Semiclassical $p$-branes in hyperbolic space

TL;DR

This paper investigates one-loop quantum corrections to $p$-branes in hyperbolic spaces, comparing path integral and Wheeler-DeWitt approaches to assess the validity of semiclassical approximations and analyze divergences.

Contribution

It provides a detailed comparison of quantization methods for $p$-branes in hyperbolic space and classifies divergences related to one-loop corrections using heat kernel regularization.

Findings

Divergences are classified by their geometric nature.

Semiclassical approximation validity depends on divergence behavior.

Heat kernel expansion reveals compatibility of divergences with semiclassical methods.

Abstract

The one-loop effects to the Dirac action of $p$-branes in a hyperbolic background from the path integral and the solution of the Wheeler-DeWitt equation are analysed. The objective of comparing the equivalent quantization procedures is to study in detail the validity of the semiclassical approximation and divergences associated to one-loop corrections. This is in line with a bottom-up approach to holographic Wilson loops. We employ the heat kernel regularization method for both quantization procedures and we study in great detail one-loop corrections to geodesics in a 2-dimensional hyperbolic space and semi-spheres in a 3-dimensional hyperbolic space. We show that the divergences, given by the high energy expansion of the heat kernel, can be classified by their compatibility with the semiclassical approximation and geometric nature.