Subleading Microstate Counting of AdS 4 Black Hole Entropy Leo Pando Zayas University of Michigan Great Lake Strings Conference Chicago, April 13, 2018 1711.01076, J. Liu, LPZ, V. Rathee and W. Zhao JHEP 1801 (2018) 026, J. Liu, LPZ, V. Rathee and W. Zhao JHEP 1708 (2017) 023, A. Cabo-Bizet, V. Giraldo-Rivera, LPZ 1712.01849, A. Cabo-Bizet, U. Kol, LPZ, I. Papadimitriou, V. Rathee Leo Pando Zayas (University of Michigan) Black Hole Entropy Geat Lakes Strings 1 / 35
Motivation S = k B c 3 A 4 G N � A confluence of thermodynamical, relativistic, gravitational, and quantum aspects. Hydrogen atom of QG. [Strominger-Vafa]. An explicit example in AdS 4 /CFT 3 : The large- N limit of the topologically twisted index of ABJM correctly reproduces the leading term in the entropy of magnetically charged black holes in asymptotically AdS 4 spacetimes [Benini-Hristov-Zaffaroni]. Extended also to: dyonic black holes, black holes with hyperbolic horizons and black holes in massive IIA theory. Agreement has been shown beyond the large N limit by matching the coefficient of log N [Liu-PZ-Rathee-Zhao] (Beyond Bohr energies). Leo Pando Zayas (University of Michigan) Black Hole Entropy Geat Lakes Strings 2 / 35
Outline The Topological Twisted Index of ABJM Theory beyond large N (logarithmic corrections). Magnetically Charged Asymptotically AdS 4 Black Holes. Logarithmic Corrections in Quantum Supergravity The quantum entropy formula for asymptotically AdS black holes. Conclusions Leo Pando Zayas (University of Michigan) Black Hole Entropy Geat Lakes Strings 3 / 35
The Index ABJM Theory ABJM: A 3d Chern-Simons-matter theory with U ( N ) k × U ( N ) − k gauge group with opposite integer levels. The matter sector contains four complex scalar fields Φ I , ( I = 1 , 2 , 3 , 4) in the bifundamental representation ( N , ¯ N ) , together with their fermionic partners. The theory is superconformal and has N = 6 supersymmetry generically but for k = 1 , 2 , the symmetry is enhanced to N = 8 . The global symmetry that is manifest in the N = 2 notation is SU (2) 1 , 2 × SU (2) 3 , 4 × U (1) T × U (1) R . Φ 1 , Φ 2 N − k N k Φ 3 , Φ 4 Leo Pando Zayas (University of Michigan) Black Hole Entropy Geat Lakes Strings 4 / 35
The Index The Topologically Twisted Index of ABJM Theories The topologically twisted index for three dimensional N = 2 field theories was defined in [Benini-Zaffaroni] (Honda ‘15, Closset ‘15) by evaluating the supersymmetric partition function on S 1 × S 2 with a topological twist on S 2 . Hamiltonian: The supersymmetric partition function of the twisted theory, Z ( n a , ∆ a ) = Tr ( − 1) F e − βH e iJ a ∆ a . It depends on the fluxes, n a , through H and on the chemical potentials ∆ a . The topologically twisted index for N ≥ 2 supersymmetric theories on S 2 × S 1 can be computed via supersymmetric localization. The supersymmetric localization computation of the topologically twisted index can be extended to theories defined on Σ g × S 1 . Leo Pando Zayas (University of Michigan) Black Hole Entropy Geat Lakes Strings 5 / 35
