THERMODYNAMICS OF ABELIAN GAUGE FIELDS IN REAL HYPERBOLIC SPACES

Abstract

We work with N-dimensional compact real hyperbolic space XΓ 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 XΓ to be the quotient space by that action: XΓ = Γ∖M = Γ∖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 𝔏p 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 XΓ. The spectral zeta function related to the operator 𝔏p, considering only the coexact 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 functions 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.