76 2. THEORIES, LAGRANGIANS AND COUNTERTERMS

Let me explain more precisely what I mean by saying there is a small L

asymptotic expansion

Ii,k[L]

j∈Z≥0

gr(L)Φr.

Without loss of generality, we can require that the local action functionals

Φr appearing here are homogeneous of degree k in the field e ∈ E .

Recall that A is the global sections of some bundle of algebras A on a

manifold with corners X. Let Ax denote the fibre of A at x ∈ X. For every

element α ∈ A , let αx ∈ Ax denote the value of α at x.

The statement that there is such an asymptotic expansion means that

there is a non-decreasing sequence dR ∈ Z, tending to infinity, such that for

all R, for all fields e ∈ E , for all x ∈ X,

lim

L→0

L−dR

αx Ii,k[L](e) −

R

r=0

gr(L)Φr(e) = 0

in the finite dimensional vector space Ax.

Then, as before, the theorem is:

Theorem 13.4.3. The space T

(n+1)(E

) has the structure of a principal

Oloc(E , A ) bundle over T

(n)(E

), in a canonical way. Further, T

(0)(E

)

is canonically isomorphic to the space Oloc(E

+

, A ) of A -valued local action

functionals on E which are at least cubic modulo the ideal I ⊂ A .

Further, the choice of renormalization scheme gives rise to a section

T

(n)(E

) → T

(n+1)(E

) of each torsor, and so a bijection between T

(∞)(E

)

and the space

Oloc(E

+

, A )[[ ]]

of local action functionals with values in A , which are at least cubic modulo

and modulo the ideal I ⊂ A .

Proof. The proof is essentially the same as before. The extra diﬃcul-

ties are of two kinds: working with an auxiliary parameter space X intro-

duces extra analytical diﬃculties, and working with quadratic terms in our

interaction forces us to use Artinian induction with respect to the powers of

the ideal I ⊂ A .

For simplicity, I will only give the proof when the effective interactions

I[L] are all at least cubic modulo . The argument in the general case is the

same, except that we also must perform Artinian induction with respect to

the powers of the ideal I ⊂ A .

As before, we will prove the renormalization scheme dependent ver-

sion of the theorem, saying that there is a bijection between T

(∞)(E

) and

Oloc(E

+

, A )[[ ]]. The renormalization scheme independent formulation is an

easy corollary.

Let us start by showing how to construct a theory associated to a local

interaction

I =

iI(i,k)

∈ Oloc(E , A )[[ ]].