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∞-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∞-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.2010 Mathematics Subject Classification: 53D17, 17B55 .Recall that Multi-Dirac structures were recently introduced by Vankerschaver, Yoshimura and Marsden [33]. They are the geometric structures that allow to describe the implicit Euler-Lagrange equations (equations of motion) of a large class of field theories, which include the treatment of non-holonomic constraints. By the above equivalence, higher Dirac structures thus acquire a field-theoretic motivation. Further, since higher Dirac structures are simpler to handle than Multi-Dirac structures (which contain some redundancy in their definition), we expect our work to be useful in the context of field theory too.The second part of the paper is concerned with the algebraic structure on the observables, which turns out to be an L ∞ -algebra. Further, we investigate an L ∞ -algebra that can be associated to a manifold without any geometric structure on it, except for a (possibly vanishing) closed differential form defining a twist. Recall that a closed 2-form on a manifold M (a 2-cocycle for the Lie algebroid T M ) can be used to obtain a Lie algebroid structure on E 0 = T M × R [14, §1.1], so the sections of the latter form a Lie algebra. Recall also that Roytenberg and Weinstein [32] associated a Lie 2-algebra to every Courant algebroid (in particular to E 1 = T M ⊕ T * M with Courant bracket twisted by a closed 3-form). Recently Getzler [18] gave an algebraic construction which extends Roytenberg and Weinstein's proof.Applying Getzler's result in a straightforward way one can extend the above results to all E p 's.Our main results in the second part of the paper ( §5- §9) are:• Thm. 6.7: the observables associated to an isotropic, involutive subbundle of E p form a Lie p-algebra. • Prop. 8.1 and Prop. 8.4: to E p = T M ⊕ ∧ p T * M and to a closed p + 2-form H on M , one can associate a Lie p + 1-algebra extending the H-twisted Courant bracket. • Thm. 7.1: there is a morphism (with one dimensional kernel) from the Lie algebra associated to E 0 and a closed 2-form into the Lie 2-algebra associated to the Courant algebroid E 1 = T M ⊕ T * M with the untwisted Courant bracket.Rogers [28] observed that there is an injective morphism -which can be interpreted as a prequantization...

Associated to any manifold equipped with a closed form of degree > 1 is an 'L∞-algebra of observables' which acts as a higher/homotopy analog of the Poisson algebra of functions on a symplectic manifold. In order to study Lie group actions on these manifolds, we introduce a theory of homotopy moment maps. Such a map is a L∞-morphism from the Lie algebra of the group into the observables which lifts the infinitesimal action. We establish the relationship between homotopy moment maps and equivariant de Rham cohomology, and analyze the obstruction theory for the existence of such maps. This allows us to easily and explicitly construct a large number of examples. These include results concerning group actions on loop spaces and moduli spaces of flat connections. Relationships are also established with previous work by others in classical field theory, algebroid theory, and dg geometry. Furthermore, we use our theory to geometrically construct various L∞algebras as higher central extensions of Lie algebras, in analogy with Kostant's quantization theory. In particular, the so-called 'string Lie 2-algebra' arises this way. Contents

We consider the problem of deforming simultaneously a pair of given structures. We show that such deformations are governed by an L-infinity algebra, which we construct explicitly. Our machinery is based on Th. Voronov's derived bracket construction. In this paper we consider only geometric applications, including deformations of coisotropic submanifolds in Poisson manifolds, of twisted Poisson structures, and of complex structures within generalized complex geometry. These applications can not be, to our knowledge, obtained by other methods such as operad theory.Comment: 32 pages. Results in Section 2 improved (Lemma 2.6 and Corollaries 2.20, 2.22). Corollary 2.5 and Corollary 2.11 added. Final version, accepted for publicatio