Modular properties of massive scalar partition functions

TL;DR

This paper calculates exact thermal partition functions for massive scalar fields on certain flat spacetimes, revealing an SL(q+1,Z) symmetry and connections to Maass-Jacobi forms, with implications for Bose-Einstein condensation in compactified dimensions.

Contribution

It provides the first exact computation of scalar partition functions on these backgrounds, uncovering modular symmetries and novel mathematical structures.

Findings

Partition functions exhibit SL(q+1,Z) symmetry.

For q=1, results relate to massive Maass-Jacobi forms.

Bose-Einstein condensation behavior is altered by small dimensions.

Abstract

We compute the exact thermal partition functions of a massive scalar field on flat spacetime backgrounds of the form $\mathbb R^{d-q}\times \mathbb T^{q+1}$ and show that they possess an ${\rm SL}(q+1,\mathbb Z)$ symmetry. Non-trivial relations between equivalent expressions for the result are obtained by doing the computation using functional, canonical and worldline methods. For $q=1$, the results exhibit modular symmetry and may be expressed in terms of massive Maass-Jacobi forms. In the complex case with chemical potential for ${\rm U}(1)$ charge turned on, the usual discussion of relativistic Bose-Einstein condensation is modified by the presence of the small dimensions.