Journal of Symplectic Geometry
Volume 10 (2012)
$L_∞$-algebras and higher analogues of Dirac structures and Courant algebroids
Pages: 563 – 599
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 et al.
We associate an $L_∞$-algebra of observables to every higher Dirac structure, extending work of Baez et al. on multisymplectic forms. Further, applying a recent result of Getzler, we associate an $L_∞$-algebra to any manifold endowed with a closed differential form $H$, via a higher analogue of split Courant algebroid twisted by $H$. Finally, we study the relations between the $L_∞$-algebras appearing above.