72 2. THEORIES, LAGRANGIANS AND COUNTERTERMS
(1) A super vector bundle E over the field R or C on M, equipped with a
direct sum decomposition E = E1⊕E2 into the spaces of propagating
and non-propagating fields, respectively. We will denote the space
of smooth global sections of E or Ei by E , Ei respectively.
We will let
E1
!
= Γ(M, E1

Dens(M)).
There is an inclusion
E1
!
E1
∨.
(2) An even, A -linear, order two differential operator
DE1 : E1 A E1 A
(where the tensor product is the completed projective tensor prod-
uct).
DE1 must be a generalized Laplacian, which means that the sym-
bol
σ(DE1 ) Γ(T
∗M,
Hom(E, E)) A
must be the identity on E times a smooth family of Riemannian
metrics
g
C∞(T ∗M)

C∞(X).
(Recall that
C∞(X)
A is a subalgebra, as A is the global sections
of a bundle of algebras on X).
(3) A differential operator
D : E1
!
E1.
This operator is required to be symmetric: the formal adjoint
(D
)∗
: E1
!
E1
is required to be equal to D .
(4) Let
DE1

: E1
!

E1!
be the formal adjoint of DE1 . We require that
D DE1

= DE1 D .
We will abuse notation and refer to the entirely of the data of a free
theory on M as E .
The most basic example of this definition is the free scalar field the-
ory, as considered in chapter 2. There, the space E1 of fields is
C∞(M).
The space E2 of non-interacting fields is 0. The operator DE1 is the usual
positive-definite Laplacian operator
C∞(M)

C∞(M).The
operator D is
the identity, where we have used the Riemannian volume element to trivi-
alize the bundle of densities on M, and so to identify E1 with
E1.!
More interesting examples will be presented in chapter 5, when we con-
sider the Batalin-Vilkovisky formalism. The use of graded vector bundles
will be essential in the BV formalism.
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