Localization of N=4 Superconformal Field Theory on S^1 x S^3 and Index

TL;DR

This paper interprets the ${ m N}=4$ superconformal index geometrically as a partition function on a deformed background and computes it using localization, revealing its relation to flat connections and matrix integrals.

Contribution

It provides a geometric interpretation of the ${ m N}=4$ superconformal index and applies localization to compute it as a matrix integral.

Findings

Superconformal index expressed as a partition function on a Scherk-Schwarz deformed background.

Localization reduces the computation to flat connections and a matrix integral.

Critical points correspond to zero modes of the gauge field.

Abstract

We provide the geometrical meaning of the ${\cal N}=4$ superconformal index. With this interpretation, the ${\cal N}=4$ superconformal index can be realized as the partition function on a Scherk-Schwarz deformed background. We apply the localization method in TQFT to compute the deformed partition function since the deformed action can be written as a $\delta_\epsilon$-exact form. The critical points of the deformed action turn out to be the space of flat connections which are, in fact, zero modes of the gauge field. The one-loop evaluation over the space of flat connections reduces to the matrix integral by which the ${\cal N}=4$ superconformal index is expressed.