Vector bundle valued differential forms on $\mathbb NQ$-manifolds

Luca Vitagliano

Vol. 283 (2016), No. 2, 449–482

DOI: 10.2140/pjm.2016.283.449

Abstract

Geometric structures on $\mathbb{N}Q$-manifolds, i.e., nonnegatively graded manifolds with a homological vector field, encode nongraded geometric data on Lie algebroids and their higher analogues. A particularly relevant class of structures consists of vector bundle valued differential forms. Symplectic forms, contact structures and, more generally, distributions are in this class. We describe vector bundle valued differential forms on nonnegatively graded manifolds in terms of nongraded geometric data. Moreover, we use this description to present, in a unified way, novel proofs of known results, and new results about degree-one $\mathbb{N}Q$-manifolds equipped with certain geometric structures, namely symplectic structures, contact structures, involutive distributions (already present in literature), locally conformal symplectic structures, and generic vector bundle valued higher order forms, in particular presymplectic and multisymplectic structures (not yet present in literature).

Keywords

graded manifolds, $\mathbb{N}\mkern-1mu Q$-manifolds, vector bundle valued differential forms, Lie algebroids, Spencer operators

Mathematical Subject Classification 2010

Primary: 53D17, 58A50

Milestones

Received: 14 April 2015

Revised: 14 January 2016

Accepted: 1 March 2016

Published: 22 June 2016