Supersymmetric localization and black holes microstates Seyed - PowerPoint PPT Presentation

Supersymmetric localization and black holes microstates Seyed Morteza Hosseini Kavli IPMU YITP (Kyoto), August 19-23 Strings and Fields 2019 Seyed Morteza Hosseini (Kavli IPMU) 1 / 26

Introduction Black holes have more lessons in store for us! S BH = Area Bekenstein-Hawking entropy: . 4 G N The number of black hole microstates d micro should then be given by d micro = e S BH . But where are the microstates accounting for the black hole entropy? String theory provides a precise statistical mechanical interpretation of S BH for a class of asymptotically flat black holes. [Strominger, Vafa’96] Black holes → bound states of D-branes! Seyed Morteza Hosseini (Kavli IPMU) 2 / 26

Introduction No similar results for AdS d +1 > 4 black holes was known until recently! [Benini, Hristov, Zaffaroni’15] Holography + supersymmetric localization Black hole entropy → counting states in the dual CFT This talk I will review recent progress for AdS d +1 BHs in diverse dimensions. Seyed Morteza Hosseini (Kavli IPMU) 3 / 26

Basics Stringy BPS black holes SCFT d on S d − 1 × R t I KN-AdS black holes ↔ SCFT d on M d − 1 × R t II magnetic AdS black holes ↔ ◮ Case I has to rotate . ◮ Case II is topologically twisted and can be static. ◮ Characterized by nonzero magnetic fluxes for the graviphoton/R-symmetry: � F ∈ 2 π Z . C⊂M d − 1 Most manifest in AdS 4 black holes w/ horizon AdS 2 × S 2 . [Romans’92] Seyed Morteza Hosseini (Kavli IPMU) 4 / 26

Counting microstates BPS partition function � Z (∆ I , ω i ) = Tr Q =0 e i(∆ I Q I + ω i J i ) = d micro ( Q I , J i ) e i(∆ I Q I + ω i J i ) . Q I ,J i ◮ It counts states w/ the same susy, charges, and angular momenta. ◮ S BH ( Q I , J i ) = log d micro ( Q I , J i ) , � d micro ( Q I , J i ) = e S BH ( Q I ,J i ) = Z (∆ I , ω i ) e − i(∆ I Q I + ω i J i ) . ∆ I , ω i Saddle point approximation (large charges) S BH ( Q I , J i ) ≡ I (∆ I , ω i ) = log Z (∆ I , ω i ) − i(∆ I Q I + ω i J i ) . ◮ ∂ I (∆ I , ω i ) = ∂ I (∆ I , ω i ) = 0 . ∂ ∆ I ∂ω i Seyed Morteza Hosseini (Kavli IPMU) 5 / 26

Counting microstates Problem AdS BHs preserve only two real supercharges while we have efficient tools for counting states preserving four.. Seyed Morteza Hosseini (Kavli IPMU) 6 / 26

Counting microstates Problem AdS BHs preserve only two real supercharges while we have efficient tools for counting states preserving four.. Witten index (supersymmetric partition function) M d − 1 × S 1 (∆ I , ω i ) = Tr H M d − 1 ( − 1) F e − β {Q , Q † } e i(∆ I Q I + ω i J i ) . Z susy ◮ Superconformal index for SCFTs on S d − 1 × S 1 [Romelsberger’05; Kinney, Maldacena, Minwalla, Raju’05] ◮ Topologically twisted index for SCFTs on twisted M d − 1 × S 1 [Okuda, Yoshida’12; Nekrasov, Shatashvili’14; Gukov, Pei’15; Benini, Zaffaroni’15] Lower bound on entropy. Index = entropy if there are no large cancellations between bosonic and fermionic ground states. [Arguments for some asymptotically flat black holes by Sen’09] Seyed Morteza Hosseini (Kavli IPMU) 6 / 26

