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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 ! , E ! c , 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 , E v c 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 ).