60 2. THEORIES, LAGRANGIANS AND COUNTERTERMS

where the limit is taken in the topological vector space

O(C∞(M ),C∞((0,

∞)L)).

(2) The gi(ε) appearing in this asymptotic expansion have a finite order

pole at zero: for each i there is a k such that limε→0

εkgi(ε)

= 0.

(3) Each Ψi appearing in the asymptotic expansion above has a small

L asymptotic expansion of the form

Ψi

∞

j=0

fi,j(L)Φi,j

where the Φi,j are local action functionals, that is, elements of

Ol(C∞(M

)); and each fi,k(L) is a smooth function of L ∈ (0, ∞).

This theorem is proved in Appendix 1. All of the hard work required to

construct counterterms is encoded in this theorem. The theorem is proved

by using the small t asymptotic expansion for the heat kernel to approximate

each wγ(P (ε, L),I) for small ε.

9.4. These results allow us to extract the singular part of the finite-

dimensional integral wγ(P (ε, L),I). Of course, the notion of singular part

is not canonical, but depends on a choice.

Definition 9.4.1. Let P((0, 1))≥0 ⊂ P((0, 1)) be the subspace of those

functions f of ε which are periods and which admit a limit as ε → 0.

A renormalization scheme is a complementary subspace

P((0, 1))

0

⊂ P((0, 1))

to P((0, 1))≥0.

Thus, once we have chosen a renormalization scheme we have a direct

sum decomposition

P((0, 1)) = P((0, 1))≥0 ⊕ P((0, 1)) 0.

A renormalization scheme is the data one needs to define the singular

part of a function in P((0, 1)).

Definition 9.4.2. If f ∈ P((0, 1)), define the singular Sing(f) of f to

be the projection of f onto P((0, 1)) 0.

9.5. We can now use this definition to extract the singular part of the

functions wγ(P (ε, L),I). As before, let us think of wγ(P (ε, L),I) as a distri-

bution on M

k.

Then, Theorem 9.3.1 shows that wγ(P (ε, L),I) has a small

ε asymptotic expansion of the form

wγ(P (ε, L),I)

∞

i=0

gi(ε)Φi

where the gi(ε) are periods and the Φi ∈

Hom(C∞(M k),C∞((0,

∞)L)).

Theorem 9.3.1 also implies that there exists an N ∈ Z≥0 such that, for all

n N, gn(ε) admits an ε → 0 limit.