Gauge theories on manifolds with boundary
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 parametrix and the heat-kernel in the leading approximation are explicitly constructed. As a result, all 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, that 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. Correspondingly, some unusual formulae for the heat-kernel diagonal are also obtained.Keywords
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