Asymptotic behaviours of the heat kernel in covariant perturbation theory

TL;DR

This paper analyzes the asymptotic behaviors of the heat kernel trace in covariant perturbation theory, calculating nonlocal form factors up to third order and exploring implications for the analyticity of the effective action in quantum field theory.

Contribution

It provides a detailed calculation of nonlocal curvature invariants and form factors up to third order, and examines their asymptotic behaviors to inform the analyticity of the effective action.

Findings

Derived third-order nonlocal form factors.

Characterized early-time and late-time asymptotics.

Established criteria for analyticity of the effective action.

Abstract

The trace of the heat kernel is expanded in a basis of nonlocal curvature invariants of $N$th 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 behaviours of the trace of the heat kernel are presented with this accuracy. The late-time behaviour 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.