Magnetic AdS black holes Black holes in M-theory on AdS 4 × S 7 : [Cacciatori, Klemm’08; Dall’Agata, Gnecchi’10; Hristov, Vandoren’10; Halmagyi14; Hristov, Katmadas, Toldo’18] ◮ Preserve two real supercharges (1/16 BPS) ◮ Four electric and magnetic charges ( p a , q a ) under U(1) 4 ⊂ SO(8), one angular momentum J in AdS 4 . ◮ Only seven independent parameters: 4 � p a = 2 − 2 g . twisting condition : a =1 together with a charge constraint for having a regular horizon. ◮ S BH = O ( N 3 / 2 ) . ◮ We focus on J = 0. ◮ Near horizon AdS 2 × Σ g . Seyed Morteza Hosseini (Kavli IPMU) 7 / 26

Magnetic AdS black holes Setting all q a = 0 � √ S BH ( p ) = 2 π 3 N 3 / 2 F 2 + Θ , 4 � � F 2 ≡ 1 p a p b − 1 Θ ≡ ( F 2 ) 2 − 4 p 1 p 2 p 3 p 4 . p 2 a , 2 4 a =1 a<b ◮ Attractor mechanism: � S BH ( p a , q a ) = i p a ∂ � W (∆ a ) � − i∆ a q a crit. . � ∂ ∆ a W (∆ a ) = − 2i √ ∆ 1 ∆ 2 ∆ 3 ∆ 4 . ◮ g-sugra prepotential: � ◮ � a ∆ a = 2 π : scalar fields at the horizon. [Ferrara, Kallosh, Strominger’ 06; Cacciatori, Klemm’08; Dall’Agata, Gnecchi’10] Seyed Morteza Hosseini (Kavli IPMU) 8 / 26

Holographic setup ABJM on S 2 × R w/ a twist on S 2 � � W = Tr A 1 B 1 A 2 B 2 − A 1 B 2 A 2 B 1 , A 2 A 1 ∆ 1 + ∆ 2 + ∆ 3 + ∆ 4 = 2 π , N + k N − k B 1 B 2 U(1) R × SU(2) 1 × SU(2) 2 × U(1) top . ◮ Magnetic background for global symmetries: Landau levels on S 2 . ◮ Twisting condition: � 4 a =1 p a = 2 . D µ ǫ = ∂ µ ǫ + 1 4 ω ab µ γ ab ǫ + i V µ ǫ = ∂ µ ǫ ���� i 4 ω ab µ γ ab ǫ = constant on S 2 . Seyed Morteza Hosseini (Kavli IPMU) 9 / 26

Holographic microstates counting A topologically twisted index H p ,σ β ( v a , p a ) = Tr H S 2 ( − 1) F e − βH e i � 4 a =1 ∆ a Q a . Z S 2 × S 1 QM [Benini, Zaffaroni; 1504.03698] ◮ ∆ a : chemical potentials for flavor symmetry charges Q a . ◮ σ a : real masses. ◮ only states with 0 = H − σ a J a contribute. ◮ electric charges q a can be introduced using ∆ a . ◮ can be computed using supersymmetric localization . The index is a holomorphic function of v a with v a = ∆ a + i βσ a . σ a = 0 . Seyed Morteza Hosseini (Kavli IPMU) 10 / 26

Supersymmetric localization Consider a supersymmetric gauge theory on a compact manifold M . Partition function � D φ e − S [ φ ] . Z M ≡ Euclidean Feynman path integral = ◮ φ : the set of fields in the theory. ◮ S [ φ ]: the action functional. Seyed Morteza Hosseini (Kavli IPMU) 11 / 26

