Asymptotic behaviors of the heat kernel in covariant perturbation theory

The trace of the heat kernel is expanded in a basis of nonlocal curvature invariants of nth order. The coefficients of this expansion (the nonlocal form factors) are calculated to third order in the curvature inclusive. The early‐time and late‐time asymptotic behaviors of the trace of the heat kernel are presented with this accuracy. The late‐time behavior gives the criterion of analyticity of the effective action in quantum field theory. The latter point is exemplified by deriving the effective action in two dimensions.