Non-Laplace type Operators on Manifolds with Boundary

TL;DR

This paper develops a systematic method to explicitly compute heat trace coefficients for non-Laplace type operators on manifolds with boundary, which are relevant in advanced physical theories involving non-commutative metrics.

Contribution

It introduces a new approach for calculating heat kernel asymptotics for non-Laplace operators, including explicit formulas for the first two coefficients.

Findings

Constructed the heat kernel parametrix for non-Laplace operators.

Derived explicit formulas for the first two heat trace coefficients.

Established a framework for systematic calculation of heat kernel asymptotics.

Abstract

We study second-order elliptic partial differential operators acting on sections of vector bundles over a compact manifold with boundary with a non-scalar positive definite leading symbol. Such operators, called non-Laplace type operators, appear, in particular, in gauge field theories, string theory as well as models of non-commutative gravity theories, when instead of a Riemannian metric there is a matrix valued self-adjoint symmetric two-tensor that plays the role of a ``non-commutative'' metric. It is well known that there is a small-time asymptotic expansion of the trace of the corresponding heat kernel in half-integer powers of time. We initiate the development of a systematic approach for the explicit calculation of these coefficients, construct the corresponding parametrix of the heat equation and compute explicitly the first two heat trace coefficients.