Journal of Symplectic Geometry

Volume 10 (2012)

Number 4

$L_∞$-algebras and higher analogues of Dirac structures and Courant algebroids

Pages: 563 – 599

DOI: http://dx.doi.org/10.4310/JSG.2012.v10.n4.a4

Author

Marco Zambon (Departamento de Matemáticas, Universidad Autónoma de Madrid, Spain)

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 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.

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