84 2. THEORIES, LAGRANGIANS AND COUNTERTERMS

14.5. The bundle E1 of propagating fields is a direct summand of the

bundle E of all fields. Thus, the fake heat kernel Kt can be viewed as an

element of the space

E

!

⊗ E ⊗

C∞(R

0) ⊗ A .

The operator D : E1

!

→ E1 extends to an operator E

!

→ E . For 0 ε

L ∞, let us define the fake propagator by

P (ε, L) =

L

ε

σ

(

(D ⊗ 1)Kt

)

dt ∈

Sym2

E ⊗ A ,

where σ : E

⊗2

→

Sym2

E is the symmetrization map.

We would like to define a theory, for the fake heat kernel Kt, to be a

collection of effective interactions

I[L] ∈ Op

+(Ec,

A )[[ ]]

satisfying the renormalization group equation defined using the fake prop-

agator P (ε, L). Recall that the subscript p in the expression

Op(Ec, A )

indicates that we are looking at functionals which are distributions with

proper support. A function in O(Ec, A ) is in Op(Ec, A ) if its Taylor com-

ponents, which are A -valued distributions on M

n,

have support which is a

proper subset of M

n

× X.

In order to do this, we need to know that the renormalization group

flow is well defined. Thus, we need to check that if we construct the

weights attached to graphs using the propagators P (ε, L) and interactions

I ∈ Op(Ec, A

)+[[

]].

Lemma 14.5.1. Let γ be a connected graph with at least one tail. Let

P ∈ E ⊗ E ⊗ A have proper support. Suppose for each vertex v of γ we have

a continuous linear map

Iv :

Ec⊗H(v)

→ A

which has proper support.

Then,

wγ(P, {Iv}) :

Ec⊗T (γ)

→ A

is well defined, and is a continuous linear map with proper support.

Proof. Let f ∈

Ec⊗T (γ)

. The expression wγ(P, {Iv})(f) is defined by

contracting the tensor

f ⊗e∈E(γ) Pe ∈ E

⊗H(γ)

given by putting a propagator P on each edge of γ and f at the tails of γ,

with the distribution

⊗v∈V (γ)Iv :

Ec⊗H(γ)

→ A .

Neither quantity has compact support. However, the restrictions we placed

on the supports of f, P and each Iv means that the intersection of the

support of ⊗Iv with that of f ⊗ Pe is a compact subset of M

H(γ).

Thus, we

can contract ⊗Iv with f ⊗ Pe to get an element of A .