sum.

This indicates an underlying integral structure to quantum gravity. Ifwe have two very close but different integers, the bulk theory will beable to distinguish the two. This would be never evident fromsemiclassical Bekenstein-Hawking formula.

It is intriguing that two AdS2 bulk theories which may have verydifferent field content but which yield the same integer W (q, p) areexpected to be dual since they will be both dual to the same CFT1.This is rare but possible.

Summary and Outlook

Outlook

On a philosophical note, our computation indicates that the quantumgravity in the bulk is as fundamental as the boundary field theory,with its own rules of computation. It is an exact dual descriptionrather than a coarse-grained description. It is worth exploring theAdS2/CFT1 duality further including correlation functions.

It is remarkable that a bulk functional integral can reproduce aninteger. The fact that the bulk is able to ‘see’ this quantization maybe relevant for information retrieval.

One may use the wealth of data from microscopic counting to learnabout the nonperturbative rules of the functional integral of quantumgravity and about the structure of the string effective action. It isuseful to have explicit answers to compare with.

Summary and Outlook

Outlook

Summary and Outlook

Outlook

Localization & Exact Quantum Entropy · 2012. 10. 9. · Wald entropy can incorporate the...

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