Modularity in $d > 2$ free conformal field theory

TL;DR

This paper derives new closed-form expressions for the partition functions of free conformally-coupled scalars on specific manifolds, revealing modular properties linked to elliptic Gamma functions and offering geometric interpretations within superconformal field theory.

Contribution

It introduces novel closed-form formulas for free scalar partition functions using elliptic Gamma functions and explores their modular and geometric properties.

Findings

Partition functions expressed via elliptic Gamma functions.

Identification of modular properties of these functions.

Geometric interpretation of modularity in superconformal field theory.

Abstract

We derive new closed form expressions for the partition functions of free conformally-coupled scalars on $S^{2D-1}\times S^1$ which resum the exact high-temperature expansion. The derivation relies on an identification of the partition functions, analytically continued in chemical potentials and temperature, with multiple elliptic Gamma functions. These functions satisfy interesting modular properties, which we use to arrive at our expressions. We describe a geometric interpretation of the modular properties of multiple elliptic Gamma functions in the context of superconformal field theory. Based on this, we suggest a geometric interpretation of the modular property in the context of the free scalar CFT in even dimensions and comment on extensions to odd dimensions and free fermions.