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\(L_\infty \)-algebras and higher analogues of Dirac sturctures and Courant algebroids. (English) Zbl 1260.53134

Summary: 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 \(L_\infty\)-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 \(L_\infty\)-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 \(L_\infty\)-algebras appearing above.

MSC:

53D17 Poisson manifolds; Poisson groupoids and algebroids
17B66 Lie algebras of vector fields and related (super) algebras
16W25 Derivations, actions of Lie algebras
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