Modular properties of massive scalar partition functions

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 $\mathrm{SL}(q+1,\mathbb{Z})$ 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 $\mathrm{U}(1)$ charge turned on, the usual discussion of relativistic Bose-Einstein condensation is modified by the presence of the small dimensions.

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