Higher vector bundles and multigraded symplectic manifolds
Abstract
A natural explicit condition is given ensuring that an action of the multiplicative monoid of nonnegative 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, multigraded (super)manifolds are canonically associated with higher vector bundles that is an equivalence of categories. Of particular interest are symplectic multigraded 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" TE≃TE^{∗}, identifying the cotangent bundles of all higher vector bundles in duality. The symplectic multigraded 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 BatalinVilkovisky formalisms. This is also a natural framework for studying nfold Lie algebroids and related structures.
 Publication:

Journal of Geometry and Physics
 Pub Date:
 September 2009
 DOI:
 10.1016/j.geomphys.2009.06.009
 arXiv:
 arXiv:math/0702772
 Bibcode:
 2009JGP....59.1285G
 Keywords:

 Mathematics  Differential Geometry;
 58A50;
 53D05 (Primary);
 53D17;
 58C50;
 17B62;
 17B63;
 18D05
 EPrint:
 27 pages, minor corrections, to appear in J. Geom. Phys