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.
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