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)SK

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

CX

∞,

satisfying the following properties:

(1) A is locally free of finite rank as a CX ∞-module. 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 = CX

∞,

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 CX

∞-modules.

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.