Janusz Grabowski, Mikolaj Rotkiewicz
Higher Vector Bundles and Multi--Graded Symplectic Manifolds
Preprint series: ESI preprints

MSC:

58A50 Supermanifolds and graded manifolds, See also {14A22,

58F05 Hamiltonian and Lagrangian systems; symplectic geometry, See also {70Hxx, 81S10}

58C50 Analysis on supermanifolds or graded manifolds

17B99 None of the above but in this section

18D05 Double categories, $2$-categories, bicategories, hypercategories

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 homoteties 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" $\dts
E\simeq\dts E^*$, 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 D.~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.


Keywords: double vector bundles, graded super-manifolds, Lie bialgebroids, Courant algebroids, Drinfeld doubles, BRST formalism