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.