Semiclassical $p$-branes in hyperbolic space

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.