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
