13. VECTOR-BUNDLE VALUED FIELD THEORIES 75

Note that Ol(E , A ) is not a closed subspace. However, as we will see in

Appendix 2, Oloc(E , A ) has a natural topology making it into a complete

nuclear space, and a module over A in the symmetric monoidal category of

nuclear spaces.

Our interactions will be elements of

Ol(E , A )[[ ]].

We would like to allow our interactions to have quadratic and linear terms

modulo . However, we require that these quadratic terms are accompanied

by elements of the nilpotent ideal

I = Γ(X, I) ⊂ A

(recall that A /I =

C∞(X)).

If we don’t impose this condition, we will

encounter infinite sums.

Thus, let us denote by

O+(E

, A )[[ ]] ⊂ O(E , A )[[ ]]

the subset of those functionals which are at least cubic modulo the ideal

generated by I and .

Then, the renormalization group operator

W (P (ε, L),I)

= log

(

exp( ∂P (ε,L)) exp(I/ )

)

:

O+(E

, A )[[ ]] →

O+(E

, A )[[ ]]

is well-defined.

Because we now allow quadratic and linear interaction terms modulo

, the Feynman graph expansion of this expression involves univalent and

bivalent genus 0 vertices. However, each such vertex is accompanied by an

element of the ideal I of A . Since this ideal is nilpotent, there is a uniform

bound on the number of such vertices that can occur, so there are no infinite

sums.

Definition 13.4.2. A theory is given by a collection of even elements

I[L] ∈

O+(E

,

C∞((0,

∞)L) ⊗ A )[[ ]],

such that

(1) The renormalization group equation

I[L ] = W

(

P (L, L ),I[L]

)

holds.

(2) Each I(i,k)[L] has a small L asymptotic expansion

I(i,k)[L](e) Ψr(e)fr(L)

where Ψr ∈ Ol(E ) are local action functionals.

Let T

(∞)(E

) denote the space of such theories, and let T

(n)(E

) denote

the space of theories defined modulo

n+1,

so that T

(∞)(E

) = lim

←−

T

(n)(E

).