Notes on massless scalar field partition functions, modular invariance and Eisenstein series

TL;DR

This paper computes the partition function of a massless scalar field on specific manifolds, revealing its modular invariance properties and expressing it via Eisenstein series, with implications for temperature duality and geometric dependence.

Contribution

It provides a novel explicit expression for the scalar field partition function using Eisenstein series and explores its modular properties and dualities in various geometric settings.

Findings

Partition function expressed as Eisenstein series with SL(2,Z) symmetry.

High/low temperature duality extended to non-zero chemical potential.

Modular covariance linked to full SL(2,Z) invariance and geometric dependence.

Abstract

The partition function of a massless scalar field on a Euclidean spacetime manifold $\mathbb{R}^{d-1}\times\mathbb{T}^2$ and with momentum operator in the compact spatial dimension coupled through a purely imaginary chemical potential is computed. It is modular covariant and admits a simple expression in terms of a real analytic SL$(2,\mathbb{Z})$ Eisenstein series with $s=(d+1)/2$. Different techniques for computing the partition function illustrate complementary aspects of the Eisenstein series: the functional approach gives its series representation, the operator approach yields its Fourier series, while the proper time/heat kernel/world-line approach shows that it is the Mellin transform of a Riemann theta function. High/low temperature duality is generalized to the case of a non-vanishing chemical potential. By clarifying the dependence of the partition function on the geometry of the torus, we discuss how modular covariance is a consequence of full SL$(2,\mathbb{Z})$ invariance. When the spacetime manifold is $\mathbb{R}^p\times\mathbb{T}^{q+1}$, the partition function is given in terms of a SL$(q+1,\mathbb{Z})$ Eisenstein series again with $s=(d+1)/2$. In this case, we obtain the high/low temperature duality through a suitably adapted dual parametrization of the lattice defining the torus. On $\mathbb{T}^{d+1}$, the computation is more subtle. An additional divergence leads to an harmonic anomaly.