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 ℝd−1 × 𝕋2 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 ℝp × 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.

Type IIB at eight derivatives: insights from Superstrings, Superfields and Superparticles

“…In Type IIB, SL(2, Z) invariance dictates that the coefficients of higher-derivative terms are written in terms of modular forms. The appearance of the modular function f 0 , the non-holomorphic Eisenstein series of weight 3/2, was first observed in [15,18], while other non-holomorphic modular forms f w first appeared in [16,19], see also [20] for a more recent discussion. For R 4 , leading order D-instanton calculations [21,22] confirmed this result.…”

Phase transition of photons and gravitons in a Casimir box

“…From a theoretical viewpoint, the exact results derived are closely related to those that appear when generalizing modular invariance from two [22,23] to higher spacetime dimensions [24][25][26][27], combined with the technique of going back to real rather than purely imaginary chemical potential [28][29][30].…”

Gravitons in a Casimir box

“…With α ‰ 0, this has been discussed instance in [2]. The detailed derivation by a variety of different methods of the expressions provided below can be found in [17].…”

“…In order to do so, we follow recent work on the finite temperature electromagnetic Casimir effect and modular invariance in this context [16] to show that this scalar field should satisfy periodic boundary on an interval of double the length of the separation of the plates. This allows us in section 5 to simply infer the exact result for the partition function for gravitons, with a chemical potential for suitably modified spin angular momentum turned on, from the well-studied case of a massless scalar field with linear momentum in the compact direction turned on [2][3][4]17], together with modular properties and consistent high and low temperature expansions.…”