The topologically twisted index on S 2 × T n and black hole entropy James Liu University of Michigan 13 April 2018 JTL, L. A. Pando Zayas, V. Rathee and W. Zhao arXiv:1707.04197, arXiv:1711.01076 Junho Hong and JTL, arXiv:1804.04592 JTL, L. A. Pando Zayas and S. Zhou, in progress GLSC 2018
Supersymmetric partition functions from localization ◮ Localization is a very powerful tool for computing supersymmetric partition functions and observables – S n partition functions, Wilson loop observables – S n × S 1 partition functions and supersymmetric indices ◮ Generically, the partition function takes the form � Z susy = d Φ Z classical Z 1 − loop Z non − perturbative JTL
Supersymmetric partition functions from localization ◮ Localization is a very powerful tool for computing supersymmetric partition functions and observables – S n partition functions, Wilson loop observables – S n × S 1 partition functions and supersymmetric indices ◮ Generically, the partition function takes the form � Z susy = d Φ Z classical Z 1 − loop Z non − perturbative ◮ We explore the connection between the topologically twisted index on S 2 × S 1 and AdS 4 black hole entropy and S 2 × T 2 and AdS 5 black string microstates [F. Benini, K. Hristov and A. Zaffaroni, arXiv:1511.04085] JTL
Magnetically charged AdS solutions ◮ AdS/CFT allows us to compare observables on both sides of the duality partition function on S n Global AdS ↔ partition function on S n − 1 × S 1 Black holes in AdS ↔ partition function on S n − 2 × T 2 ↔ Black strings in AdS JTL
Magnetically charged AdS solutions ◮ AdS/CFT allows us to compare observables on both sides of the duality partition function on S n Global AdS ↔ partition function on S n − 1 × S 1 Black holes in AdS ↔ partition function on S n − 2 × T 2 ↔ Black strings in AdS ◮ Consider a magnetic BPS black hole with spherical horizon boundary near horizon AdS 2 × S 2 AdS AdS 4 − → ↓ ↓ S 2 × S 1 S 1 CFT − → JTL
The topologically twisted index on S 2 × S 1 ◮ The topologically twisted index was introduced by F. Benini and A. Zaffaroni, arXiv:1504.03698 ◮ Take a magnetic BPS black hole in AdS 4 What do we do on the field theory side? – Background R symmetry flux on S 2 – This cancels the curvature of S 2 ⇒ partial topological twist – The index may be computed using localization JTL
The topologically twisted index on S 2 × S 1 ◮ The topologically twisted index was introduced by F. Benini and A. Zaffaroni, arXiv:1504.03698 ◮ Take a magnetic BPS black hole in AdS 4 What do we do on the field theory side? – Background R symmetry flux on S 2 – This cancels the curvature of S 2 ⇒ partial topological twist – The index may be computed using localization ◮ This topologically twisted index is conjectured to count the black hole microstates [Benini, Hristov, Zaffaroni] – Many general features are now known – Extended to dyonic black holes, black holes with hyperbolic horizons, magnetic black strings, . . . JTL
Building blocks of the S 2 × S 1 index ◮ Consider three-dimensional N = 2 Chern-Simons-matter theories on S 2 × S 1 ◮ The index receives contributions from: • Vector multiplets: dx i � x k m i � (1 − x α ) Z vector = i 2 π ix i α ∈ G i • Chiral multiplets: � x µ/ 2 y µ f / 2 � µ ( m )+ µ f ( n ) − q +1 � Z chiral = 1 − x µ y µ f µ ∈ R ◮ These elements can be combined to construct the index for various models JTL
Counting black hole microstates ◮ Given a magnetically charged AdS black hole, we can construct the topologically twisted index in the field theory dual and evaluate it in the large- N limit JTL
Counting black hole microstates ◮ Given a magnetically charged AdS black hole, we can construct the topologically twisted index in the field theory dual and evaluate it in the large- N limit ◮ We consider the following examples 1. Magnetic black holes in M-theory on AdS 4 × S 7 / Z k Dual to ABJM theory 2. Magnetic black holes in massive IIA on AdS 4 × S 6 Dual to N = 2 Chern-Simons-matter theory 3. Magnetic black strings in IIB on AdS 5 × S 5 Dual to N = 4 super-Yang-Mills JTL
