Nonperturbative methods for calculating the heat kernel

TL;DR

This paper introduces a covariant algebraic method for approximating the heat kernel in strongly curved manifolds, utilizing Lie algebra structures to derive closed-form formulas for the heat kernel diagonal.

Contribution

It presents a novel covariant algebraic approach that simplifies heat kernel calculations by leveraging Lie algebra structures and low-order derivatives of background fields.

Findings

Derived closed-form formulas for the heat kernel diagonal.

Established a finite-dimensional Lie algebra structure for differential operators.

Provided generating functions for heat kernel coefficients in symmetric spaces.

Abstract

We study the low-energy approximation for calculation of the heat kernel which is determined by the strong slowly varying background fields in strongly curved quasi-homogeneous manifolds. A new covariant algebraic approach, based on taking into account a finite number of low-order covariant derivatives of the background fields and neglecting all covariant derivatives of higher orders, is proposed. It is shown that a set of covariant differential operators together with the background fields and their low-order derivatives generates a finite dimensional Lie algebra. This algebraic structure can be used to present the heat semigroup operator in the form of an average over the corresponding Lie group. Closed covariant formulas for the heat kernel diagonal are obtained. These formulas serve, in particular, as the generating functions for the whole sequence of the Hadamard-\-Minakshisundaram-\-De~Witt-\-Seeley coefficients in all symmetric spaces.