Quantum Effective Action in Spacetimes with Branes and Boundaries

TL;DR

This work develops a covariant background-field framework for quantum brane theories by exploiting a Neumann-Dirichlet duality that extends beyond tree level. It shows that, at one loop, the bulk determinant with generalized Neumann boundary conditions factorizes into the Dirichlet bulk determinant and a brane-to-brane determinant, yielding an additive decomposition of the effective action: $\Gamma_{\text{1-loop}} = \tfrac{1}{2}\mathrm{Tr}_D\ln F + \tfrac{1}{2}\mathrm{tr}\ln \boldsymbol{F}^{\text{brane}}$. This duality also provides a new, universal route to calculate boundary (surface) terms in the heat-kernel expansion for Robin and oblique boundary conditions, by relating Neumann surface densities to their Dirichlet counterparts. The paper confirms the approach with simple one- and two-dimensional examples and outlines a path to multi-loop generalizations and gauge-theory extensions, with potential impact on brane-induced gravity and related cosmological models.

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.