Derivative expansion of the heat kernel in curved space

TL;DR

This paper develops a detailed covariant derivative expansion of the heat kernel in curved space-time, applicable to various gauge and scalar fields, extending existing formulas and providing explicit results up to fourth order.

Contribution

It introduces a fourth-order covariant derivative expansion of the heat kernel in curved space, including extensions of Chan's formula and applications to the bosonic effective action.

Findings

Explicit fourth-order heat kernel expressions in curved space

Extended Chan's formula for curved space-time

Computed bosonic effective action to fourth order

Abstract

The heat kernel in curved space-time is computed to fourth order in a strict expansion in the number of covariant derivatives. The computation is made for arbitrary non abelian gauge and scalar fields and for the Riemann connection in the coordinate sector. The expressions obtained hold for arbitrary tensor representations of the matter field. Complete results are presented for the diagonal matrix elements and for the trace of the heat kernel operator. In addition, Chan's formula is extended to curved space-time. As a byproduct, the bosonic effective action is also obtained to fourth order.