Aspects of Higher-Abelian Gauge Theories at zero and finite temperature: Topological Casimir effect, duality and Polyakov loops

TL;DR

This paper explores higher-abelian gauge theories on compact manifolds, focusing on their topological Casimir effect, duality properties, and implications for Polyakov loops at zero and finite temperatures.

Contribution

It introduces a generalized framework for higher-abelian Maxwell theories, including a topological action, and applies it to analyze the topological Casimir effect and dualities on compact manifolds.

Findings

Derived exact free energy expressions at finite temperature.

Identified topological contributions to vacuum energy and their temperature dependence.

Provided explicit thermodynamic functions on n-dimensional tori.

Abstract

Higher-abelian gauge theories associated with Cheeger-Simons differential characters are studied on compact manifolds without boundary. The paper consists of two parts: First the functional integral formulation based on zeta function regularization is revisited and extended in order to provide a general framework for further applications. A field theoretical model - called extended higher-abelian Maxwell theory - is introduced, which is a higher-abelian version of Maxwell theory of electromagnetism extended by a particular topological action. This action is parametrized by two non-dynamical harmonic forms and generalizes the $\theta$-term in usual gauge theories. In the second part the general framework is applied to study the topological Casimir effect in higher-abelian gauge theories at finite temperature at equilibrium. The extended higher-abelian Maxwell theory is discussed in detail and an exact expression for the free energy is derived. A non-trivial topology of the background space-time modifies the spectrum of both the zero-point fluctuations and the occupied states forming the thermal ensemble. The vacuum (Casimir) energy has two contributions: one related to the propagating modes and the second one related to the topologically inequivalent configurations of higher-abelian gauge fields. In the high temperature limit the leading term is of Stefan-Boltzmann type and the topological contributions are suppressed. With a particular choice of parameters extended higher-abelian Maxwell theories of different degrees are shown to be dual. On the $n$-dimensional torus we provide explicit expressions for the thermodynamic functions in the low- and high temperature regimes, respectively. Finally, the impact of the background topology on the two-point correlation function of a higher-abelian variant of the Polyakov loop operator is analyzed.