exact quantum entropy, mock modular forms and … quantum entropy, mock modular forms and holography...

Exact quantum entropy, mock modular forms andholography

Sameer Murthy

ITF and Spinoza Institute, Utrecht

Indo-Israeli strings meetingFeb 7, 2012

Sameer Murthy (UU) Quantum entropy and mock modular forms Feb 7, 2012, Jerusalem 1 / 23

Exact quantum entropy

log(d(Qi )) =A(Qi )

4+ O(1/Qi ) ,

?= log(W (Qi )) .

Computation of W using localization

The Wilson line expectation value in N = 2 supergravity localizes to afinite dimensional integral

W (q, p) =

∫MQ

exp(Sren(φ, p, q)

)[dφ] ; (1)

Sren = −πφ · q + ImF (φ+ ip) .


Plan of talk

Compute W (Q) for explicit example and compare with d(Q).

Modular forms, and AdS3 holography.

Wall crossing problem, and solution using mock modular forms.

Based on

Atish Dabholkar, Joao Gomes, S.M. arXiv:1012.0265 and arXiv:1111.1161.

S.M., Satoshi Nawata “Which AdS3 configurations contribute to the SCFT2

elliptic genus?” arXiv:1112.4844.

Atish Dabholkar, S.M., Don Zagier “Quantum black holes, wall crossing andmock modular forms,” to appear.

Kathrin Bringmann, S.M. “On the positivity of black hole degeneracies,” toappear.


Outline

1 Exact computation of black hole entropy

2 Modular forms and AdS3 holography

3 Wall-crossing and mock modular forms


1/8-BPS black holes in N = 8 string theory

1/8 BPS dyonic states, preserve four supersymmetries.

Charges (Q,P); U-duality invariant ∆ = Q2P2 − (Q.P)2 ≡ 4n − r 2.

For the N = 2 reduction of the N = 8 theory that we consider,nv = 7, and the exact prepotential is

F (X ) = −1

2

X 1CabX aX b

X 0, a, b = 2, . . . , 7 .

Cab =(

0 11 0

)⊗ 13×3 is the intersection matrix of the two cycles of T 4.

Apply localization: solutions parameterized by (φ0, φ1, · · · , φ7).

Change to duality invariant variables (σ, τ1, · · · , τ7).


Quantum entropy of 1/8-BPS black holes

Compute renormalized action:

Sren =

(σ +

π2∆

4σ

)− π τ2

2.

Compute induced measure as a function of the variables:power of σ.

Plug into (1) and integrate over the eight variables:Gaussian integrals over τ a.We obtain A. Dabholkar, J. Gomes, S.M. arXiv:1111.1161.

W (∆) =

∫dσ

σ9/2exp

(σ + π2∆/4σ

)= I7/2(π

√∆) .


Comparison with data

∆ d(∆) eπ√

∆

3 8 230.76

4 12 535.49

7 39 4071.93

8 56 7228.35

11 152 33506.1

12 208 53252.3

15 513 192401.0

log(d)∆→∞−→ π

√∆ = SBH .


Comparison with data

∆ d(∆) W (∆) eπ√

∆

3 8 7.97 230.76

4 12 12.20 535.49

7 39 38.99 4071.93

8 56 55.72 7228.35

11 152 152.04 33506.1

12 208 208.46 53252.3

15 513 512.96 192401.0

200 430369461412872


Outline





Microscopic degeneracies

The degeneracies of the 1/8-BPS dyonic states d(n, r) in the type IIstring theory on a T 6 can be computed using a D-branerepresentation. e.g D1-D5-p-monopole system in Type IIBMaldacena, Moore, Strominger [1999], Shih, Strominger, Yin [2005].

Microstate degeneracies d(n, r) = (−1)r+1c(n, r).The generating function is (with q ≡ e2πiτ and y ≡ e2πiz),

A(τ, z) ≡ ϑ1(τ, z)2

η(τ)6=∑n,r

c(n, r) qn y r .

The functions ϑ1 and η are familiar to string theorists.


