13. VECTOR-BUNDLE VALUED FIELD THEORIES 77

We will assume that I(0,k) = 0 if k 3.

The argument is essentially the same as the argument we gave earlier.

We will perform induction on the set Z≥0 × Z≥0 with the lexicographical

order: (i, k) (r, s) if i r or if i = r and k s.

Suppose, by induction, we have constructed counterterms

I(i,k)(ε)

CT

∈ Oloc(E ,

C∞((0,

∞)ε) ⊗ A )

for all (i, k) (I, K). The I(i,k)

CT

are supposed, by induction, to have the

following properties:

(1) Each I(i,k)(ε)

CT

is homogeneous of degree k as a function of the field

e ∈ E .

(2) Each I(i,k)(ε)

CT

is required to be a finite sum

I(i,k)(ε)

CT

= gr(ε)Φr

where gr(ε) ∈

C∞((0,

∞)ε) and Φr ∈ Oloc(E , A ). Each gr(ε) is

required to have a finite order pole at 0; that is, limε→0

εkgr(ε)

= 0

for some k 0.

(3) Recall that A is the space of global sections of a vector bundle A

on X. For any element α ∈ A, let αx ∈ Ax denote its value at

x ∈ X.

We require that, for all L ∈ (0, ∞) and all x ∈ X, the limit

lim

ε→0

W(r,s)

⎛

⎝P

(ε, L),I −

(i,k)≤(r,s)

iI(i,k)(ε)⎠CT

⎞

x

exists in the topological vector space Hom(E

⊗r,

R) ⊗ Ax. Here,

Hom(E

⊗r,

R) is given the strong topology (i.e. the topology of uni-

form convergence on bounded subsets).

Now we need to construct the next counterterm I(I,K)(ε).

CT

We would like

to define

I(I,K)(ε)

CT

= Singε W(I,K)

⎛

⎝P

(ε, L),I −

(r,s) (I,K)

rI(r,s)CT

(ε)⎠

⎞

.

In order to be able to define the singular part like this, we need to know

that

W(I,K)

⎛

⎝P

(ε, L),I −

(r,s) (I,K)

rI(r,s)CT (ε)⎠

⎞

has a nice small ε asymptotic expansion. The required asymptotic expansion

is provided by the following theorem, proved in Appendix 1.

Theorem 13.4.4. For all graphs γ, and all I ∈ Oloc(E , A )[[ ]], there

exist local action functionals Φr ∈ Oloc(E , A

⊗C∞((0,

∞)L) and functions gr