Higher vector bundles and multi-graded symplectic manifolds

Abstract

A natural explicit condition is given ensuring that an action of the multiplicative monoid of non-negative reals on a manifold F comes from homotheties of a vector bundle structure on F, or, equivalently, from an Euler vector field. This is used in showing that double (or higher) vector bundles present in the literature can be equivalently defined as manifolds with a family of commuting Euler vector fields. Higher vector bundles can be therefore defined as manifolds admitting certain $\N^n$-grading in the structure sheaf. Consequently, multi-graded (super)manifolds are canonically associated with higher vector bundles that is an equivalence of categories. Of particular interest are symplectic multi-graded manifolds which are proven to be associated with cotangent bundles. Duality for higher vector bundles is then explained by means of the cotangent bundles as they contain the collection of all possible duals. This gives, moreover, higher generalizations of the known `universal Legendre transformation' T*E->T*E*, identifying the cotangent bundles of all higher vector bundles in duality. The symplectic multi-graded manifolds, equipped with certain homological Hamiltonian vector fields, lead to an alternative to Roytenberg's picture generalization of Lie bialgebroids, Courant brackets, Drinfeld doubles and can be viewed as geometrical base for higher BRST and Batalin-Vilkovisky formalisms. This is also a natural framework for studying n-fold Lie algebroids and related structures.Comment: 27 pages, minor corrections, to appear in J. Geom. Phy

The category of Z2n-supermanifolds

“…Note that canonical symplectic structures on iterated bundles like T * TM are multihomogeneous (see [GR09]), so one can develop the concept of Courant brackets as homological multi-homogeneous Hamiltonians on such Z n 2 -supermanifolds (see [Roy02]). Observe eventually that the appearance of canonical superizations of n-fold vector bundles as canonical models for Z n 2 -supergeometry, which is a variant of the Batchelor-Gawedzki theorem in standard supergeometry, corresponds well with the fact that classical mechanics has recently been recognized as inextricably associated with some double vector bundle structures and their morphisms (e.g.…”

“…As changes of coordinates respect the multidegree, all coordinate changes are polynomial in nonzero degrees and all products which appear in these polynomials are products of coordinates with degrees having disjoint supports (otherwise we would get degrees with some entries > 1), therefore commuting in the Z n 2 -setting (cf. [GR09]). The problem of ordering disappears and the coordinate changes in the even n-fold vector bundle E remain valid and consistent also for its Z n 2 -superization.…”

Towards integration on colored supermanifolds

“…A simpler but equivalent denition is due to [GR09]. In contrast to the more common extraction of the scalar multiplication from addition in a topological vector space, the authors construct the additions out of the multiplications by scalars, and, more precisely, out of an action on a manifold E of the multiplicative monoid of nonnegative real numbers.…”

Brackets

“…[37,38]) is a simplification of the original categorical concept of a double vector bundle due to Pradines [85], see also [67,52].…”