10. CONSTRUCTING LOCAL COUNTERTERMS 61

Denote the N

th

partial sum of the asymptotic expansion by

ΨN (ε) =

N

i=0

gi(ε)Φi.

Then, we can define the singular part of wγ(P (ε, L),I) simply by

Singε wγ(P (ε, L),I) = Singε ΨN (ε) =

N

i=0

(Singε gi(ε)) Φi.

This singular part is independent of N, because if N is increased the function

ΨN (ε) is modified only by the addition of functions of ε which are periods

and which tend to zero as ε → 0.

Theorem 9.3.1 implies that Singε wγ(P (ε, L),I) has the following prop-

erties.

Theorem 9.5.1. Let I ∈

Oloc(C∞(M

))[[ ]] be a local functional, and let

γ be a connected stable graph.

(1) Singε wγ(P (ε, L),I) is a finite sum of the form

Singε wγ(P (ε, L),I) = fi(ε)Φi

where

Φi ∈

Oloc(C∞(M ),C∞((0,

∞)L)),

and

fi ∈ P((0, 1))

0

are purely singular periods.

(2) The limit

lim

ε→0

(wγ(P (ε, L),I) − Singε wγ(P (ε, L),I))

exists in the topological vector space

Oloc(C∞(M ),C∞((0,

∞)L)).

(3) Each Φi appearing in the finite sum above has a small L asymptotic

expansion

Φi

∞

j=0

fi,j(L)Ψi,j

where Ψi,j ∈

Oloc(C∞(M

)) is local, and fi,j(L) is a smooth function

of L ∈ (0, ∞).

10. Constructing local counterterms

10.1. The heart of the proof of theorem A is the construction of local

counterterms for a local interaction I ∈

Oloc(C∞(M

)). This construction is

simple and inductive, without the complicated graph combinatorics of the

BPHZ algorithm.

The theorem on the existence of local counterterms is the following.