The Index General form of the Index Background: A R = 1 ds 2 = R 2 ( dθ 2 + sin 2 θdφ 2 ) + β 2 dt 2 , 2 cos θdφ. The index can be expressed as a contour integral: � � Z ( n a , y a ) = Z int ( x, m ; n a , y a ) . m ∈ Γ h C Z int meromorphic form, Cartan-valued complex variables x = e i ( A t + iβσ ) , lattice of magnetic gauge fluxes Γ h . Flavor magnetic fluxes n and fugacities y a = e i ( A a t + iβσ a ) . Localization: Z int = Z class Z one − loop . class = x km , Z gauge (1 − x α ) ( idu ) r , r – rank of the E.G.: Z CS 1 − loop = � α ∈ G gauge group, α – roots of G and u = A t + iβσ . Leo Pando Zayas (University of Michigan) Black Hole Entropy Geat Lakes Strings 6 / 35
The Index The topologically twisted index for ABMJ theory: 4 1 2 N 2 n a − 1 � � Z ( y a , n a ) = y det B × a a =1 I ∈ BAE � � � � � N i =1 x N x N 1 − x i 1 − ˜ x i i ˜ � i i � = j x j ˜ x j x j ) 1 − n a . � N � a =1 , 2 (˜ x j − y a x i ) 1 − n a � a =3 , 4 ( x i − y a ˜ i,j =1 Contour integral → Evaluation (Poles): e iB i = e i ˜ B i = 1 ˜ x j x j ˜ N (1 − y 3 x i )(1 − y 4 x i ) e iB i = x k � , i ˜ x j x j ˜ (1 − y − 1 x i )(1 − y − 1 x i ) j =1 1 2 ˜ x j x j ˜ N (1 − y 3 x i )(1 − y 4 x i ) e i ˜ B j = ˜ x k � . j ˜ x j x j ˜ (1 − y − 1 x i )(1 − y − 1 x i ) i =1 1 2 The 2 N × 2 N matrix B is the Jacobian relating the { x i , ˜ x j } variables to the { e iB i , e i ˜ B j } variables Leo Pando Zayas (University of Michigan) Black Hole Entropy Geat Lakes Strings 7 / 35
The Index Algorithmic Summary Recall, the chemical potentials ∆ a according to y a = e i ∆ a , and change of variables x i = e iu i , ˜ x j = e i ˜ u j . N � 1 − e i (˜ uj − ui +∆ a ) � � 1 − e i (˜ uj − ui − ∆ a ) � � � � 0 = ku i − i log log − 2 πn i , − a =3 , 4 a =1 , 2 j =1 N 1 − e i (˜ uj − ui +∆ a ) � 1 − e i (˜ uj − ui − ∆ a ) � � � � � � 0 = k ˜ u j − i log log − 2 π ˜ n j . − i =1 a =3 , 4 a =1 , 2 The topologically twisted index: (i) solve these equations for { u i , ˜ u j } ; (ii) insert the solutions into the expression for Z . Leo Pando Zayas (University of Michigan) Black Hole Entropy Geat Lakes Strings 8 / 35
The Index The large- N limit In the large- N limit, the eigenvalue distribution becomes continuous, and the set { t i } may be described by an eigenvalue density ρ ( t ) . u i = iN 1 / 2 t i + π − 1 u i = iN 1 / 2 t i + π + 1 2 δv ( t i ) , ˜ 2 δv ( t i ) , Figure: Eigenvalues for ∆ a = { 0 . 4 , 0 . 5 , 0 . 7 , 2 π − 1 . 6 } and N = 60 . Leo Pando Zayas (University of Michigan) Black Hole Entropy Geat Lakes Strings 9 / 35
The Index Description of the eigenvalue distribution. Figure: The eigenvalue density ρ ( t ) and the function δv ( t ) for ∆ a = { 0 . 4 , 0 . 5 , 0 . 7 , 2 π − 1 . 6 } and N = 60 , compared with the leading order expression. Re log Z = − N 3 / 2 n a � � 2∆ 1 ∆ 2 ∆ 3 ∆ 4 3 ∆ a a Leo Pando Zayas (University of Michigan) Black Hole Entropy Geat Lakes Strings 10 / 35
