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 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.

N=1 and N=2 pure supergravities on a manifold with boundary

“…At the full supergravity level, boundary contributions were considered from several authors, using different approaches, and in particular in [24][25][26][27][28][29][30][31][32][33][34]. While in [30][31][32][33] boundary conditions on the fields are imposed, in [24][25][26][27][28][29] it is pointed out that the supergravity action should be invariant under local supersymmetry without imposing Dirichlet boundary conditions on the fields, in contrast to the Gibbons-Hawking prescription [3].…”

“…While in [30][31][32][33] boundary conditions on the fields are imposed, in [24][25][26][27][28][29] it is pointed out that the supergravity action should be invariant under local supersymmetry without imposing Dirichlet boundary conditions on the fields, in contrast to the Gibbons-Hawking prescription [3]. The explicit construction of an AdS supergravity theory with a boundary and with no boundary conditions on the fields was achieved in reference [24][25][26][27][28][29] using superconformal tensor calculus, in the particular case of N = 1, D = 3 (off-shell) supergravity.…”

Quantum Gravity: Unification of Principles and Interactions, and Promises of Spectral Geometry

“…But both of them vanish if and only if the metric solves the vacuum Einstein equations in four dimensions. In Euclidean quantum gravity, if one uses the de Donder gauge-fixing functional, one finds a boundary operator formally analogous to the one in equation (6.1) but imcompatible with the strong ellipticity of the boundary value problem [5]. Only recently, in [45,46], has one found a viable way out in the particular case of the Euclidean 4-ball, and more work is in order on this key issue, which has also implications for singularity avoidance in quantum cosmology, as we mentioned in Section 3.…”

Gribov Problem for Gauge Theories: A Pedagogical Introduction

“…(6) and (7), supplemented by the following asymptotic conditions on the D matrices (the ξ α functions no longer vanish at spatial infinity):…”

“…(6). In the very definition of gauge fixing at perturbative level, one requires that each orbit should intersect the gaugefixing surface only once, so as to ensure uniqueness of the potentials satisfying dynamical equations for a given choice of gauge fixing.…”