88 2. THEORIES, LAGRANGIANS AND COUNTERTERMS

(2) This limit defines a collection of elements I[L] ∈ Op

+(E

, A )[[ ]],

satisfying the renormalization group equation and locality axiom,

and so defines a theory.

(3) The counterterms Ii,k

CT

(ε) do not depend on the choice of a fake heat

kernel.

(4) If U ⊂ M is an open subset, then the counterterms for I restricted

to U are the restrictions to U of the counterterms Ii,k

CT

(ε) for I.

Thus, counterterms define a map of sheaves

Oloc(Ec,

+

A )[[ ]] → Oloc(Ec,

+

A ⊗alg P((0, ∞)ε) 0)[[ ]].

Proof. The proof of the first two statements is identical to the proof

of the corresponding statement on a compact manifold, and so is mostly

omitted. One point is worth mentioning briefly, though: the counterterms

are defined to be the singular parts of the small ε asymptotic expansion of

the weight wγ(P (ε, L),I) attached to a graph γ. One can ask whether such

asymptotic expansions exist when we use a fake heat kernel rather than an

actual heat kernel.

The asymptotic expansion, as constructed in Appendix 1, only relies on

the small t expansion of the heat kernel Kt, and thus exists whenever the

propagator is constructed from a kernel with such a small t expansion.

The third clause is proved using the uniqueness of the counterterms, as

follows. Suppose that Kt, Kt are two fake heat kernels, with associated fake

propagators P (ε, L), P (ε, L). Let

ICT

(ε),

ICT

(ε) denote the counterterms

associated to the two different fake heat kernels. We need to show that they

are the same.

Note that

lim

ε→0

W P (0,L) − P (0,L),W

(

P (ε, L),I −

ICT

(ε)

)

exists. We can write the expression inside the limit as

W P (0,ε) − P (0,ε),W P (ε, L),I −

ICT

(ε) .

Note that P (0,ε) − P(0,ε) tends to zero, with all derivatives, faster than

any power of ε. Also, W P (ε, L),I −

ICT

(ε) has a small ε asymptotic

expansion in terms of functions of ε of polynomial growth at the origin.

From these two facts it follows that

lim

ε→0

W P (ε, L),I −

ICT

(ε)

exists.

Uniqueness of the counterterms implies that

ICT

(ε) =

ICT

(ε).

The fourth clause can be proved easily using independence of the coun-

terterms of the fake heat kernel.