Thermodynamics of Abelian Gauge Fields in Real Hyperbolic Spaces

Abstract

We work with $N-$dimensional compact real hyperbolic space $X_Γ$ with universal covering $M$ and fundamental group $Γ$. Therefore, $M$ is the symmetric space $G/K$, where $G=SO_1(N,1)$ and $K=SO(N)$ is a maximal compact subgroup of $G$. We regard $Γ$ as a discrete subgroup of $G$ acting isometrically on $M$, and we take $X_Γ$ to be the quotient space by that action: $X_Γ=Γ\backslash M = Γ\backslash G/K$. The natural Riemannian structure on $M$ (therefore on $X$) induced by the Killing form of $G$ gives rise to a connection $p-$form Laplacian ${\frak L}_p$ on the quotient vector bundle (associated with an irreducible representation of K). We study gauge theories based on abelian $p-$forms on the real compact hyperbolic manifold $X_Γ$. The spectral zeta function related to the operator ${\frak L}_p$, considering only the co-exact part of the $p-$forms and corresponding to the physical degrees of freedom, can be represented by the inverse Mellin transform of the heat kernel. The explicit thermodynamic fuctions related to skew-symmetric tensor fields are obtained by using the zeta-function regularization and the trace tensor kernel formula (which includes the identity and hyperbolic orbital integrals). Thermodynamic quantities in the high and low temperature expansions are calculated and new entropy/energy ratios established.