Gauge theories on manifolds with boundary

TL;DR

This paper investigates boundary-value problems for Laplace-type operators on manifolds with boundary, focusing on strong ellipticity, heat kernel construction, and applications to gauge theories, including Yang-Mills and quantum gravity.

Contribution

It formulates strong ellipticity conditions for gauge theories with boundary and constructs heat kernels, extending previous work and analyzing non-elliptic cases like quantum gravity.

Findings

Strong ellipticity conditions are established for gauge theories.

Heat kernels are explicitly constructed in leading approximation.

Quantum gravity boundary-value problem is not strongly elliptic.

Abstract

The boundary-value problem for Laplace-type operators acting on smooth sections of a vector bundle over a compact Riemannian manifold with generalized local boundary conditions including both normal and tangential derivatives is studied. The condition of strong ellipticity of this boundary-value problem is formulated. The resolvent kernel and the heat kernel in the leading approximation are explicitly constructed. As a result, the previous work in the literature on heat-kernel asymptotics is shown to be a particular case of a more general structure. For a bosonic gauge theory on a compact Riemannian manifold with smooth boundary, the problem is studied of obtaining a gauge-field operator of Laplace type, jointly with local and gauge-invariant boundary conditions, which should lead to a strongly elliptic boundary-value problem. The scheme is extended to fermionic gauge theories by means of local and gauge-invariant projectors. After deriving a general condition for the validity of strong ellipticity for gauge theories, it is proved that for Euclidean Yang--Mills theory and Rarita--Schwinger fields all the above conditions can be satisfied. For Euclidean quantum gravity, however, this property no longer holds, i.e. the corresponding boundary-value problem is not strongly elliptic. Some non-standard local formulae for the leading asymptotics of the heat-kernel diagonal are also obtained. It is shown that, due to the absence of strong ellipticity, the heat-kernel diagonal is non-integrable near the boundary.