14. FIELD THEORIES ON NON-COMPACT MANIFOLDS 85

The resulting linear map

E

⊗T (γ)

→ A

f → wγ(P, {Iv})(f)

is easily seen to have proper support.

Note that this lemma is false if the graph γ has no tails. This is the reason

why we only consider functionals on E modulo constants, or equivalently,

without a constant term.

Corollary 14.5.2. The renormalization group operator

Op

+(Ec,

A )[[ ]] → Op

+(Ec,

A )[[ ]]

I → W (P (ε, L),I)

is well-defined.

As always, Op

+(Ec,

A )[[ ]] refers to the subspace of Op(Ec, A )[[ ]] con-

sisting of elements which are at least cubic modulo and the ideal Γ(X, m) ⊂

A .

Proof. The renormalization group operator is defined by

W (P (ε, L),I) =

γ

1

Aut)γ)

g(γ)wγ(P

(ε, L),I).

The sum is over connected stable graphs; and, as we are working with func-

tionals on E modulo constants, we only consider graphs with at least one

tail. Lemma 14.5.1 shows that each wγ(P (ε, L),I) is well-defined.

14.6. Now we can define the notion of theory on the manifold M, using

the fake propagator P (ε, L).

Definition 14.6.1. A theory is a collection {I[L] | L ∈ R 0} of ele-

ments of Op

+(Ec,

A )[[ ]] which satisfy

(1) The renormalization group equation,

I[L] = W (P (ε, L),I[ε])

(2) The asymptotic locality axiom: there a small L asymptotic expan-

sion

I[L] fi(L)Ψi

in terms of local action functionals Ψi ∈ Oloc(Ec,

+

A )[[ ]]. We will

assume that the functions fi(L) appearing in this expansion have

at most a finite order pole at L = 0; that is, we can find some n

such that limL→0

Lnfi(L)

= 0.

Let T

(n)(E

, A ) denote the set of theories defined modulo

n+1,

and let

T

(∞)(E

, A ) denote the set of theories defined to all orders in .