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.
