Quantum Fields and Extended Objects in Space-Times with Constant Curvature Spatial Section

TL;DR

This paper reviews mathematical techniques like heat-kernel expansion and $$-regularization for quantum fields on curved space-times with constant curvature, discussing exact solutions, physical applications, and extensions to string theory.

Contribution

It provides new exact solutions, detailed analysis of hyperbolic geometry, and a novel representation for one-loop superstring free energy in curved space-times.

Findings

Exact solutions of the heat-kernel equation on constant curvature manifolds

Analysis of vacuum energy and one-loop renormalization in hyperbolic space-times

A new representation for one-loop superstring free energy

Abstract

The heat-kernel expansion and $\zeta$-regularization techniques for quantum field theory and extended objects on curved space-times are reviewed. In particular, ultrastatic space-times with spatial section consisting in manifold with constant curvature are discussed in detail. Several mathematical results, relevant to physical applications are presented, including exact solutions of the heat-kernel equation, a simple exposition of hyperbolic geometry and an elementary derivation of the Selberg trace formula. With regards to the physical applications, the vacuum energy for scalar fields, the one-loop renormalization of a self-interacting scalar field theory on a hyperbolic space-time, with a discussion on the topological symmetry breaking, the finite temperature effects and the Bose-Einstein condensation, are considered. Some attempts to generalize the results to extended objects are also presented, including some remarks on path integral quantization, asymptotic properties of extended objects and a novel representation for the one-loop (super)string free energy.