14. FIELD THEORIES ON NON-COMPACT MANIFOLDS 89

14.11. Now we can prove Proposition 14.8.1 and Theorem 14.9.1. The

proof is easy once we know that counterterms exist.

Let us start by regarding the set of theories T

(∞)(E

, A ) as just a set,

and not as arising from a sheaf on M. (After all, we have not yet proved

Proposition 14.8.1, so we do not know that we have a presheaf of theories).

Lemma 14.11.1. Let us choose a renormalization scheme, and a fake

heat kernel. Then the map of sets

Oloc(Ec,

+

A )[[ ]] → T

(∞)(E

, A )

I → {I[L] = lim

ε→0

W

(

P (ε, L),I −

ICT

(ε)

)

}

is a bijection.

Proof. This is proved by the usual inductive argument.

Now we can prove Proposition 14.8.1, which we restate here for conve-

nience.

Proposition 14.11.2. Let U ⊂ M be an open subset.

Given any theory

{I[L] ∈ Op

+(E

, A )[[ ]]}

on M, for any fake heat kernel, U, there is a unique theory

{IU [L] ∈ Op

+(Γ(U,

E |U ), A )[[ ]]}

with the property that the small L asymptotics of IU [L] is the restriction to

U of the small L asymptotics of I[L].

Proof. Uniqueness is obvious, as any two theories on U with the same

small L asymptotic expansions must coincide.

For existence, we will use the bijection between theories and Lagrangians

which arises from the choice of a renormalization scheme. We can as-

sume that the theory {I[L]} on M arises from a local action functional

I ∈ Oloc(Ec,

+

A )[[ ]]. Then, we define IU [L] to be the theory on U associated

to the restriction of I to U.

It is straightforward to check that, with this definition, I[L] and IU [L]

have the same small L asymptotics.

It follows from the proof of this lemma that the map

Oloc(Ec,

+

A )[[ ]] → T

(∞)(E

, A )

of sets actually arises from a map of presheaves on M. Since Oloc(Ec,

+

A )[[ ]]

is actually a sheaf on M, it follows that T

(∞)(E

, A ) is also a sheaf, and

similarly for T

(n)(E

, A ).

The remaining statements of Theorem 14.9.1 are proved in the same way

as the corresponding statements on a compact manifold.