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.