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

Understand

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.