L-infinity algebras and higher analogues of Dirac structures and Courant algebroids

Abstract

We define a higher analogue of Dirac structures on a manifold M. Under a regularity assumption, higher Dirac structures can be described by a foliation and a (not necessarily closed, non-unique) differential form on M, and are equivalent to (and simpler to handle than) the Multi-Dirac structures recently introduced in the context of field theory by Vankerschaver, Yoshimura and Marsden. We associate an L-infinity algebra of "observables" to every higher Dirac structure, extending work of Baez, Hoffnung and Rogers on multisymplectic forms. Further, applying a recent result of Getzler, we associate an L-infinity algebra to any manifold endowed with a closed differential form H, via a "higher analogue of Courant algebroid twisted by H". Finally, we study the relations between the L-infinity algebras appearing above.