The Witten index counts the difference in the number of bosonic and fermionic states of a quantum mechanical system. The Schur index, which can be defined for theories with at least N = 2 supersymmetry in four dimensions is a particular refinement of the index, dependent on one parameter q serving as the fugacity for a particular set of charges which commute with the hamiltonian and some supersymmetry generators. This index has a known expression for all Lagrangian and some non-Lagrangian theories as a finite dimensional integral or a complicated infinite sum. In the case of N = 4 SYM with gauge group U(N ) we rewrite this as the partition function of a gas of N non interacting and translationally invariant fermions on a circle. This allows us to perform the integrals and write down explicit expressions for fixed N as well as the exact all orders large N expansion.

The holographic renormalization group flows associated with marginally relevant operators are analysed. The associated perturbative and non-perturbative β-functions are calculated and the consistent scalar potentials are identified. The presence of a Zamolodchikov metric in the multiscalar case is shown to affect β-functions starting at two loops.

Abstract:We study the Schur index of 4-dimensional N = 2 circular quiver theories. We show that the index can be expressed as a weighted sum over partition functions describing systems of free Fermions living on a circle. For circular SU(N ) quivers of arbitrary length we evaluate the large N limit of the index, up to exponentially suppressed corrections. For the single node theory (N = 4 SYM) and the two node quiver we are able to go beyond the large N limit, and obtain the complete, all orders large N expansion of the index, as well as explicit finite N results in terms of elliptic functions.

We present a simple construction of solutions to the supersymmetric higher spin theory based on solutions to bosonic theories. We illustrate this for the case of the Didenko-Vasiliev solution in arXiv:0906.3898, for which we have found a striking simplification where the higher-spin connection takes the vacuum value. Studying these solutions further, we check under which conditions they preserve some supersymmetry in the bulk, and when they are compatible with the boundary conditions conjectured to be dual to certain 3d SUSY Chern-Simons-matter theories. We perform the analysis for a variety of theories with N = 2, N = 3, N = 4 and N = 6 and find a rich spectrum of 1/4, 1/3 and 1/2-BPS solutions.