We study the Cardy-like asymptotics of the 4d N = 4 index and demonstrate the existence of partially deconfined phases where the asymptotic growth of the index is not as rapid as in the fully deconfined case. We then take the large-N limit after the Cardy-like limit and make a conjecture for the leading asymptotics of the index. While the Cardy-like behavior is derived using the integral representation of the index, we demonstrate how the same results can be obtained using the Bethe ansatz type approach as well. In doing so, we discover new non-standard solutions to the elliptic Bethe ansatz equations including continuous families of solutions for SU(N) theory with N ≥ 3. We argue that the existence of both standard and continuous non-standard solutions has a natural interpretation in terms of vacua of N = 1 * theory on R 3 × S 1 .

We examine the topologically twisted index of N = 4 super-Yang-Mills with gauge group SU(N) on T 2 ×S 2 , and demonstrate that it receives contributions from multiple sectors corresponding to the freely acting orbifolds T 2 /Z m × Z n where N = mn. After summing over these sectors, the index can be expressed as the elliptic genus of a twodimensional N = (0, 2) theory resulting from Kaluza-Klein reduction on S 2. This provides an alternate path to the 'high-temperature' limit of the index, and confirms the connection to the right-moving central charge of the N = (0, 2) theory.

We find a family of complete non-linear Kaluza-Klein reduction ansätze from type IIB supergravity to Romans' 6D F (4) gauged supergravity in the bosonic sector. The reduction is over a sphere S 2 and a Riemann surface Σ, and depends on a pair of arbitrary locally holomorphic functions A ± on Σ. This family of reductions is inspired by the recent construction of 1/2 BPS supersymmetric warped AdS 6 solutions of IIB supergravity that depend on these same functions A ± .

We systematically study various sub-leading structures in the superconformal index of $$ \mathcal{N} $$ N = 4 supersymmetric Yang-Mills theory with SU(N) gauge group. We concentrate in the superconformal index description as a matrix model of elliptic gamma functions and in the Bethe-Ansatz presentation. Our saddle-point approximation goes beyond the Cardy-like limit and we uncover various saddles governed by a matrix model corresponding to SU(N) Chern-Simons theory. The dominant saddle, however, leads to perfect agreement with the Bethe-Ansatz approach. We also determine the logarithmic correction to the superconformal index to be log N, finding precise agreement between the saddle-point and Bethe-Ansatz approaches in their respective approximations. We generalize the two approaches to cover a large class of 4d $$ \mathcal{N} $$ N = 1 superconformal theories. We find that also in this case both approximations agree all the way down to a universal contribution of the form log N. The universality of this last result constitutes a robust signature of this ultraviolet description of asymptotically AdS5 black holes and could be tested by low-energy IIB supergravity.

We construct necessary and sufficient geometric conditions for a class of AdS 2 solutions of M-theory with, at least, minimal supersymmetry to exist. We generalize previous results in the literature for N = (2, 0) supersymmetry in AdS 2 to N = (1, 0). When the solution can be locally described as AdS 2 × Σ g × SE 7 with Σ g a Riemann surface of genus g and SE 7 a seven-dimensional Sasaki-Einstein manifold, we clarify and unify various solutions present in the literature. In the case of SE 7 = Q 1,1,1 we find a new solution with baryonic and mesonic charges turned on simultaneously.A There are no AdS 2 solutions with Spin(7)-structure 32 B No AdS 3 solutions within SU(4)-structure AdS 2 class 33 C Killing spinor approach 35