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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 E ! 1 = Γ(M, E∨ 1 ⊗ Dens(M)). There is an inclusion E ! 1 ⊂ E ∨ 1 . (2) An even, A -linear, order two differential operator DE 1 : E1 ⊗ A → E1 ⊗ A (where the tensor product is the completed projective tensor prod- uct). DE 1 must be a generalized Laplacian, which means that the sym- bol σ(DE 1 ) ∈ 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 : E ! 1 → E1. This operator is required to be symmetric: the formal adjoint (D )∗ : E ! 1 → E1 is required to be equal to D . (4) Let DE ∗ 1 : E1 ! → E1! be the formal adjoint of DE 1 . We require that D DE ∗ 1 = DE 1 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 DE 1 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 E ! 1 . 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.