Elliptic aspects of statistical mechanics on spheres

TL;DR

This paper extends previous work on elliptic properties of thermodynamic quantities on odd spheres to orbifold factors, introducing modular covariant derivatives and expressing key quantities in terms of elliptic functions.

Contribution

It introduces a modular covariant derivative framework and expresses the specific heat and free energy on spheres in elliptic terms, broadening the understanding of elliptic aspects in statistical mechanics.

Findings

Specific heat on odd spheres can be expressed using three functions.

Free energy on the circle can be written elliptically.

Behavior under modular transformations is characterized by a covariant derivative.

Abstract

Our earlier results on the temperature inversion properties and the ellipticisation of the finite temperature internal energy on odd spheres are extended to orbifold factors of odd spheres and then to other thermodynamic quantities, in particular to the specific heat. The behaviour under modular transformations is facilitated by the introduction of a modular covariant derivative and it is shown that the specific heat on any odd sphere can be expressed in terms of just three functions. It is also shown that the free energy on the circle can be written elliptically.