Black Hole Entropy from 5D Twisted Itamar Yaakov University of - PowerPoint PPT Presentation

Black Hole Entropy from 5D Twisted Indices Black Hole Entropy from 5D Twisted Itamar Yaakov University of Indices Tokyo - Kavli IPMU based on: Introduction S. M. Hosseini, I.Y. and A. Zaffaroni - in preparation Calculation in the CFT An example Itamar Yaakov Conclusion University of Tokyo - Kavli IPMU Workshop on Supersymmetric Localization and Holography: Black Hole Entropy and Wilson Loops International Center for Theoretical Physics (ICTP), Trieste July 13, 2018

Black hole entropy Black Hole Entropy from Identification of the microstates contributing to the entropy of 5D Twisted Indices black holes is a long standing problem since the work of Itamar Yaakov Bekenstein and Hawking. University of Tokyo - Kavli ◮ String theory has had some success at computing the IPMU entropy of supersymmetric black holes [Strominger and Introduction Vafa (1996)]. Calculation in the CFT ◮ For black holes in AdS, the AdS/CFT correspondence An example gives, in principle, a way of counting the microstates Conclusion using the dual conformal field theory. ◮ Attempts have been made to do this by counting operators preserving some fraction of supersymmetry in N = 4 super-Yang-Mills, but with only partial success [Kinney, Maldacena, Minwalla, and Raju (2005), Grant, Grassi, Kim, and Minwalla (2008)].

Twisted partition functions vs black hole entropy Benini, Hristov, and Zaffaroni matched the partition function Black Hole Entropy from on twisted S 2 × S 1 to the entropy of a 4d black hole [Benini, 5D Twisted Indices Hristov, and Zaffaroni (2015)] Itamar Yaakov � University of I ( s , ˆ ∆) ≡ log Z ( s , ˆ q I ˆ Tokyo - Kavli ∆) − i ∆ I = S BH ( s , q ) IPMU I Introduction ◮ The field theory model is ABJM [Aharony, Bergman, Calculation in the CFT Jafferis, Maldacena (2008)]: A Lagrangian, maximally An example supersymmetric SCFT in 3d. Conclusion ◮ The entropy is represented by the finite part of the partition function, not an anomaly. ◮ The partition function is computable at large N using an effective twisted superpotential and its Bethe Ansatz Equations. ◮ The comparison is to an AdS 4 supersymmetric black hole [Cacciatori and Klemm (2010)].

Aspects of the BHZ calculation Black Hole Entropy from Several technical aspects of the BHZ calculation seem crucial 5D Twisted Indices to its success relative to previous endeavors Itamar Yaakov ◮ The index is topological and all of the states University of Tokyo - Kavli contributing are regarded as ground states. IPMU ◮ The one loop contributions to the effective action for the Introduction matrix model are simple. Calculation in the CFT ◮ The flavor symmetries are manifest. An example ◮ The complex scalar modulus is integrated over a contour Conclusion given by the Jeffrey-Kirwan prescription. ◮ There are supersymmetric fluxes for dynamical and background gauge fields. ◮ There is an equivariant deformation , which represents a black hole with angular momentum, but it can be turned off.

A five dimensional analogue We are using the same approach in 5d, by considering an Black Hole Entropy from appropriate theory on a manifold of the type M 4 × S 1 where 5D Twisted Indices M 4 is toric Kähler, which shares many of the features of BHZ Itamar Yaakov ◮ There is a twisted topological partition function amenable University of Tokyo - Kavli to localization. IPMU ◮ There are gravity solutions with which to compare. Introduction ◮ The theory lives in an odd dimension and the finite part Calculation in the CFT of the partition function is expected to be universal. An example ◮ There is an equivariant deformation which can be turned Conclusion off [Nekrasov (2002)]. ◮ There are fluxes and a contour prescription for the evaluation of the matrix model [Nekrasov (2006), Bawani,Bonelli,Ronzani,Tanzini (2014), Bershtein, Bonelli, Ronzani, Tanzini (2015)]. Many of the necessary calculations have already been done [citations on this slide + Källén and Zabzine (2012), Jafferis and Pufu (2012)].

Toward an entropy formula in five dimensions Black Hole Entropy from 5D Twisted There are a number of differences from the three dimensional Indices case Itamar Yaakov University of ◮ We can consider more complicated topologies. Tokyo - Kavli IPMU ◮ There is a Lagrangian theory with N = 2 supersymmetry, Introduction but it is not conformal. The strong coupling limit is Calculation in believed to represent a 6d (2 , 0) SCFT. the CFT ◮ Instanton contributions are present, but presumably go An example Conclusion away at leading order at large N . There are also some technical challenges ◮ The integration contour and sum over fluxes is not well understood. ◮ The correct analogue of the Bethe Ansatz Equations is not clear.

