Non-Laplace type Operators on Manifolds with Boundary

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.

arXiv commentIntroduction and bibliography significantly expanded. 39 pages, no figures, to be published in: Proceedings of the Workshop ``Krzysztof Wojciechowski 50 years - Analysis and Geometry of Boundary Value Problems'', Roskilde, Denmark, 20-22 May, 2005; Minor cosmetic changes in version 3