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 DE1 : E1 → E1.
We are interested in the heat kernel for the Laplacian ΔE1 . On a compact
manifold M, this is unique, and is an element of
Kt ∈ E1
⊗ E1 ⊗
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
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 ∈ E1
⊗ E1 ⊗
0) ⊗ A
with the following properties.
(1) Kt extends, at t = 0 to a distribution. Thus, Kt extends to an
Further, K0 is the kernel for the identity map E1 → E1.
(2) The support
Supp Kt ⊂ M × M × R
is proper. Recall that this means that both projection maps from
Supp Kt to M × R
× X are proper.
(3) The heat kernel Kt satisfies the heat equation up to exponentially
small terms in t. More precisely,
Kt +DE1 Kt extends to a smooth
Kt + DE1 Kt ∈ E1
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
t− dim M/2e−x−y
Let me explain more carefully about what I mean by a small t asymptotic
expansion. We will let
y)t− dim M/2e−x−y