Superconformal indices at large N and the entropy of AdS₅ × SE₅ black holes

Abstract

The large N limit of the four-dimensional superconformal index was computed and successfully compared to the entropy of a class of AdS5 black holes only in the particular case of equal angular momenta. Using the Bethe ansatz formulation, we compute the index at large N with arbitrary chemical potentials for all charges and angular momenta, for general four-dimensional conformal theories with a holographic dual. We conjecture and bring some evidence that a particular universal contribution to the sum over Bethe vacua dominates the index at large N. For SYM, this contribution correctly leads to the entropy of BPS Kerr–Newman black holes in AdS5 × S 5 for arbitrary values of the conserved charges, thus completing the microscopic derivation of their microstates. We also consider theories dual to AdS5 × SE5, where SE5 is a Sasaki–Einstein manifold. We first check our results against the so-called universal black hole. We then explicitly construct the near-horizon geometry of BPS Kerr–Newman black holes in AdS5 × T 1,1, charged under the baryonic symmetry of the conifold theory and with equal angular momenta. We compute the entropy of these black holes using the attractor mechanism and find complete agreement with the field theory predictions.

Sub-leading structures in superconformal indices: subdominant saddles and logarithmic contributions

“…Under shifting by τ the argument of Z(u; ∆, τ ) we have the following property: 17) where the function Q k (u; ∆, τ ) measures the lack of periodicity of Z(u; ∆, τ ) in the variable u k under shifting by τ and is defined in [23] as:…”

“…This solution was also used by Benini and Milan in their discussion of the SCI [3] and it was further extended in [15,16] where it was shown that it furnishes a solution for a generic class of N = 1 supersymmetric field theories. More recently, the BA approach to the index based on the basic solution was extended to include two different angular velocities [17], thus covering the most general type of asymptotically AdS 5 × S 5 black holes. Our results for the BA approach goes beyond the leading N 2 order in the large-N limit and takes the form I(τ ; ∆) = I(τ ; ∆) Basic BA + (contribution from other BA solutions)…”

The SCI of $$ \mathcal{N} $$ = 4 USp(2Nc) and SO(Nc) SYM as a matrix integral

“…with the constraint ∆ 1 + ∆ 2 − τ − 1 − η 2 = 0 (which is derived from (5.4)). The overall constant κ is fixed as κ = 1 8 (see the discussion in [35]). The entropy is computed in terms of the charges Q 1,2 and angular momentum J of the dual black hole.…”

The Bethe-Ansatz approach to the $$ \mathcal{N} $$ = 4 superconformal index at finite rank

“…The BA approach has been introduced for a generic 4d N = 1 supersymmetric gauge theories in [28] based on insightful observations made in [29,30]. It was then applied to the N = 4 SU(N ) SYM theory with p = q in [3] and, more recently, to the same theory with p = h a and q = h b where a, b ∈ N and gcd(a, b) = 1 [31]. Some discussion of the BA approach to the SCI of a large class of 4d N = 1 supersymmetric quiver gauge theories was presented to leading order in [12,13] and a systematic sub-leading study was presented recently in [15].…”