We investigate the short-time expansion of the heat kernel of a Laplace type
operator on a compact Riemannian manifold and show that the lowest order term
of this expansion is given by the Fredholm determinant of the Hessian of the
energy functional on a space of finite energy paths. This is the asymptotic
behavior to be expected from formally expressing the heat kernel as a path
integral and then (again formally) using Laplace's method on the integral. We
also relate this to the zeta determinant of the Jacobi operator, which is
another way to assign a determinant to the Hessian of the energy functional. We
consider both the near-diagonal asymptotics as well as the behavior at the cut
locus.Comment: 37 pages, restructured introductio
