$L_∞$-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∞-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...

The Hamilton-Pontryagin principle and multi-Dirac structures for classical field theories

“…Our concept of multi-Dirac structures also includes, and is in fact equivalent to, the so-called higher-order Dirac structures of Ref. 45.…”

Dirac structures for higher analogues of Courant algebroids

“…We refer to [1] for the proof of (14). Conversely, if L is a (p, k)-isotropic subspace satisfying conditions (1) and (2),…”

“…Hence, L cannot be a linear (p, k)-Dirac structure. The construction of η has already been given in [1]. Let us recall it here.…”

“…See [1] for more details about the relation between Dirac structures of order p and L ∞ -algebras. For any closed p + 1-form ω, the graph of the induced bundle map ω : T M → Λ p T * M , which we denote by G ω , is a Dirac structure of order p.…”

“…The p-Courant bracket ·, · p on the space of sections of T p is given by (1), for all X + ξ, Y + η ∈ Γ(T p ). This bracket is obviously skew-symmetric.…”