12. PROOF OF THE PARAMETRIX FORMULATION 69

The following lemma now shows that T

(n+1)

→ T

(n)

is a torsor for

Oloc(C∞(M

)).

Lemma 11.1.1. Let I, J ∈ T

(n+1)

be theories which agree in T

(n).

Then,

the functional

I0[L] + δ

−(n+1)(I[L]

− J[L]) ∈

O(C∞(M

))

satisfies the classical renormalization group equation modulo

δ2,

and so de-

fines an element of

TI0[L](T

(0))

∼

=

Oloc(C∞(M

)).

Note that

−(n+1)(I[L]

− J[L]) is well-defined as I[L] and J[L] agree

modulo

n+1.

Proof. This is a simple calculation.

12. Proof of the parametrix formulation of the main theorem

In this page we will prove the equivalence of the definition of theory

based on arbitrary parametrices, explained in Section 8 with the definition

based on the heat kernel. Since this result is not used elsewhere in the book,

I will not give all the details.

Thus, suppose we have a theory in the heat kernel sense, given by a fam-

ily I[L] of effective interactions satisfying the renormalization group equation

and the locality axiom. If P is a parametrix, let us define a functional I[P ]

by

I[P ] = W (P − P (0,L),I[L]) ∈

O(C∞(M

))[[ ]].

Since P (0,L) and P are both parametrices for the operator D

+m2,

the

difference between them is smooth. Thus, W (P − P (0,L),I[L]) is well-

defined.

Lemma 12.0.1. The collection of effective interactions {I[P ]}, defined

for each parametrix P , defines a theory using the parametrix definition of

theory.

Proof. To prove this, we need to verify the following.

(1) If P is another parametrix, then

I[P ] = W

(

P − P, I[P ]

)

(this is the version of the renormalization group equation for the

definition of theory based on parametrices.

(2) By choosing a parametrix P with support close to the diagonal, we

can make the distribution

Ii,k[P ] ∈ D(M

k)Sk

on M

k

supported as close as we like to the small diagonal.