14. FIELD THEORIES ON NON-COMPACT MANIFOLDS 87
again defined using any fake heat kernel on U, with the property that the
small L asymptotic expansion of IU [L] is the restriction to U of the small L
asymptotic expansion of I[L].
The existence of this restriction map shows that there is a presheaf on
M which assigns to an open subset U ⊂ M the set T
(n)(Γ(U,
EU )), A ) of
theories on U. We will denote this presheaf by T (n)(E , A ).
14.9. Now we are ready to state the main theorem.
Theorem 14.9.1. (1) The presheaves T
(n)(E
, A ) (of theories de-
fined modulo
n+1)
and T
(∞)(E
, A ) (of theories defined to all order
in ) on M are sheaves.
(2) There is a canonical isomorphism between the sheaf T
(0)(E
, A )
and the sheaf of local action functionals I ∈ Oloc(Ec, A ) which are
at least cubic modulo the ideal Γ(X, m) ⊂ A .
(3) For n 0, T
(n)(E
, A ) is, in a canonical way, a torsor over
T
(n−1)(E
, A ) for the sheaf of abelian groups Oloc(Ec, A ), in a
canonical way.
(4) Choosing a renormalization scheme leads to a map
T
(n)(E
, A ) → T
(n+1)(E
, A )
of sheaves for each n, which is a section. Thus, the choice of a
renormalization scheme leads to an isomorphism of sheaves
T
(∞)(E
, A )
∼
=
Oloc(Ec,
+
A )[[ ]]
on M.
14.10. The proof of the theorem, and of Proposition 14.8.1, is along the
same lines as before, using counterterms.
Proposition 14.10.1. Let us fix a renormalization scheme, and a fake
heat kernel Kt on M.
Let
I =
iIi,k
∈ Oloc(Ec,
+
A )[[ ]].
Then:
(1) there is a unique series of counterterms
ICT
(ε) =
iIi,k CT
(ε)
where
Ii,k
CT
(ε) ∈ Oloc(E , A ) ⊗alg P((0, ∞))
0
is purely singular as a function of ε, with the property that the limit
lim
ε→0
W
(
P (ε, L),I −
ICT
(ε)
)
exists, for all L.
