Elliptic functions and temperature inversion symmetry on spheres

TL;DR

This paper explores temperature inversion symmetry and modular invariance in finite temperature quantum field theories on spheres, revealing relationships between energies across dimensions and connections to elliptic functions.

Contribution

It demonstrates that energies in conformally invariant theories on spheres can be expressed as power series across dimensions and linked to elliptic function theory at special temperatures.

Findings

Energy expressions as power series in different dimensions

Finite terms at specific elliptic function temperatures

Examples illustrating the theoretical relationships

Abstract

Finite temperature boson and fermion field theories on ultrastatic space-times with a d-sphere spatial section are discussed with one eye on the questions of temperature inversion symmetry and modular invariance. For conformally invariant theories it is shown that the total energy at any temperature for any dimension, d, is given as a power series in the d=3 and d=5 energies, for scalars, and the d=1 and d=3 energies for spinors. Further, these energies can be given in finite terms at specific temperatures associated with singular moduli of elliptic function theory. Some examples are listed and numbers given.