Covariant techniques for computation of the heat kernel

TL;DR

This paper introduces a covariant, algorithmic method for computing heat kernel coefficients of elliptic operators on Riemannian manifolds, enabling explicit calculations and analysis of their structure, including the first explicit computation of the fourth coefficient.

Contribution

A new covariant and algorithmic technique for computing heat kernel coefficients on manifolds of arbitrary dimension and signature, including explicit formulas and structural insights.

Findings

Explicit computation of the fourth heat kernel coefficient.

Derivation of generating functions for covariantly constant and low-derivative terms.

Analysis of the structure of heat kernel coefficients for symmetric spaces.

Abstract

The heat kernel associated with an elliptic second-order partial differential operator of Laplace type acting on smooth sections of a vector bundle over a Riemannian manifold, is studied. A general manifestly covariant method for computation of the coefficients of the heat kernel asymptotic expansion is developed. The technique enables one to compute explicitly the diagonal values of the heat kernel coefficients, so called Hadamard-Minackshisundaram-De Witt-Seeley coefficients, as well as their derivatives. The elaborated technique is applicable for a manifold of arbitrary dimension and for a generic Riemannian metric of arbitrary signature. It is very algorithmic, and well suited to automated computation. The fourth heat kernel coefficient is computed explicitly for the first time. The general structure of the heat kernel coefficients is investigated in detail. On the one hand, the leading derivative terms in all heat kernel coefficients are computed. On the other hand, the generating functions in closed covariant form for the covariantly constant terms and some low-derivative terms in the heat kernel coefficients are constructed by means of purely algebraic methods. This gives, in particular, the whole sequence of heat kernel coefficients for an arbitrary locally symmetric space.