Partition functions for supersymmetric gauge theories on spheres

TL;DR

This paper reviews localization techniques for supersymmetric gauge theories on spheres, analyzing partition functions and expressing them through Bell polynomials and spectral functions, advancing computational methods in the field.

Contribution

It introduces a novel approach to express supersymmetric partition functions using Bell polynomials and spectral functions, extending localization techniques to various sphere geometries.

Findings

Partition functions can be written as series of Bell polynomials.

Spectral functions provide a new representation of partition functions.

Method applies to both even- and odd-dimensional spheres.

Abstract

In this paper we briefly review the main idea of the localization technique and its extension suitable in supersymmetric gauge field theory. We analyze the partition function of the vector multiplets with supercharges and its blocks on the even- and odd-dimensional spheres and squashed spheres. We exploit so-called Fa\`a di Bruno's formula and show that multipartite partition functions can be written in the form of expansion series of the Bell polynomials. Applying the restricted specialization argument we show that $q$-infinite-product representation of partition functions admits presentation in terms of the Patterson-Selberg (or the Ruelle-type) spectral functions.