The $\mathcal{N}=2$ Schur index from free fermions

TL;DR

This paper expresses the $ ext{N}=2$ Schur index of circular quiver theories as a sum over free fermion systems, providing large and finite N expansions and explicit formulas.

Contribution

It introduces a novel representation of the Schur index as a sum over free fermions and computes detailed large N and finite N expansions for specific theories.

Findings

Index expressed as a sum over free fermion partition functions

Large N limit evaluated for arbitrary quiver length

Complete finite N results obtained for certain theories

Abstract

We study the Schur index of 4-dimensional $\mathcal{N}=2$ 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 ($\mathcal{N}=4$ 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.