Functional determinants, Index theorems, and Exact quantum black hole entropy

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1 Functional determinants, Index theorems, and Exact quantum black hole entropy Sameer Murthy King s College London Strings 2015 Bengaluru, June 25, 2015

2 BPS black hole entropy is a detailed probe of aspects of quantum gravity Black hole entropy underlying ensemble of microscopic states

3 BPS black hole entropy is a detailed probe of aspects of quantum gravity Black hole entropy Macroscopic underlying ensemble of microscopic states Microscopic g s N 1 g s N 1 g s N N

4 BPS black hole entropy is a detailed probe of aspects of quantum gravity Black hole entropy Macroscopic underlying ensemble of microscopic states Microscopic g s N 1 g s N 1 g s N N Q: How many microstates are associated to a BH?

5 Aim of talk In a theory of 4d N=2 supergravity, coupled to vector multiplets and hyper multiplets, with generic higher derivative couplings, compute the perturbatively exact formula for the quantum gravitational entropy of a 1/2 BPS black hole.

6 Aim of talk In a theory of 4d N=2 supergravity, coupled to vector multiplets and hyper multiplets, with generic higher derivative couplings, compute the perturbatively exact formula for the quantum gravitational entropy of a 1/2 BPS black hole. Based on: S.M., V. Reys, arxiv: , B. de Wit, S.M., V. Reys, in progress. and on the work of: Sen (Quantum entropy function program), Ooguri,Strominger,Vafa (OSV), Cardoso, de Wit, Mohaupt; Banerjee, Dabholkar, David, Denef, Gaiotto, Gomes,Gupta, Hama, Hosomichi, Jatkar, Lal, Mahapatra, Mandal, Moore, Pestun, Pioline, Shih, Yin,

7 Near-horizon region is an independent quantum system, fixed by charges 4d BPS black hole with charges (q I,p I ) ds 2 = v (r 2 1) d 2 + dr2 r 2 1 F I rt = e I, F I = p I sin + v d 2 +sin 2 d 2 X I = X I r Attractor geometry: (Ferrara, Kallosh, Strominger 95) L 0 Euclidean AdS 2 S 2 J 0

8 Quantum BPS black hole entropy is an AdS 2 functional integral (Sen 08) exp(s qu BH (q I)) Z AdS2 (q I )= exp iq I A I reg AdS 2 Saddle point evaluation S qu BH = Sclass BH + Attractor entropy including local higher-derivative corrections. (Cardoso, de Wit, Mohaupt, 98, ) Leading logarithmic one-loop corrections systematized. 2 (Sen + Banerjee, Gupta, Mandal, 10-14, Larsen, Keeler, Lisbão 14, 15).

9 The functional integral localizes onto the solutions of the off-shell BPS equations Duistermaat-Heckmann, Atiyah-Singer-Bott, Berligne-Vergne, Witten (1980s), Pestun 07 Z AdS2 (q I ) Z M dµ O e S L 0 Euclidean AdS 2 S 2 J 0 Superalgebra Q 2 = L 0 J 0 H I := Z M dµ O e S

10 The functional integral localizes onto the solutions of the off-shell BPS equations Duistermaat-Heckmann, Atiyah-Singer-Bott, Berligne-Vergne, Witten (1980s), Pestun 07 Z Z AdS2 (q I ) dµ O e S M J 0 L 0 Euclidean AdS 2 S 2 Superalgebra Q 2 = L 0 J 0 H Z I( )= M dµ O e S+ QV, V = X Z d 4 x Q I(0) = I(1) = Z M Q dµ Q O e S Z 1-loop (QV) = {Q =0}

11 Localization in supergravity (A.Dabholkar, J.Gomes, S.M. 10, 11, 14) 1. Describe the supergravity theory in a formalism which admits off-shell supersymmetry transformations. 2. Find all solutions of localization equations Q =0, subject to AdS 2 S 2 boundary conditions. 3. Evaluate full supergravity action on these solutions (including all higher derivative terms). 4. Compute quantum measure on the solution space.

12 The fields of N=2 conformal supergravity Weyl multiplet coupled to n v +1 (de Wit, van Holten, Van Proeyen 80) U(1) vector multiplets X I! (x) X I, (I =0,,n v ), g µ! (x) 2 g µ e K i(x I FI XI F I ) appears in the action as p ge K R + (conformal compensator) Can choose gauge e K =1 or p g =1

13 The fields of N=2 conformal supergravity Weyl multiplet coupled to n v +1 (de Wit, van Holten, Van Proeyen 80) U(1) vector multiplets X I! (x) X I, (I =0,,n v ), g µ! (x) 2 g µ e K i(x I FI XI F I ) appears in the action as p ge K R + (conformal compensator) Can choose gauge e K =1 or p g =1 In this gauge, measures the size of e K(XI ) AdS 2

