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 review and discuss in detail type II string theory on $AdS_3\times S^3\times T^4$ and its $1+1$ dimensional superconformal field theory dual, emphasizing the string theoretic aspects of this duality. For one unit of NS-NS 5-brane flux ($Q_5=1$), this string theory has been suggested to be dual to a grand-canonical ensemble of $T^{4N}/S_N$ free symmetric orbifold CFTs, and one goal will be to understand how this is consistent and how it fits with the description of other values of $Q_5$. I will discuss how the strong coupling limit of the NS-NS string theory arises (even at large $N$) in the free orbifold description, and why this limit does not have a weakly coupled R-R description. The dual CFT includes (for all values of $Q_5$) an extra $T^4$ factor that is decoupled from perturbative string theory, and I will discuss how this appears and how it is coupled to the CFT. Based on 2406.14605 with Erez Urbach.

Many gauge theories, such as 3+1d and 2+1d SU(N) gauge theories with N_f massless Dirac fermions in the fundamental representation, exhibit "conformal windows" - a range of values of N_f for which the theory flows to a non-trivial conformal field theory at low energy. It is interesting to ask how these windows end and transition into phases with confinement and with spontaneous breaking of the flavor symmetry. I will begin by reviewing what is known about 3+1 dimensional QCD, focusing on the large N and N_f limit and on the conformal window there, and I will review the standard "merger scenario" for the end of its conformal window. I will then move on to QCD in 2+1 dimensions, and discuss its phase diagram as a function of the mass and of the discrete parameters N, N_f and the Chern-Simons level k, focusing again on the limit of large N, N_f, k. I will argue that this theory also has a conformal window (corresponding to a second order phase transition when changing the mass) but that it does not always end at a merger, and I will suggest two other possibilities for the end of general conformal windows. The different scenarios can be distinguished by the behavior of the theory as the boundary of the conformal window is approached from either side. Based on work in progress with Thomas Dumitrescu, Zohar Komargodski and Jiangyuan Qian.

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.