ar X iv : h ep - t h / 93 03 02 8 v 2 8 M ar 1 99 3 The trace of the heat kernel on a compact hyperbolic 3 - orbifold

The heat coefficients related to the Laplace-Beltrami operator defined on the hyperbolic compact manifold H3/Γ are evaluated in the case in which the discrete group Γ contains elliptic and hyperbolic elements. It is shown that while hyperbolic elements give only exponentially vanishing corrections to the trace of the heat kernel, elliptic elements modify all coefficients of the asymptotic expansion, but the Weyl term, which remains unchanged. Some physical consequences are briefly discussed in the examples. PACS numbers: 02.90.+p, 11.10.-z, 05.30.Jp In the last decades, there has been a great deal of investigations on the properties of interacting quantum field theories in curved space-time. Several techniques have been employed. Among these, we mention the background-field method within path-integral approach [1], which is very useful in dealing with the one-loop approximation and which permits to evaluate the one-loop effective action. As a consequence, all physical interesting quantities (one-loop effective potential, vacuum energy, quantum anomalies and so on) can be derived in a straightforward way. The one-loop effective action, as derived from the path-integral, is an ill defined quantity, being related to the determinant of the fluctuation operator. Many regularization schemes have been proposed. One of the most promising, which works very well in a curved manifold too, is the so called “zeta-function regularization” [2], by which one can define the determinant of the fluctuation operator and therefore the regularized one-loop effective action. On a generic Riemannian manifold M, the zeta-function related to the Laplace-Beltrami operator is generally unknown, but nevertheless, some physical interesting quantities can be related to its Mellin inverse transform, which has a computable asymptotic expansion (heat kernel expansion). The situation is really better on manifolds with constant curvature where ∗ Email: [email protected], [email protected], itnvax::cognola † Email: [email protected], [email protected], itnvax::vanzo