QUANTUM-FIELDS AND EXTENDED OBJECTS IN SPACE-TIMES WITH CONSTANT CURVATURE SPATIAL SECTION

The heat-kernel expansion and zeta-regularization techniques for quant um field theory and extended objects on curved space-times are reviewe d. In particular, ultrastatic space-times with spatial section consist ing in manifold with constant curvature are discussed in detail. Sever al mathematical results, relevant to physical applications are present ed, including exact solutions of the heat-kernel equation, a simple ex position of hyperbolic geometry and an elementary derivation of the Se lberg trace formula. With regard to the physical applications, the vac uum energy for scalar fields, the one-loop renormalization of a self-i nteracting scalar field theory on a hyperbolic space-time, with a disc ussion on the topological symmetry breaking, the finite-temperature ef fects and the Bose-Einstein condensation, are considered. Some attempt s to generalize the results to extended objects are also presented, in cluding some remarks on path-integral quantization, asymptotic propert ies of extended objects and a novel representation for the one-loop (s uper)string free energy.