Modular forms and Jacobi forms

A(τ, z) is a Jacobi form of weight −2 and index 1.

A Jacobi form of weight k and index m is a holomorphic functionϕ(τ, z) from H× C to C which is modular in τ and elliptic in z :

ϕ(aτ + b

cτ + d,

z

cτ + d) = (cτ + d)k e

2πimcz2

cτ+d ϕ(τ, z) ∀(

a bc d

)∈ SL2(Z)

and under the translations of z by Zτ + Z as

ϕ(τ, z + λτ + µ) = e−2πim(λ2τ+2λz)ϕ(τ, z) ∀ `, m ∈ Z .

These properties are highly constraining. A finite number of coefficientsenough to fix the function.


What this buys us: (1) Classification and calculability

The Jacobi properties leads to the theta expansion of A:

A(τ, z) = h0(τ)ϑ1,0(τ, z) + h1(τ)ϑ1,1(τ, z) ,

with

h0(τ) = −ϑ1,1(τ, 0)

η6(τ)= −2− 12q − 56q2 − 208q3 . . .

h1(τ) =ϑ1,0(τ, 0)

η6(τ)= q−

14(1 + 8q + 39q2 + . . .

).

ϑm,`(τ, 0) =∑r∈Z

q(2mr+`)2/4m ,1

η(τ)= q−1/24

∞∏n=1

1

(1− qn).

The coefficients d(∆) can easily be read off to be positive integers,as conjectured by Sen.


What this buys us: (2) Convergent expansions

Due to its modular symmetries, the degeneracy admits an exact expansion

C (∆) = 2π(π

2

)7/2∞∑

c=1

c−9/2 Kc(∆) I7/2

(π√∆

c

), (2)

called the Hardy-Ramanujan-Rademacher expansion. Here,

Iρ(z) ∼ ez is the modified Bessel function of index ρ.

the Kloosterman sum

Kc(∆) :=∑

−c≤d<0;(d,c)=1

e2πi dc

(∆/4) e2πi ac

(−1/4) M(γc,d)−1r1 ,

with(

a bc d

)∈ S`2(Z), and M(γ) is the multiplier system,

defined by M(T ) =(

1 00 −i

), M(S) =

√i2

(1 11 −1

).


Meaning of modular symmetries: AdS3/CFT2

f (τ) partition function (elliptic genus) of SCFT2. τ is the modularparameter (complex structure) of the boundary torus.

Holographic dual is finite temperature AdS3.

The black hole can be thought of as momentum excitations of aneffective string wound around a circle. This has an associatednear-horizon AdS3 geometry. Very-near horizon AdS2.

Compute exact f (τ) in gravitational theory? Use localization, firststep in S.M., S.Nawata arXiv:1112.4844.Deformation which connects the AdS2 to AdS3 boundary.


Outline





More general situation: N = 2 theories of gravity

Very generically, for a given set of charges, other solutions in gravitywith same charges (Multi-centered black hole bound states).

These solutions decay on crossing co-dimension one surfaces inmoduli space (walls), so index jumps.

Can one characterize the partition function of single centered blackholes ?

AdS3 situation

One can analyze these N = 2 systems when the radius of the circle islarge, and one has an AdS3 black string.

Two types of behavior possible – multi-centered BHs inside AdS3 orseparate AdS3 throats. de Boer, Denef, El-Showk, Messamah, van den Bleeken 0802.2257.


Breakdown of modular symmetry

In the absence of D6-brane charge, answer is the latter. The solutionflows to a single black string with one AdS3 throat, and there areother multi-throat AdS3s.

What then is the partition function of the single AdS3 throat?

Since one has thrown away some part of the spectrum, modularsymmetry is broken.

On the other hand, one expects modularity from AdS3/CFT2.Left as a puzzle in arxiv:0802.2257.

A description of the SCFT known, but not exactly solvable in general(MSW CFT) Maldacena, Strominger, Witten, hep-th/9711053.


N = 4 string theory in four dimensions

Type II string theory on K 3× T 2. States labelled by T-dualityinvariants (Q2/2,P2/2,Q · P) ≡ (m, n, `).U-duality invariant ∆ = 4mn − `2.

