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Higher-abelian gauge theories associated with Cheeger-Simons differentialcharacters are studied on compact manifolds without boundary. The paperconsists of two parts: First the functional integral formulation based on zetafunction regularization is revisited and extended in order to provide a generalframework for further applications. A field theoretical model - called extendedhigher-abelian Maxwell theory - is introduced, which is a higher-abelianversion of Maxwell theory of electromagnetism extended by a particulartopological action. This action is parametrized by two non-dynamical harmonicforms and generalizes the $\theta$-term in usual gauge theories. In the secondpart the general framework is applied to study the topological Casimir effectin higher-abelian gauge theories at finite temperature at equilibrium. Theextended higher-abelian Maxwell theory is discussed in detail and an exactexpression for the free energy is derived. A non-trivial topology of thebackground space-time modifies the spectrum of both the zero-point fluctuationsand the occupied states forming the thermal ensemble. The vacuum (Casimir)energy has two contributions: one related to the propagating modes and thesecond one related to the topologically inequivalent configurations ofhigher-abelian gauge fields. In the high temperature limit the leading term isof Stefan-Boltzmann type and the topological contributions are suppressed. Witha particular choice of parameters extended higher-abelian Maxwell theories ofdifferent degrees are shown to be dual. On the $n$-dimensional torus we provideexplicit expressions for the thermodynamic functions in the low- and hightemperature regimes, respectively. Finally, the impact of the backgroundtopology on the two-point correlation function of a higher-abelian variant ofthe Polyakov loop operator is analyzed.

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