Off-diagonal heat-kernel expansion and its application to fields with differential constraints

TL;DR

This paper derives the off-diagonal heat-kernel expansion for Laplace operators with gauge connections on compact manifolds, analyzing its implications for fields with differential constraints and applications to quantum gravity.

Contribution

It provides the first computation of the off-diagonal heat-kernel expansion including gauge connections up to third order in curvatures, and studies its effects on constrained vector fields.

Findings

Seeley-deWitt coefficients develop singularities for constrained fields

Singularities vanish for flat or Einstein metrics

Results inform gravitational functional renormalization group calculations

Abstract

The off-diagonal heat-kernel expansion of a Laplace operator including a general gauge-connection is computed on a compact manifold without boundary up to third order in the curvatures. These results are used to study the early-time expansion of the traced heat-kernel on the space of transverse vector fields satisfying the differential constraint $D^\mu v_\mu = 0$. It is shown that the resulting Seeley-deWitt coefficients generically develop singularities, which vanish if the metric is flat or satisfies the Einstein condition. The implications of our findings for the evaluation of the gravitational functional renormalization group equation are briefly discussed.