14.11. Now we can prove Proposition 14.8.1 and Theorem 14.9.1. The
proof is easy once we know that counterterms exist.
Let us start by regarding the set of theories T
, A ) as just a set,
and not as arising from a sheaf on M. (After all, we have not yet proved
Proposition 14.8.1, so we do not know that we have a presheaf of theories).
Lemma 14.11.1. Let us choose a renormalization scheme, and a fake
heat kernel. Then the map of sets
A )[[ ]] T
, A )
I {I[L] = lim
P (ε, L),I
is a bijection.
Proof. This is proved by the usual inductive argument.
Now we can prove Proposition 14.8.1, which we restate here for conve-
Proposition 14.11.2. Let U M be an open subset.
Given any theory
{I[L] Op
, A )[[ ]]}
on M, for any fake heat kernel, U, there is a unique theory
{IU [L] Op
E |U ), A )[[ ]]}
with the property that the small L asymptotics of IU [L] is the restriction to
U of the small L asymptotics of I[L].
Proof. Uniqueness is obvious, as any two theories on U with the same
small L asymptotic expansions must coincide.
For existence, we will use the bijection between theories and Lagrangians
which arises from the choice of a renormalization scheme. We can as-
sume that the theory {I[L]} on M arises from a local action functional
I Oloc(Ec,
A )[[ ]]. Then, we define IU [L] to be the theory on U associated
to the restriction of I to U.
It is straightforward to check that, with this definition, I[L] and IU [L]
have the same small L asymptotics.
It follows from the proof of this lemma that the map
A )[[ ]] T
, A )
of sets actually arises from a map of presheaves on M. Since Oloc(Ec,
A )[[ ]]
is actually a sheaf on M, it follows that T
, A ) is also a sheaf, and
similarly for T
, A ).
The remaining statements of Theorem 14.9.1 are proved in the same way
as the corresponding statements on a compact manifold.
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