This (blackboard) talk is based on 2104.13932 and on work in progress with Francesco Benini, Ohad Mamroud and Paolo Milan. I will begin by briefly reviewing the superconformal index of the d=4 N=4 SU(N) supersymmetric Yang-Mills theory, how it is related (in the large N limit) to counting black hole microstates, and how it can be computed. I will then review a specific way to compute the index called the Bethe ansatz expansion, and describe the known solutions to the Bethe ansatz equations, and what they contribute to the index in the large N limit, including both perturbative and non-perturbative terms in 1/N. The index is related to the partition function of N=4 SYM on S^3xS^1, and in the large N limit this should be related by the AdS/CFT correspondence to a sum over Euclidean gravity solutions with appropriate asymptotic behavior. I will show that each known Bethe ansatz contribution arises from a specific supersymmetric (complex) black hole solution, which reproduces both its perturbative and its non-perturbative behavior (the latter comes from wrapped Euclidean D3-branes). A priori there are many more gravitational solutions than Bethe ansatz contributions, but we show that by considering the non-perturbative effects, the extra solutions are ruled out, leading to a precise match between the solutions on both sides.---- Part of the London Integrability Journal Club. Please register at integrability-london.weebly.com if you are a new participant. The link will be emailed on Tuesday.

I will start by discussing the 4d N=4 supersymmetric Yang-Mills theory on two spaces with a boundary - a half-line (times three infinite dimensions) and four dimensional anti-de Sitter space. I will review in detail the possible choices of boundary conditions for this theory, and then exhibit the solutions of string theory (supergravity) that are holographically dual to the N=4 SYM theory on these spaces. The same methods used to obtain these duals can also be used to find duals for 3d N=4 and 4d N=2 superconformal field theories that appear in brane configurations, and I will discuss these solutions as well.