Casimir energy of a massive field in a genus-1 surface

TL;DR

This paper calculates the Casimir energy for a massive scalar field on a genus-1 surface, providing an explicit analytic expression that depends on the surface's parameters and the field's mass, using zeta function regularization.

Contribution

It presents the first exact calculation of Casimir energy for a massive field on a general genus-1 surface, explicitly showing dependence on Teichmüller parameters.

Findings

Derived an explicit analytic formula for Casimir energy on a genus-1 surface.

Demonstrated dependence of Casimir energy on surface parameters and mass.

Clarified the role of renormalization in defining the Casimir energy.

Abstract

We review the definition of the Casimir energy steming naturally from the concept of functional determinant through the zeta function prescription. This is done by considering the theory at finite temperature and by defining then the Casimir energy as its energy in the limit $T\to 0$. The ambiguity in the coefficient $C_{d/2}$ is understood to be a result of the necessary renormalization of the free energy of the system. Then, as an exact, explicit example never calculated before, the Casimir energy for a massive scalar field living in a general $(1+2)$-dimensional toroidal spacetime (i.e., a general surface of genus one) with flat spatial geometry ---parametrized by the corresponding Teichm\"uller parameters--- and its precise dependence on these parameters and on the mass of the field is obtained under the form of an analytic function.