Non-Perturbative Heat Kernel Asymptotics on Homogeneous Abelian Bundles

Abstract

We study the heat kernel for a Laplace type partial differential operator acting on smooth sections of a complex vector bundle with the structure group G × U(1) over a Riemannian manifold M without boundary. The total connection on the vector bundle naturally splits into a G-connection and a U(1)-connection, which is assumed to have a parallel curvature F. We find a new local short time asymptotic expansion of the off-diagonal heat kernel U(t|x, x′) close to the diagonal of M × M assuming the curvature F to be of order t −1. The coefficients of this expansion are polynomial functions in the Riemann curvature tensor (and the curvature of the G-connection) and its derivatives with universal coefficients depending in a non-polynomial but analytic way on the curvature F, more precisely, on tF. These functions generate all terms quadratic and linear in the Riemann curvature and of arbitrary order in F in the usual heat kernel coefficients. In that sense, we effectively sum up the usual short time heat kernel asymptotic expansion to all orders of the curvature F. We compute the first three coefficients (both diagonal and off-diagonal) of this new asymptotic expansion.