13. VECTOR-BUNDLE VALUED FIELD THEORIES 71 Lemma 12.0.2. Let {I(r,s)[P ]} and {I (r,s) [P ]} be two parametrix theories, defined for all (r, s) ≤ (I, K). Suppose that I(r,s)[P ] = I (r,s) [P ] if (r, s) (I, K). Then, J(I,K) = I(I,K)[P ] − I (I,K) [P ] is a local functional, that is, an element of Oloc(C∞(M )). Proof. Note that the renormalization group equation implies that the functional J(I,K) is independent of P . The locality axiom implies that J(I,K) is supported on the small diagonal of M K . Further, J(I,K) has smooth first derivative. Any distribution J ∈ D(M K )S K which is supported on the small diagonal and which has smooth first derivative is a local functional. 13. Vector-bundle valued field theories We would like to have a bijection between theories and Lagrangians for a more general class of field theories. The most general set-up we will need is when the fields are sections of some vector bundle on a manifold and the interactions depend smoothly on some additional supermanifold. In this section we will explain how to do this on a compact manifold. Definition 13.0.1. A nilpotent graded manifold is the following data: (1) A smooth manifold with corners X, (2) A sheaf A of commutative superalgebras over the sheaf of algebras C∞, X satisfying the following properties: (1) A is locally free of finite rank as a C∞-module. X In other words, A is the sheaf of sections of some super vector bundle on X. (2) A is equipped with an ideal I such that A/I = C∞, X and Ik = 0 for some k 0. The ideal I, its powers Il, and the quotient sheaves A/Il, are all required to be locally free sheaves of C∞-modules. X The algebra Γ(X, A) of C∞ global sections of A will be denoted by A . Everything in this section will come in families, parameterized by a nilpotent graded manifold (X, A). 13.1. We are interested in vector-valued theories on a compact manifold M. As in the case of scalar field theories, we will fix the data of the free theory, which gives us our propagator and then consider possible interacting theories which deform this. The following definition aims to be broad enough to capture all of the free field theories used in this book, and in future applications. Unfortunately, it is not particularly transparent. Definition 13.1.1. A free theory on a manifold M consists of the fol- lowing data.

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