The Heat Kernel Expansion

Abstract

Our operator Q = D², the square of the spin-c Dirac operator, has scalar principal symbol. So for the discussion of the asymptotic expansion of its heat kernel, we may restrict ourselves to the case that Q is a second order differential operator, acting on sections of a complex vector bundle F over a d-dimensional Riemannian manifold (M, β), with principal symbol given by

$$\sigma _{\it{Q}} (\xi)\,=\,\beta ^{-1}\,(\xi,\,\xi)\, \cdot \,1,\,\,\,\,\xi\,\in\,\rm{T}^{*}\,\it{M}.$$

(8.1)