The $\mathcal{N}$ = 1 superconformal index for class $\mathcal{S}$ fixed points

Abstract

We investigate the superconformal index of four-dimensional superconformal field theories that arise on coincident M5 branes wrapping a holomorphic curve in a local Calabi-Yau three-fold. The structure of the index is very similar to that which appears in the special case preserving $\mathcal{N}$ = 2 supersymmetry. We first compute the index for the fixed points that admit a known four-dimensional ultraviolet description and prove infrared equivalence at the level of the index for all such constructions. These results suggest a formulation of the index as a two-dimensional topological quantum field theory that generalizes the one that computes the $\mathcal{N}$ = 2 index. The TQFT structure leads to an expression for the index of a much larger family of $\mathcal{N}$ = 1 class S fixed points in terms of the index of the $\mathcal{N}$ = 2 theories. Calculations of simple quantities with the index suggests a connection between these families of fixed points and the mathematics of SU(2) Yang-Mills theory on the wrapped curve.