The Index Beyond Large N : Numerical Fits ∆ 1 ∆ 2 ∆ 3 f 1 f 2 f 3 π/ 2 π/ 2 π/ 2 3 . 0545 − 0 . 4999 − 3 . 0466 π/ 4 π/ 2 π/ 4 4 . 2215 − 0 . 0491 n 1 − 0 . 4996 + 0 . 0000 n 1 − 4 . 1710 − 0 . 2943 n 1 − 0 . 1473 n 2 − 0 . 0491 n 3 +0 . 0000 n 2 + 0 . 0000 n 3 +0 . 0645 n 2 − 0 . 2943 n 3 0 . 3 0 . 4 0 . 5 7 . 9855 − 0 . 2597 n 1 − 0 . 4994 − 0 . 0061 n 1 − 9 . 8404 − 0 . 9312 n 1 − 0 . 5833 n 2 − 0 . 6411 n 3 − 0 . 0020 n 2 − 0 . 0007 n 3 − 0 . 0293 n 2 + 0 . 3739 n 3 0 . 4 0 . 5 0 . 7 6 . 6696 − 0 . 1904 n 1 − 0 . 4986 − 0 . 0016 n 1 − 7 . 5313 − 0 . 6893 n 1 − 0 . 4166 n 2 − 0 . 4915 n 3 − 0 . 0008 n 2 − 0 . 0001 n 3 − 0 . 1581 n 2 + 0 . 2767 n 3 Numerical fit for: Re log Z = Re log Z 0 + f 1 N 1 / 2 + f 2 log N + f 3 + · · · The values of N used in the fit range from 50 to N max where N max = 290 , 150 , 190 , 120 for the four cases, respectively. The index is independent of the magnetic fluxes in the special case ∆ a = { π/ 2 , π/ 2 , π/ 2 , π/ 2 } Leo Pando Zayas (University of Michigan) Black Hole Entropy Geat Lakes Strings 11 / 35
The Index In the large- N limit, the k = 1 index takes the form F = − N 3 / 2 n a � � + N 1 / 2 f 1 (∆ a , n a ) 2∆ 1 ∆ 2 ∆ 3 ∆ 4 3 ∆ a a − 1 2 log N + f 3 (∆ a , n a ) + O ( N − 1 / 2 ) , where F = Re log Z . The leading O ( N 3 / 2 ) term [BHZ], and exactly reproduces the Bekenstein-Hawking entropy of a family of extremal AdS 4 magnetic black holes admitting an explicit embedding into 11d supergravity, once extremized with respect to the flavor and R -symmetries. The − 1 2 log N term [Liu-PZ-Rathee-Zhao]. Leo Pando Zayas (University of Michigan) Black Hole Entropy Geat Lakes Strings 12 / 35
The Index Topologically twisted index on Riemann surfaces The topologically twisted index can be defined on Riemann surfaces with arbitrary genus. There is a simple relation between the index on Σ g × S 1 and that on S 2 × S 1 : F S 2 × S 1 ( n a , ∆ a ) = (1 − g ) F Σ g × S 1 ( n a 1 − g , ∆ a ) . Since the coefficient of the logarithmic term in F S 2 × S 1 does not depend on n a we simply have F Σ g × S 1 ( n a , ∆ a ) = · · · − 1 − g log N + · · · . 2 Leo Pando Zayas (University of Michigan) Black Hole Entropy Geat Lakes Strings 13 / 35
One-loop Supergravity AdS 4 /CFT 3 Holographically, ABJM describes a stack of N M2-branes probing a C 4 / Z k singularity, whose low energy dynamics are effectively described by 11 dimensional supergravity. The index is computed for ABJM theory with a topological twist, equivalently, fluxes on S 2 . On the gravity side it corresponds to microstate counting of magnetically charged asymptotically AdS 4 black holes. Leo Pando Zayas (University of Michigan) Black Hole Entropy Geat Lakes Strings 14 / 35
One-loop Supergravity Supergravity solution A solution of N = 2 gauged sugra with prepotential √ X 0 X 1 X 2 X 3 coming from M theory on AdS 4 × S 7 with F = − 2 i U (1) 4 ∈ SO (8) . Background metric : � 2 dr 2 � c ds 2 = − e K ( X ) dt 2 + e −K ( X ) � 2 +2 e −K ( X ) r 2 d Ω 2 g r − 2 . 2 g r � c g r − 2 g r Magnetic charges θφ = − n a F a F 1 √ 2 sin θ, tr = 0 . Leo Pando Zayas (University of Michigan) Black Hole Entropy Geat Lakes Strings 15 / 35
Recommend
More recommend
