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

We study nearly extreme black holes with nearly AdS 2 horizon geometry in various settings inspired by string theory. Our focus is on the scales of the nAdS 2 region and their relation to microscopic theory. These scales are determined by a generalization of the attractor mechanism for extremal black holes and realized geometrically as the normal derivatives along the extremal attractor flow. In some cases the scales are equivalently determined by the charge dependence of the extremal attractor by itself. Our examples include near extreme black holes in D ≥ 4 dimensions, AdS boundary conditions, rotation, and 5D black holes on the non-BPS branch. A 4D N = 2 Ungauged Supergravity 34 B Solution of the Attractor Equations for the ST (N ) model. 35 B.1 Time-like Solutions: Q a X a = 0 36 B.2 Space-like Solutions: Q a X a = 0 36 C Deriving Black Hole Solutions for "Space-like" Charge Vectors 37-1 -1 In [20] a length scale was introduced as L there = 1 2 πL here . The normalization was chosen so L there coincides with the "long string scale" in the UV theory, at least in the simplest cases. The normalization L here is advantageous presently because it generalizes to any dimension.

We investigate the Bethe-Ansatz approach to the superconformal index of $$ \mathcal{N} $$ N = 4 supersymmetric Yang-Mills with SU(N) gauge group in the context of finite rank, N. We explicitly explore the role of the various types of solutions to the Bethe-Ansatz Equations in recovering the exact index for N = 2, 3. We classify the Bethe-Ansatz Equations solutions as standard (corresponding to a freely acting orbifold T2/ℤm× ℤn) and non-standard. For N = 2, we find that the index is fully recovered by standard solutions and displays an interesting pattern of cancellations. However, for N ≥ 3, the standard solutions alone do not suffice to reconstruct the index. We present quantitative arguments in various regimes of fugacities that highlight the challenging role played by the continuous families of non-standard solutions.

We study the large N limit of some supersymmetric partition functions of the U(N)k × U(N)−k ABJM theory computed by supersymmetric localization. We conjecture an explicit expression, valid to all orders in the large N limit, for the partition function on the U(1) × U(1) invariant squashed sphere in the presence of real masses in terms of an Airy function. Several non-trivial tests of this conjecture are presented. In addition, we derive an explicit compact expression for the topologically twisted index of the ABJM theory valid at fixed k to all orders in the 1/N expansion. We use these results to derive the topologically twisted index and the sphere partition function in the ’t Hooft limit which correspond to genus g type IIA string theory free energies to all orders in the α′ expansion. We discuss the implications of our results for holography and the physics of AdS4 black holes.