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 .
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