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 ⊗

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 ∈ E1

!

⊗ 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 +DE1 Kt extends to a smooth

section

d

dt

Kt + DE1 Kt ∈ E1

!

⊗ E1

!

⊗

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/2e−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

Kt

N

= ψ(x,

y)t− dim M/2e−x−y

2

/t

N

i=0

tiΦi(x,

y)