10. CONSTRUCTING LOCAL COUNTERTERMS 63

Figure 5. Explanation of why I1,1

CT

(ε, L) is independent of

L.

Since I1,1

CT

(ε, L) is independent of L, to verify that it is local we only

need to examine the behaviour of W1,1 (P (ε, L),I) at small L. Theorem

9.5.1 implies that Singε W1,1 (P (ε, L),I) has a small L asymptotic expansion

in terms of local action functionals. Therefore, since we know I1,1

CT

(ε, L) is

independent of L, it follows that it is local.

Now that we know I1,1

CT

(ε, L) is independent of L, we will normally drop

L from the notation.

10.3. The next step is to construct I1,2

CT (ε, L). However, it is just as

simple to construct directly the general counterterm Ii,k

CT

(ε, L). Let us lex-

icographically order the set Z≥0 × Z≥0, so that (i, k) (j, l) if i j or if

i = j and k l. Let us write

W

(i,k)

(P, I) =

(j,l) (i,k)

jWj,l

(P, I) .

We can write this expression in terms of stable graphs, as follows. Let Γ

(i,k)

denote the set of stable graphs with genus smaller than i, or with genus equal

to i and fewer than k external edges. Then,

W

(i,k)

(P, I) =

γ∈Γ

(i,k)

g(γ)

|Aut γ|

wγ(P, I).

Let us assume, by induction, we have constructed counterterms Ij,l

CT

(ε)

for all (j, l) (i, k), with the following properties.