14 The space of off-shell BPS solutions In vector multiplet sector, scalar fields go off-shell: X I (r) I r AdS 2 (A.Dabholkar, J.Gomes, S.M. 10) In the gravity multiplet sector, only solution is AdS 2 S 2. (R.Gupta, S.M. 12)

15 The space of off-shell BPS solutions In vector multiplet sector, scalar fields go off-shell: X I (r) (A.Dabholkar, J.Gomes, S.M. 10) In the gravity multiplet sector, only solution is AdS 2 S 2. (R.Gupta, S.M. 12) Localization manifold labelled by one parameter for each vector multiplet X I. I X I (r = 1) I r AdS 2

16 The space of off-shell BPS solutions In vector multiplet sector, scalar fields go off-shell: X I (r) (A.Dabholkar, J.Gomes, S.M. 10) In the gravity multiplet sector, only solution is AdS 2 S 2. (R.Gupta, S.M. 12) Localization manifold labelled by one parameter for each vector multiplet X I. I X I (r = 1) Classical measure for scalar manifold I AdS 2 Y I dx I M(X I ) has been computed. Does not scale with charges. r (Cardoso, de Wit, Mahapatra 12)

17 Action of supergravity theory Chiral superspace action is governed by holomorphic prepotential F (X I ), computable in string theory. Takes a simple form on localization manifold.

18 Action of supergravity theory Chiral superspace action is governed by holomorphic prepotential F (X I ), computable in string theory. Takes a simple form on localization manifold. Only holomorphic chiral-superspace integrals in the action contribute (Non-renormalization). (S.M., V.Reys, 13)

19 Action of supergravity theory Chiral superspace action is governed by holomorphic prepotential F (X I ), computable in string theory. Takes a simple form on localization manifold. Only holomorphic chiral-superspace integrals in the action contribute (Non-renormalization). (S.M., V.Reys, 13) Z pert AdS 2 (q, p) = exp Z Y nv I=0 d I M( I ) QV q I I +4 Im F I + ip I Z 1-loop ( I )

20 Action of supergravity theory Chiral superspace action is governed by holomorphic prepotential F (X I ), computable in string theory. Takes a simple form on localization manifold. Only holomorphic chiral-superspace integrals in the action contribute (Non-renormalization). (S.M., V.Reys, 13) Z pert AdS 2 (q, p) = exp Z Y nv I=0 d I M( I ) QV q I I +4 Im F I + ip I Z 1-loop ( I )

21 1-loop determinant is a function of dynamical scale of quantum supergravity The length scale at the center of AdS 2 `2AdS = e K( I ) depends on the position in the localization manifold.

22 1-loop determinant is a function of dynamical scale of quantum supergravity The length scale at the center of AdS 2 `2AdS = e K( I ) depends on the position in the localization manifold. Result (S.M., V. Reys, arxiv: ) QV Z 1-loop ( I )=exp a 0 K( I ), a 0 = X multiplets a multiplet 0

23 1-loop determinant is a function of dynamical scale of quantum supergravity The length scale at the center of AdS 2 `2AdS = e K( I ) depends on the position in the localization manifold. Result (S.M., V. Reys, arxiv: ) QV Z 1-loop ( I )=exp a 0 K( I ), a 0 = X multiplets a multiplet 0 Index theorem a vector 0 = a hyper 0 = 1 12 (Result for vector multiplets also in Ito, Gupta, Jeon ) (Graviton multiplet in progress with B. de Wit, V. Reys)

24 Sketch of computation X Q QX Separate fields into doublets of Q Q Q

25 Sketch of computation X Q QX Separate fields into doublets of Q D 10 D 01 Q Q

26 Sketch of computation X Q QX Separate fields into doublets of Q D 10 D 01 Q Q Determinant of b Q b V =(X, Kb X)+(,K f ) related by linear algebra to determinant of Q 2 = H

27 Sketch of computation X Q QX Separate fields into doublets of Q D 10 D 01 Q Q Determinant of b Q b V =(X, Kb X)+(,K f ) related by linear algebra to determinant of Q 2 = H Eigenvalues of H encoded in index ind(d 10 )(t) =Tr X e iht Tr e iht = X n a(n) e it n

28 The index is captured by fixed points of H Atiyah-Bott index theorem ind(d 10 )(t) = X x=fixed pts of H Tr X, ( 1) F e iht (ex = Hx) Fixed points of H = L 0 J 0 J 0 L 0 Euclidean AdS 2 S 2

29 The index is captured by fixed points of H Atiyah-Bott index theorem ind(d 10 )(t) = X x=fixed pts of H Tr X, ( 1) F e iht (ex = Hx) Fixed points of H = L 0 J 0 ind vec (D 10 )= X n 1 4ne itn/`ads J 0 L 0 Euclidean AdS 2 S 2

30 The index is captured by fixed points of H Atiyah-Bott index theorem ind(d 10 )(t) = X x=fixed pts of H Tr X, ( 1) F e iht (ex = Hx) Fixed points of H = L 0 J 0 ind vec (D 10 )= X n 1 4ne itn/`ads Degeneracies Eigenvalues L 0 Euclidean AdS 2 S 2 J 0 Regularize determinant using zeta functions.