M-theory on AdS 4 × S 7 / Z k ◮ The field theory dual is ABJM theory Chern-Simons-matter with U ( N ) k × U ( N ) − k gauge groups and bi-fundamental matter A i , B j ◮ The topologically twisted index is given by 1 � � dx i � 1 − x i � � � x k m i Z ( y a , n a ) = i ( N !) 2 2 π ix i x j m , ˜ i i � = j m � � d ˜ x i � 1 − ˜ x i � x − k ˜ � m i ˜ i 2 π i ˜ ˜ x i x j i i � = j � xi m i − ˜ m j − n a +1 � � ˜ m j − m i − n b +1 ˜ � 1 / 2 � 1 / 2 xi ya yb � � � xj ˜ xi 1 − xi ˜ xj ya xj ˜ 1 − ya xi a i , j i , j b ◮ The index can be evaluated from the Jeffrey-Kirwan residue JTL
Eigenvalue distribution ◮ Single solution to the BAE up to permutations Solution for ∆ a = { 0 . 3 , 0 . 4 , 0 . 5 , 2 π − 1 . 2 } and N = 50 JTL
Eigenvalue distribution ◮ Single solution to the BAE up to permutations Solution for ∆ a = { 0 . 3 , 0 . 4 , 0 . 5 , 2 π − 1 . 2 } and N = 50 ◮ Large- N behavior ℜ log Z ∼ f 0 N 3 / 2 + f 1 N 1 / 2 − 1 2 log N + · · · ◮ Subleading terms are difficult to extract analytically Tails in the distribution lead to complications JTL
Massive IIA theory on AdS 4 × S 6 ◮ The dual field theory is an N = 2 Chern-Simons-matter theory with SU ( N ) k gauge group and adjoint matter X , Y , Z [Guarino, Jafferis and Varela, arXiv:1504.08009] ◮ Here the topologically twisted index is given by Z ( y a , n a ) = ( − 1) N � � dx i � 1 − x i � � � x k m i i N ! 2 π ix i x j i i � = j m � xi m i − m j + n a +1 � 1 / 2 ya � � xj 1 − xi ya xj i , j a ◮ Once again, the index is evaluated from the Jeffrey-Kirwan residue JTL
Eigenvalue distribution ◮ Single solution to the BAE up to permutations Solution for ∆ a = { 0 . 2 , 0 . 7 , 2 π − 0 . 9 } and N = 50 JTL
Eigenvalue distribution ◮ Single solution to the BAE up to permutations Solution for ∆ a = { 0 . 2 , 0 . 7 , 2 π − 0 . 9 } and N = 50 ◮ Large- N behavior ℜ log Z ∼ f 0 N 5 / 3 + f 1 N 2 / 3 + f 2 N 1 / 3 + f 3 log N + · · · ◮ Can we understand the subleading behavior? No tails, but still have to deal with endpoints JTL
Building blocks of the S 2 × T 2 index ◮ We now turn to the topologically twisted index on S 2 × T 2 where T 2 is parametrized by q = e 2 π i τ – Four-dimensional Yang-Mills theory on S 2 × T 2 JTL
Building blocks of the S 2 × T 2 index ◮ We now turn to the topologically twisted index on S 2 × T 2 where T 2 is parametrized by q = e 2 π i τ – Four-dimensional Yang-Mills theory on S 2 × T 2 ◮ Work in an N = 2 language • Vector multiplets: � θ 1 ( x α , q ) dx i � Z vector = ( − 1) 2 ρ ( m ) � η ( q ) 2 � 2 π ix i i η ( q ) i ∈ G α ∈ G • Chiral multiplets: � µ ( m )+ µ f ( n )+1 � i η ( q ) � Z chiral = θ 1 ( x µ y µ f , q ) µ ∈ R JTL
IIB on AdS 5 × S 5 ◮ The dual theory is N = 4 SYM with SU ( N ) gauge group One vector and three chiral multiplets in the N = 1 language ◮ The topologically twisted index is [Hosseini, Nedelin and Zaffaroni, arXiv:1611.09374] � θ 1 ( x i x j , q ) � Z ( y a , n a ) = 1 � � dx i � η ( q ) 2 � N ! 2 π ix i i η ( q ) i i � = j m � m i − m j − n a +1 � i η ( q ) � � θ 1 ( x i x j y a , q ) i , j a JTL
IIB on AdS 5 × S 5 ◮ The dual theory is N = 4 SYM with SU ( N ) gauge group One vector and three chiral multiplets in the N = 1 language ◮ The topologically twisted index is [Hosseini, Nedelin and Zaffaroni, arXiv:1611.09374] � θ 1 ( x i x j , q ) � Z ( y a , n a ) = 1 � � dx i � η ( q ) 2 � N ! 2 π ix i i η ( q ) i i � = j m � m i − m j − n a +1 � i η ( q ) � � θ 1 ( x i x j y a , q ) i , j a ◮ After evaluating the Jeffrey-Kirwan residue � 1 − n a θ 1 ( x i � x j , q ) 1 i η ( q ) � � � Z = A θ 1 ( x j det B i η ( q ) x j y a , q ) a I ∈ BAEs i � = j JTL
Solving the BAE ◮ The BAEs that we need to solve are θ 1 ( e i ( u j − u i +∆ a ) , q ) 1 = e iB i ≡ e iv � � θ 1 ( e i ( u i − u j +∆ a ) , q ) a j ◮ How do we obtain the u i ’s? JTL
Solving the BAE ◮ The BAEs that we need to solve are θ 1 ( e i ( u j − u i +∆ a ) , q ) 1 = e iB i ≡ e iv � � θ 1 ( e i ( u i − u j +∆ a ) , q ) a j ◮ How do we obtain the u i ’s? u + 2 π τ Hosseini, Nedelin, Zaffaroni obtained u j = ¯ N j in the high-temperature limit β → 0 + where τ = i β/ 2 π JTL
Multiple solutions to the BAE ◮ Evenly distributed eigenvalues ⇒ good solution for any q ! JTL
Multiple solutions to the BAE ◮ Evenly distributed eigenvalues ⇒ good solution for any q ! JTL
Multiple solutions to the BAE ◮ Evenly distributed eigenvalues ⇒ good solution for any q ! JTL
Multiple solutions to the BAE ◮ Evenly distributed eigenvalues ⇒ good solution for any q ! JTL
Multiple solutions to the BAE ◮ Evenly distributed eigenvalues ⇒ good solution for any q ! JTL
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