# Schur correlation functions on $S^3\times S^1$

## TL;DR

This paper derives the relation between the Schur limit of the superconformal index and the supercharacter of the associated chiral algebra for 4D $ ext{N}=2$ theories using localization on $S^3 imes S^1$, and extends the computation to certain correlation functions.

## Contribution

It provides a direct localization derivation of the superconformal index and extends localization techniques to include correlation functions of Schur operators, including fermionic insertions.

## Key findings

- Derived the superconformal index as a supercharacter via localization.
- Extended localization to compute correlators of Schur operators.
- Demonstrated inclusion of fermionic operator insertions in localization.

## Abstract

The Schur limit of the superconformal index of four-dimensional $\mathcal N=2$ superconformal field theories has been shown to equal the supercharacter of the vacuum module of their associated chiral algebra. Applying localization techniques to the theory suitably put on $S^3\times S^1$, we obtain a direct derivation of this fact. We also show that the localization computation can be extended to calculate correlation functions of a subset of local operators, namely of the so-called Schur operators. Such correlators correspond to insertions of chiral algebra fields in the trace-formula computing the supercharacter. As a by-product of our analysis, we show that the standard lore in the localization literature stating that only off-shell supersymmetrically closed observables are amenable to localization, is incomplete, and we demonstrate how insertions of fermionic operators can be incorporated in the computation.