Quantum Effective Action in Spacetimes with Branes and Boundaries

TL;DR

This paper develops a method to construct quantum effective actions in spacetimes with branes and boundaries by reducing Neumann problems to Dirichlet problems, extending duality concepts beyond tree level, and introducing new heat kernel techniques.

Contribution

It introduces a Neumann-Dirichlet duality extension for quantum effective actions in brane spacetimes and proposes a novel approach to heat kernel surface terms for generalized boundary conditions.

Findings

Factorization of functional determinants into Dirichlet parts and brane operators.

New method for calculating surface terms in heat kernel expansion.

Potential for multi-loop applications and universal background field methods.

Abstract

We construct quantum effective action in spacetime with branes/boundaries. This construction is based on the reduction of the underlying Neumann type boundary value problem for the propagator of the theory to that of the much more manageable Dirichlet problem. In its turn, this reduction follows from the recently suggested Neumann-Dirichlet duality which we extend beyond the tree level approximation. In the one-loop approximation this duality suggests that the functional determinant of the differential operator subject to Neumann boundary conditions in the bulk factorizes into the product of its Dirichlet counterpart and the functional determinant of a special operator on the brane -- the inverse of the brane-to-brane propagator. As a byproduct of this relation we suggest a new method for surface terms of the heat kernel expansion. This method allows one to circumvent well-known difficulties in heat kernel theory on manifolds with boundaries for a wide class of generalized Neumann boundary conditions. In particular, we easily recover several lowest order surface terms in the case of Robin and oblique boundary conditions. We briefly discuss multi-loop applications of the suggested Dirichlet reduction and the prospects of constructing the universal background field method for systems with branes/boundaries, analogous to the Schwinger-DeWitt technique.