14. FIELD THEORIES ON NON-COMPACT MANIFOLDS 83

be the N

th

partial sum of this asymptotic expansion (where we have intro-

duced a cut-off ψ(x, y) so that Kt

N

is zero outside of a small neighbourhood

of the diagonal).

Then, we require that for all compact subsets C ⊂ M × M × X,

∂t

k

(

Kt − Kt

N

)

C,l,m

=

O(tN−dim M/2−l/2−k).

where −l,m refers to the norm on Γ(M,

E∨

⊗ Dens(M)) ⊗ E ⊗ A where

we differentiate l times on M × M, m times on X, and take the supremum

over the compact subset C.

These estimates are the same as the ones satisfied by the actual heat

kernel on a compact manifold M, as detailed in (BGV92).

The fake heat kernel Kt satisfies the heat equation up to a function

which vanishes faster than any power of t. This implies that the asymptotic

expansion

t− dim M/2e−x−y

2

/t

i≥0

tiΦi(x,

y).

of Kt must be a formal solution to the heat equation (in the sense described

in (BGV92), Section 2.5). This characterizes the functions Φi(x, y) (defined

in a neighbourhood of the diagonal) uniquely.

Lemma 14.3.2. Let Kt, Kt be two fake heat kernels. Then Kt − Kt

extends across t = 0 to a smooth kernel, that is,

Kt − Kt ∈ E

!

⊗ E ⊗ A ⊗

C∞(R≥0).

Further, Kt − Kt vanishes to all orders at t = 0.

Proof. This is clear from the existence and uniqueness of the small t

asymptotic expansion.

Lemma 14.3.3. A fake heat kernel always exists.

Proof. The techniques of (BGV92) allow one to construct a fake heat

kernel by approximating it with the partial sums of the asymptotic expan-

sion.

14.4. Let us suppose that M is an open subset of a compact manifold

N, and that the free field theory E on M is restricted from one, say F , on

N.

Then, the restriction of the heat kernel for F to M is an element

Kt ∈ E1

!

⊗ E1 ⊗

C∞(R

0) ⊗ A ,

which satisfies all the axioms of a fake heat kernel except that of requiring

proper support.

If Ψ ∈

C∞(M

× M) is a smooth function, with proper support, which

takes value 1 on a neighbourhood of the diagonal, then

ΨKt ∈ E1

!

⊗ E1 ⊗

C∞(R

0) ⊗ A

is a fake heat kernel.