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.
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