The Shape of Compact Toroidal Dimensions $T^d_{\theta}$ and the Casimir Effect on $M^D\times T^d_{\theta}$ spacetime

TL;DR

This paper investigates how the shape of twisted toroidal extra dimensions affects the Casimir energy and force for scalar fields, revealing unique regularity properties and implications for stabilization in higher-dimensional models.

Contribution

It provides a detailed analysis of the Casimir effect on twisted toroidal dimensions, highlighting singularities and regularities depending on the dimension and shape parameter.

Findings

Massive scalar Casimir energy is singular for even D, depending on R.

Massless scalar Casimir energy and force are regular only in D=4.

Shape parameter θ influences energy and force, with implications for stabilization.

Abstract

We study the influence of the shape of compact dimensions to the Casimir energy and Casimir force of a scalar field. We examine both the massive and the massless scalar field. The total spacetime topology is $M^D\times T^2_{\theta}$, where $M^D$ is the $D$ dimensional Minkowski spacetime and $T^2_{\theta}$ the twisted torus described by $R_1$, $R_2$ and $\theta$. For the case $R_1=R_2$ we found that the massive bulk scalar field Casimir energy is singular for $D$=even and this singularity is $R$-dependent and remains even when the force is calculated. Also the massless Casimir energy and force is regular only for D=4 (!). This is very interesting phenomenologically. We examine the energy and force as a function of $\theta$. Also we address the stabilization problem of the compact space. We also briefly discuss some phenomenological implications.