31 Technical comments In supergravity, Q 2 = L 0 J 0 + gauge Introduce (b,c) ghost system and BRST operator Q B Introduce bq = Q + Q B with Q b 2 = H = L 0 J 0 Boundary modes ( pure gauge modes that do not vanish at the boundary) need to be treated separately. (See e.g. Banerjee, Gupta, Sen 10; Sen 11)

32 Checks QV Z 1-loop ( I )=exp a 0 K( I ), a 0 = X multiplets a multiplet 0

33 Checks QV Z 1-loop ( I )=exp a 0 K( I ), a 0 = X multiplets a multiplet 0 Large charge limit: S BH = A H 4 + a 0 log A H Agrees with the on-shell results. (Sen et al)

34 Checks QV Z 1-loop ( I )=exp a 0 K( I ), a 0 = X multiplets a multiplet 0 Large charge limit: S BH = A H 4 + a 0 log A H Agrees with the on-shell results. (Sen et al) Microscopics of N=8 string theory: Truncation of N=8 to N=2 theory. Quantum BH entropy agrees with microscopic integer degeneracies exactly! (A. Dabholkar, J. Gomes, S.M., 11 14) Can check that the 1-loop contributions of fields that are thrown away add up to zero.

35 1 2 -BPS N=2 BH entropy: where do we stand? Z pert AdS 2 (q, p) = Z Y nv I=0 d I ( 0 ) 2 exp 12 e K( ) q I I +4 ImF ( I + ip I ) = 2(n v +1 n h ) (cf. Ooguri-Strominger-Vafa 04, Denef-Moore 07, Sen, ) Note: Still need to complete graviton calculation. This assumes a classical measure that has been derived for two-derivative theories. (Cardoso, de Wit, Mahapatra 12) This formula implies a surprising cancellation of states in N=2 string theory.

36 Resolution of topological string puzzle Consider type II string theory on a CY3. The (perturbative) topological string partition function at strong topological string coupling!1 is: F top = i (2 )3 6 2 C ABC t A t B t C i 12 c 2A t A + F GW (,t A )

37 Resolution of topological string puzzle Consider type II string theory on a CY3. The (perturbative) topological string partition function at strong topological string coupling F top =!1 is: i (2 )3 6 2 C ABC t A t B t C i 12 c 2A t A + F GW (,t A ) Upon analytic continuation to weak coupling (supergravity regime), the correct expression is: ef top = i (2 )3 6 2 C ABC t A t B t C 24 log i 12 c 2A t A + F GW (,t A ) The puzzle is to interpret the log term (not associated to any order in perturbation theory). (Pioline 06, Denef-Moore 07)

38 Resolution of topological string puzzle Consider type II string theory on a CY3. The (perturbative) topological string partition function at strong topological string coupling F top =!1 is: i (2 )3 6 2 C ABC t A t B t C i 12 c 2A t A + F GW (,t A ) Upon analytic continuation to weak coupling (supergravity regime), the correct expression is: ef top = i (2 )3 6 2 C ABC t A t B t C 24 log i 12 c 2A t A + F GW (,t A ) The puzzle is to interpret the log term (not associated to any order in perturbation theory). (Pioline 06, Denef-Moore 07) Resolution: arises in 1PI effective action of supergravity. (See Dedushenko,Witten 14)

39 The exact N=2 BH entropy formula Z pert AdS 2 (q, p) = exp Z M Q n v Y I=0 d I M class ( I ) q I I +4 Im F I + ip I Z 1-loop ( I ) Z 1-loop =exp K( I ) 2 24, = 2(n v +1 n h ).

40 The exact N=2 BH entropy formula Z pert AdS 2 (q, p) = exp Z M Q n v Y I=0 d I M class ( I ) q I I +4 Im F I + ip I Z 1-loop ( I ) Z 1-loop =exp K( I ) 2 24, = 2(n v +1 n h ). M class = 2 exp( K 0 0 ( I )) 24 1 at first order in large charges. (Cardoso, de Wit, Mahapatra 12)

41 The exact N=2 BH entropy formula e Squ (q,p) = Z n v Y d I e q I I top 2 e K0 Z top 2 M Q I=0 e K0 = i(x I F 0 I XI F 0 I ), X I = I + ip I, Z top = top 2 /24e 2 if (X I ), top =4 /X 0, = 2(n v +1 n h ).

42 Why does the truncation work? We kept: zero modes of 8 vector multiplets We threw away: Non-zero modes of 8 vector multiplets (15-8) vector multiplets 10 hyper multiplets 6 gravitini multiplets 1 gravity multiplet (15 8) Total 0

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