Asymptotic expansion of the heat kernel for generalized boundary conditions

Abstract

The asymptotic expansion of the heat kernel corresponding to e- tau Delta for a second-order symmetric elliptic differential operator Delta acting on vector fields over a manifold M with a boundary is extended to generalized Neumann boundary conditions. The normal derivative of the vector fields at any point on the boundary delta M is related to the vector field at the same point acted on by a linear operator Lambda which is symmetric with respect to the natural scalar product given by the induced measure on delta M. In this paper Lambda is allowed to be a first-order differential operator defined on vector fields restricted to delta M which is motivated by calculations with open strings. The authors use a method previously developed by them which extends the DeWitt asymptotic expansion to manifolds with a boundary by including geodesic paths undergoing reflection on the boundary. The first two terms in the boundary contributions to the asymptotic expansion are calculated and they involve a non-polynomial dependence on the coefficient of the derivative term in Lambda . The leading terms in the expansion of vector and tensor fields defined by the heat kernel are also obtained. The results are applied to determining the dependence of the functional determinant of Delta on conformal rescalings of the metric in two dimensions.