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 ).
