In [16] I studied the heat kernel on compact manifolds with smooth boundary with standard (Dirichlet) boundary conditions. It is shown that the standard De Witt technique can be applied to this case too if one takes into account in the semi-classical approximation one additional geodesic reflected from the boundary and proposed a new algorithm for calculating the boundary contributions in the heat kernel asymptotic expansion. This approach enables one not only to calculate the asymptotic expansion of the functional trace of the heat kernel but also to analyze the local structure of the heat kernel near the boundary.
In [17] (together with G. Esposito and A. Yu. Kamenshchik) and [18,19] (together with G. Esposito) we studied the Euclidean quantum gravity (more generally, arbitrary gauge theories) on manifolds with boundary. The requirement of the gauge invariance of the boundary value problem leads to the fact that the boundary conditions in gauge theories depend, generally, on the gauge fixing condition. By choosing the De Witt orthogonal gauge, the gauge field operator reduces to a Laplace type operator. However, the boundary conditions become more complicated. Namely, a part of the quantum field, (in gravity - the spatial components of the metric perturbations), satisfies Dirichlet boundary conditions, i.e. vanishes on the boundary, whereas the other part of the field satisfies oblique boundary conditions (so called Grubb-Gilkey-Smith boundary conditions), when the boundary operator includes not only the usual normal derivative but also a first order tangential differential operator on the boundary. In [17] the graviton operator for the gauge invariant generalized boundary conditions is explicitly shown to be self-adjoint. We also studied the boundary conditions resulting from the axial gauge and evaluated the corresponding one-loop divergence in the case of four-dimensional flat manifold bounded by a 3-sphere.
In [20] (together with G. Esposito) we studied the heat kernel asymptotics for a Laplace type differential operator for the oblique boundary conditions within the framework of the invariance theory and functorial methods. We found a useful reduction formula, which relates the heat kernel asymptotics with different boundary conditions. In a particular case (flat manifold with flat boundary and covariantly constant matrices appearing in the boundary operator) the heat kernel asymptotics Ak/2 are computed exactly for any order.
In [19] we analyzed the general gauge theories on manifolds with boundary. We studied the oblique boundary value problem for Laplace type operators acting on smooth sections of a vector bundle over a compact Riemannian manifold with oblique local boundary conditions including tangential derivatives and formulated an explicit condition of strong ellipticity of this boundary-value problem. The parametrix and the heat-kernel in the leading approximation are explicitly constructed. In particular, we calculated the first non-trivial coefficient in the heat kernel asymptotic expansion A1/2. The previous work in the literature on heat-kernel asymptotics is shown to be a very particular case of a more general structure. Namely, the previous results are shown to be valid only in the purely Abelian case, when all the matrices appearing in the boundary operator commute. 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 (and Dirac type operators). 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 (see, also [18]), however, it is shown that this property no longer holds, i.e. the gauge invariant oblique boundary value problem in quantum gravity is not strongly elliptic. Some non-standard formulae for the heat-kernel diagonal are also obtained. The heat kernel diagonal is shown to be non-integrable near the boundary. This leads to the fact that the trace of the heat kernel, and, therefore, the zeta-function and the functional determinant are not well defined in the non-elliptic case. This result raises deep interpretative issues for Euclidean quantum gravity on manifolds with boundary.