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

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.

arXiv comment63 pages, to appear on JHEP

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