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
).
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