14. FIELD THEORIES ON NON-COMPACT MANIFOLDS 81

14.1. Now let us start defining the notion of theory on a possibly non-

compact manifold M.

As in Section 13, we will fix a nilpotent graded manifold (X, A), and a

family of free field theories on M parameterized by (X, A). The free field

theory is given by a super vector bundle E on M, whose space of global

sections will be denoted by E ; together with various auxiliary data detailed

in Definition 13.1.1.

We will use the following notation. We will let E denote the space of all

smooth sections of E, Ec the space of compactly supported sections, E the

space of distributional sections and E

c

the space of compactly supported

distributional sections. The bundle E ⊗ DensM will be denoted

E!.

We will

use the notation E

!, Ec!,

E

!

and E

!

c

to denote spaces of smooth, compactly

supported, distributional and compactly supported distributional sections

of the bundle

E!.

sections of We will let E denote the space of distribu-

tional sections of E , and E

c

denote the compactly supported distributional

sections. With this notation, E

∨

= E

!

c

,

Ecv

ee = E

!

, and so on.

14.2.

Definition 14.2.1. Let M, X be topological spaces. A subset C ⊂ M

n

×

X is called proper if each of the projection maps πi : M

n

× X → M × X is

proper when restricted to C.

We say that a section of a vector bundle on M

n

× X has proper support

if its support is a proper subset of M

n

× X.

Recall that we can identify the space O(Ec, A ) of A -valued functions

on Ec (modulo constants) with the completed symmetric algebra

O(Ec, A ) =

n0

Symn(E

!

) ⊗ A ,

where we have identified E

!

– the space of distributional sections of the

bundle

E!

on M – with

(Ec)∨.

We will let

Op(Ec, A ) ⊂ O(Ec, A )

be the subset consisting of those functionals Φ each of whose Taylor com-

ponents

Φn ∈

Symn

E

!

have proper support. We are only interested in functions on Ec modulo

constants.

Note that Op(E , A ) is not an algebra; the direct product of two properly

supported distributions does not necessarily have proper support.

Note also that every A -valued local action functional I ∈ Ol(E , A ) is

an element of Op(E , A ).