80 2. THEORIES, LAGRANGIANS AND COUNTERTERMS
Suppose that we have a local action functional
J =
(i,k) (I,K)
iJ(i,k)
Oloc(E
+
, A )[[ ]]
with associated effective interactions J[L].
Suppose, by induction, that
J(i,k)[L] = I(i,k)[L]
for all (i, k) (I, K).
We need to find some J(I,K) such that, if we set J = J +
I
J(I,K), then
J(I,K)[L] = I(I,K)[L].
We simply let
J(I,K) = I(I,K)[L] J(I,K)[L].
The renormalization group equation implies that J(I,K) is independent of L.
It is automatic that
J(I,K)[L] = I(I,K)[L].
Finally, the fact that both J(I,K)[L] and J(I,K)[L] satisfy the small L asymp-
totics axiom of a theory implies that J(I,K) is local.
14. Field theories on non-compact manifolds
A second generalization is to non-compact manifolds. On non-compact
manifolds, we don’t just have ultraviolet divergences (arising from small
scales) but infrared divergences, which arise when we try to integrate over
the non-compact manifold.
However, by imposing a suitable infrared cut-off, we will find a notion
of theory on a non-compact manifold; and a bijection between theories and
Lagrangians.
The infrared cut-off we impose is rather brutal: we multiply the propaga-
tor by a cut-off function so that it becomes supported on a small neighbour-
hood of the diagonal in M
2.
However, the notion of theory is independent
of the cut-off chosen.
We will also show that there are restriction maps, allowing one to restrict
a theory on a non-compact manifold M to any open subset U. This allows
one to define a sheaf of theories on any manifold. If we are on a compact
manifold, global sections of this sheaf are theories in the sense we defined
before.
The sheaf-theoretic statement of our main theorem asserts that this
sheaf is isomorphic to the sheaf of local action functionals. As always, this
isomorphism depends on the choice of a renormalization scheme. The renor-
malization scheme independent statement of the theorem is that the sheaf
of theories defined modulo
n+1
is a torsor over the sheaf of theories defined
modulo
n,
for the sheaf of local action functionals on M.
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