9. EXTRACTING THE SINGULAR PART 57

(2) The functionals I[P ] satisfy the following locality axiom. For any

(i, k), the support of

Supp Ii,k[P ] ⊂ M

k

can be made as close as we like to the diagonal by making the

parametrix P small. More precisely, we require that, for any open

neighbourhood U of the small diagonal M ⊂ M

k,

we can find some

P such that

Supp Ii,k[P ] ⊂ U

for all P ≤ P .

(3) Finally, the functionals I[P ] all have smooth first derivative.

Theorem 8.2.2. This definition of a quantum field theory is equivalent

to the previous one, presented in definition 7.1.2.

More precisely, if I[L] is a set of effective interactions satisfying the

heat-kernel definition of a QFT, then if we set

I[P ] = W (P − P (0,L),I[L])

for any parametrix P , the functionals I[P ] satisfy the parametrix definition

of quantum field theory presented in this section. Further, every collection

I[P ] of functionals satisfying the parametrix definition arises uniquely in

this way.

The proof of this theorem will be presented in Section 12, after the proof

of Theorem A.

9. Extracting the singular part of the weights of Feynman graphs

9.1. In order to construct the local counterterms needed for theorem A,

we need a method for extracting the singular part of the finite-dimensional

integrals wγ(P (ε, L),I) attached to Feynman graphs. This section will de-

scribe such a method, which relies on an understanding of the behaviour of

of the functions wγ(P (ε, L),I) as ε → 0. We will see that wγ(P (ε, L),I) has

a small ε asymptotic expansion

wγ(P (ε, L),I)(a) gi(ε)Φi(L, a),

where the Φi(L, a) are well-behaved functions of the field a and of L. Fur-

ther, the Φi(L, a) have a small L asymptotic expansion in terms of local

action functionals.

The functionals gi(ε) appearing in this expansion are of a very special

form: they are periods of algebraic varieties. For the purposes of this book,

the fact that these functions are periods is not essential. Thus, the reader

may skip the definition of periods without any loss. However, given the

interest in the relationship between periods and quantum field theory in the

mathematics literature (see (KZ01), for example) I felt that this point is

worth mentioning.