Supersymmetric localization Consider a supersymmetric gauge theory on a compact manifold M . Partition function � D φ e − S [ φ ] . Z M ≡ Euclidean Feynman path integral = ◮ φ : the set of fields in the theory. ◮ S [ φ ]: the action functional. Localization argument [Witten’88; Pestun’06] ◮ Let δ be a Grassmann-odd symmetry of our theories, i.e. δS = 0. ◮ Deform the theories by a δ -exact term. � D φ e − S [ φ ] − tδV , Z M ( t ) = t ∈ R > 0 . Seyed Morteza Hosseini (Kavli IPMU) 11 / 26

Supersymmetric localization Consider a supersymmetric gauge theory on a compact manifold M . Partition function � D φ e − S [ φ ] . Z M ≡ Euclidean Feynman path integral = ◮ φ : the set of fields in the theory. ◮ S [ φ ]: the action functional. Localization argument [Witten’88; Pestun’06] ◮ Let δ be a Grassmann-odd symmetry of our theories, i.e. δS = 0. ◮ Deform the theories by a δ -exact term. � D φ e − S [ φ ] − tδV , Z M ( t ) = t ∈ R > 0 . The partition function is independent of t ! � � � � ∂Z M ( t ) D φ e − S [ φ ] − tδV δV = − e − S [ φ ] − tδV V = − D φ δ = 0 . ∂t Hence we can evaluate Z M ( t ) as t → ∞ . Seyed Morteza Hosseini (Kavli IPMU) 11 / 26

Supersymmetric localization Localization locus If ( δV ) | even ≥ 0 = ⇒ the integral localizes to ( δV ) | even ( φ 0 ) = 0 . ◮ Let’s parameterize the fields around the localization locus by φ = φ 0 + t − 1 / 2 ˆ φ . ◮ For large t , we can Taylor expand the action around φ 0 : S + δV = S [ φ 0 ] + ( δV ) (2) [ˆ φ ] + O ( t − 1 / 2 ) . ◮ Gaussian integration! Localization formula � D φ 0 e − S [ φ 0 ] Z 1-loop [ φ 0 ] . Z M = ( δV ) | even =0 ◮ Z 1-loop [ φ 0 ]: the ratio of fermionic and bosonic determinants. Seyed Morteza Hosseini (Kavli IPMU) 12 / 26

A topologically twisted index Localization formula [Benini, Zaffaroni’15; Closset, Kim, Willett’16] � � 1 Z S 2 × S 1 ( p , y ) = Z int ( m , x ; p , y ) , | W | C m ∈ Γ h ◮ x = e i u , y a = e i∆ a . ◮ Classical piece: Z cl = x k m . ◮ One-loop contributions: � √ x ρ y a � ρ ( m ) − p a +1 � � Z χ Z V (1 − x α ) . 1-loop = , 1-loop = 1 − x ρ y a ρ ∈ R α ∈ G We are interested in the large N limit of the matrix integral. Seyed Morteza Hosseini (Kavli IPMU) 13 / 26

TQFT and Bethe vacua Reduction to two-dimensional theory w/ all KK modes on S 1 [Witten’92; Nekrasov, Shatashvili’09] 2D ◮ Massive theory w/ a set of discrete vacua (Bethe vacua), � � � � � i ∂ W ( x ) � Li 2 ( x ρ y a ) + . . . . W ( x, y a ) = exp x = x ∗ = 1 , � ∂x ρ ∈ R Many 3D and 4D supersymmetric partition functions can be written as a sum over Bethe vacua. [Closset, Kim, Willett’17’18] Seyed Morteza Hosseini (Kavli IPMU) 14 / 26

A topologically twisted index Bethe sum formula: � � − 1 � Z S 2 × S 1 ( p , y ) = ( − 1) rk( G ) Z int ( m = 0 , x ∗ ; p , y ) det ij ∂ i ∂ j W ( x ) . | W | x ∗ [Okuda, Yoshida’12; Nekrasov, Shatashvili’14; Gukov, Pei’15; Benini, Zaffaroni’15; Closset, Kim, Willett’17] Seyed Morteza Hosseini (Kavli IPMU) 15 / 26