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.