Brackets in the jet-bundle approach to field theory

In the first part the sh Lie structure of brackets in field theory, described in the jet bundle context along the lines suggested by Gel’fand, Dickey and Dorfman, is analyzed. In the second part, we discuss how this description allows us to find a natural relation between the Batalin-Vilkovisky antibracket and the Poisson bracket. (∗)Alexander-von-Humboldt fellow. On leave of absence from Chargé de Recherches du Fonds National Belge de la Recherche Scientifique at Univerité Libre de Bruxelles. This invited contribution summarizes the talk given by the author at the conference “Secondary Calculus and Cohomological Physics, August 24–31, 1997, Moscow, Russia”. The first part is based on work done in collaboration with R. Fulp, T. Lada and J. Stasheff [BFLS]. The second part is based on work done in collaboration with M. Henneaux [BaHe]. 1 Sh Lie structure of brackets on the horizontal complex 1.1 The horizontal complex as a resolution for local functionals In the approach of Gel’fand, Dickey and Dorfman to functionals in field theory [GeDi1, GeDi2, GeDo1, GeDo2, GeDo3] (see [Dic1] for a review), one replaces local functionals satisfying appropriate boundary conditions by equivalence classes of local functions. Let M be an n-dimensional manifold homeomorphic to bfR with coordinates denoted by x and π : E = M × V → M a trivial vector bundle of fiber dimension k over M . The coordinates of V are denoted by u. Let JE denote the infinite jet bundle of E over M with π E : J E → E and π M : J E → M the canonical projections. The vector space of smooth sections of E with compact support will be denoted ΓE. For each section φ of E, let jφ denote the induced section of the infinite jet bundle JE. The bundle π : JE = M × V ∞ → M (1.1) then has induced coordinates given by (x, u, uai , u a i1i2 , . . . , ). (1.2) Definition 1.1 A local function on JE is the pullback of a smooth function on some finite jet bundle JE, i.e., a composite JE → JE → bfR. In local coordinates, a local function L(x, u) is a smooth function in the coordinates x and the coordinates uI , where the order |I| = r of the multiindex I is less than or equal to some integer p. The space of local functions will be denoted by Loc(E)