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

The partition function of a massless scalar field on a Euclidean spacetime manifold Md-1 x 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, 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, Z) invariance. When the spacetime manifold is R-p x Tq+1, the partition function is given in terms of a SL(q + 1, 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 Td+1, the computation is more subtle. An additional divergence leads to an harmonic anomaly.

Automated summary

The study calculates the partition function of a massless scalar field on a Euclidean spacetime manifold and discusses the generalization of high/low temperature duality, as well as the modular covariance of the partition function under different geometric conditions. The results provided by the study offer valuable insights into the properties of quantum field theory in specific backgrounds.