BH phenomenology

The single-centered black hole exists everywhere in moduli space(immortal). This carries entropy

S = π√

∆ + · · · (3)

The only multi-centered configurations that contribute to the1/4 BPS index are the two-centered small black hole bound statesA.Sen hep-th/0702141; A.Dabholkar, M.Guica, S.M., S.Nampuri; arxiv:0903.2481.

Each center is a 1/2 BPS state which carry purely electric (or purelymagnetic) charges. This configuration carries entropyS = 4π

√m + 4π

√n + · · ·


Microscopic partition function

Natural boundary conditions in AdS3: fix m = P2/2,allow n = Q2/2, ` = Q.P to vary.

Partition function (supersymmetric index) is a Jacobi form of weightk = −10 and index m.

ψm(τ, z) meromorphic in z , so ambiguity in Fourier expansion.

This simple fact leads to some deep mathematics. In particular,– the theta expansion breaks down, and– the coefficients are no longer modular forms.(analyzed by S. Zwegers in 2002).


Theorem 1 – Decomposition A. Dabholkar, S.M., D. Zagier

We show that the meromorphic Jacobi form can be written uniquelyas the sum of two parts, each with characteristic properties.

ψm(τ, z) = ψFm(τ, z) + ψP

m(τ, z) .

The polar part ψP is exactly the two centered BH partition function.The finite part ψF is exactly the partition function of the singlecentered black hole c.f. attractor mechanism.

The finite part ψFm is a mock Jacobi form. ψF

m can be completed to amodular partition function ψF

m by adding a specific non-holomorphicfunction. The completion obeys the equation

τ3/22

∂ψFm(τ, τ)

∂τ=

√m

8πi

p24(m + 1)

η(τ)24

∑`mod(2m)

ϑm,`(τ)ϑm,`(τ, z) .


Theorem 2 – Optimality A. Dabholkar, S.M., D. Zagier

Mock modular forms are only defined up to the addition of truemodular forms. Is there a canonical or optimal mock Jacobi form thatI can pick for each m, which is “small”?

“Small” = slow growth, and usually interesting.

Answer – yes. For each m, we find such a canonical mock Jacobiform. (Note that there is no classification/basis yet for mock modularforms).

e.g. for m prime, the mock part is related to the generating functionof Hurwitz-Kronecker class numbers.


Application: (1) Holography

In the AdS3 setup, the correct AdS3 partition function which isinvariant all over moduli space is the mock Jacobi form.

This function (more precisely, its completion) has all the propertiesone expects of the single AdS3 throat.

The completion is modular covariant, and large gauge transformations(spectral flow) is a symmetry.

Clear statement, suggests general solution of the wall-crossingproblem.


Application: (2) Positivity

For black holes, the decomposition and optimality theorems helpprove the positivity conjecture of Sen.

E.g. Case m = 1

ψBH1 (τ, z) =

3

η(τ)24

(E4(τ) A(τ, z)− 216 H(τ, z)

)(4)

Known functions. Coefficients of H are small.

E4A : 2 + 492q + 7256q2 + 53008q3 + 287244q4 + 1262512q5 ,

H : − 1

12+

1

2q + q2 +

4

3q3 +

3

2q4 + 2q5 + 2q6 .

For the cases m = 1, 2 and for all values of (n, `), can prove positivity.K. Bringmann, S.M..


Summary: Quantum black holes and number theory

The flow is both ways – number theory is useful to organize themicroscopic data of string theory, and seems to provide a convergentexpansion for quantum gravity.

We begin to see that these results can be recovered from gravityusing localization.

Ideas in quantum black hole theory act as guides for formalmathematical results (e.g. attractor mechanism ⇒ decompositiontheorem for mock Jacobi forms).

They also provide challenging problems in number theory, some ofwhich have begun to be solved (e.g. positivity of black holedegeneracies).


exact quantum entropy, mock modular forms and … quantum entropy, mock modular forms and holography...

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