Donaldson - Witten theory A 4d N = 2 theory can be twisted: coupled to curvature on Black Hole Entropy from M 4 using a diagonal combination of the spin group 5D Twisted Indices SU (2) l × SU (2) r and the R-symmetry group SU (2) R Itamar Yaakov [Witten (1988)] University of Tokyo - Kavli ◮ A scalar supercharge Q is preserved on an arbitrary IPMU manifold. Introduction ◮ The energy-momentum tensor is Q exact. Calculation in the CFT For G = SU (2) , the theory of Q closed observables is the SWDN theory on toric Kähler manifolds cohomological Donaldson-Witten TQFT Supergravity background ◮ Computes the intersection theory on the moduli space of Localization Partition function and G -instantons on M 4 : Donaldson invariants. large N limit An example ◮ The Seiberg-Witten solution is an effective computational Conclusion tool. ◮ The low energy effective field theory approach includes a sum over SW monopoles and an integral over a moduli space: the u-plane [Moore and Witten (1997)].

The toric Kähler manifold M 4 Black Hole � n 2 , D 2 Entropy from 5D Twisted Indices m 1 � Itamar Yaakov University of σ 2 σ 1 m 2 � Tokyo - Kavli n 3 , D 3 � IPMU n 1 , D 1 � Introduction Calculation in σ d the CFT SWDN theory on toric Kähler manifolds Supergravity background Localization n d , D d � Partition function and large N limit An example Canonical construction for a metric [Guillemin (1994) and Conclusion Abreu (2003)] � G − 1 � ds 2 = G ij dx i dx j + ij dy i dy j , i, j ∈ { 1 , 2 } ◮ x i , y i coordinates on the Delzant polytope and torus.

Nekrasov’s equivariant extension I Black Hole Entropy from When M 4 admits a metric with an isometry, there is a 5D Twisted Indices refinement due to Nekrasov, using the Ω -deformation Itamar Yaakov ◮ Introduce a supercharge which squares to the isometry University of Tokyo - Kavli with vector v . IPMU ◮ On R 4 this is the setting for the Nekrasov partition Introduction function. Recall the relationship to the effective Calculation in the CFT prepotential [Nekrasov (2002)] SWDN theory on toric Kähler manifolds Supergravity 1 background log Z inst ( � a, ǫ 1 , ǫ 2 ; q ) ≈ F 0 ( � a, Λ) , Localization ǫ 1 ǫ 2 Partition function and large N limit q → Λ 2 h ∨ ( G ) − k ( R ) . An example Conclusion On a toric Kahler manifold we use the torus isometry to localize to the vertices of the polytope. We will have to use the 5d version of the Nekrasov partition function.

Nekrasov’s equivariant extension II Black Hole Entropy from The Nekrasov partition function contains much more 5D Twisted Indices information than just the effective prepotential. We can Itamar Yaakov expand in ǫ 1 , ǫ 2 University of Tokyo - Kavli IPMU 1 F 0 + ǫ 1 + ǫ 2 log Z inst ( � a, ǫ 1 , ǫ 2 ; q ) = H 1 Introduction ǫ 1 ǫ 2 ǫ 1 ǫ 2 2 Calculation in + F 1 + ( ǫ 1 + ǫ 2 ) 2 the CFT G 1 + . . . SWDN theory on toric Kähler ǫ 1 ǫ 2 manifolds Supergravity background Localization ◮ The extra terms show up in calculations on curved Partition function and large N limit manifolds. An example ◮ The expansion has been worked out for the 5d version of Conclusion the Nekrasov partition function [Göttsche, Nakajima, Yoshioka (2006)].

Nekrasov’s equivariant extension III On a compact toric Kähler manifold M 4 [Nekrasov (2006)] Black Hole Entropy from 5D Twisted � � � �  Indices k a ; q i � a + ǫ a Z M 4 = da Z inst � Itamar Yaakov University of i ∈ vertices { � k a ∈ Z N } Tokyo - Kavli � � IPMU � �  ˆ � − − − − − → da exp F 0 � a + k a c 1 ( L a ) Introduction ǫ 1 ,ǫ 2 → 0 M 4 a Calculation in { � k a ∈ Z N } the CFT � � SWDN theory � on toric Kähler manifolds � + c 1 ( M 4 ) H 1 � a + k a c 1 ( L a ) Supergravity background 2 a Localization Partition + χ ( M 4 ) F 1 ( � a ) + σ ( M 4 ) F 1 ( � function and a ) large N limit An example Conclusion ◮ I have omitted the insertion of observables. The contour for a and the exact sum are unknown. ◮ � k a is an integer flux vector and ǫ i is the action on the fixed point i .