82 2. THEORIES, LAGRANGIANS AND COUNTERTERMS 14.3. Recall that the super vector bundle E on M has additional struc- ture, as described in Definition 13.1.1. This data includes a decomposition E = E1 E2 and a generalized Laplacian DE 1 : E1 E1. We are interested in the heat kernel for the Laplacian ΔE 1 . On a compact manifold M, this is unique, and is an element of Kt E ! 1 E1 C∞(R 0 ) A . On a non-compact manifold, there are many heat kernels, corresponding to various boundary conditions. In addition, such heat kernels may grow on the boundary of the non-compact manifold in ways which are difficult to control. To remedy this, we will introduce the concept of fake heat kernel. A fake heat kernel is something which solves the heat equation but only up to the addition of a smooth kernel. Definition 14.3.1. A fake heat kernel is a smooth section Kt E ! 1 E1 C∞(R 0 ) A with the following properties. (1) Kt extends, at t = 0 to a distribution. Thus, Kt extends to an element of E ! 1 E 1 C∞(R≥0) A Further, K0 is the kernel for the identity map E1 E1. (2) The support Supp Kt M × M × R 0 × X is proper. Recall that this means that both projection maps from Supp Kt to M × R 0 × X are proper. (3) The heat kernel Kt satisfies the heat equation up to exponentially small terms in t. More precisely, d dt Kt+DE 1 Kt extends to a smooth section d dt Kt + DE 1 Kt E ! 1 E ! 1 C∞(R≥0) A which vanishes at t = 0, with all derivatives in t and on M, faster than any power of t. (4) The heat kernel Kt admits a small t asymptotic expansion which can be written, in normal coordinates x, y near the diagonal of M, in the form Kt t− dim M/2 e−x−y 2 /t i≥0 tiΦi(x, y). Let me explain more carefully about what I mean by a small t asymptotic expansion. We will let KN t = ψ(x, y)t− dim M/2 e−x−y 2 /t N i=0 tiΦi(x, y)
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