L-infinity Algebras From Multicontact Geometry

Abstract

I define higher codimensional versions of contact structures on manifolds as maximally non-integrable distributions. I call them multicontact structures. Cartan distributions on jet spaces provide canonical examples. More generally, I define higher codimensional versions of pre-contact structures as distributions on manifolds whose characteristic symmetries span a constant dimensional distribution. I call them pre-multicontact structures. Every distribution is almost everywhere, locally, a pre-multicontact structure. After showing that the standard symplectization of contact manifolds generalizes naturally to a (pre-)multisymplectization of (pre-)multicontact manifolds, I make use of results by C. Rogers and M. Zambon to associate a canonical $L_{\infty }$-algebra to any (pre-)multicontact structure. Such $L_{\infty }$-algebra is a multicontact version of the Jacobi bracket on a contact manifold. However, unlike the multisymplectic $L_{\infty }$-algebra of Rogers and Zambon, the multicontact $L_{\infty }$-algebra is always a homological resolution of a Lie algebra. Finally, I describe in local coordinates the $L_{\infty }$-algebra associated to the Cartan distribution on jet spaces.

19 pages, v2: exposition slightly changed. to appear in Diff. Geom. Appl. Comments still welcome!

References in corpus (1)

• L-infinity algebras of local observables from higher prequantum bundles

Cited by in corpus (9)

• Multisymplectic formulation of vielbein gravity. De Donder-Weyl formulation, Hamiltonian (n-1)-forms
• Multicontact formulation for non-conservative field theories
• Contact Lie systems
• Rigidity of integral coisotropic submanifolds of contact manifolds
• The Local Structure of Generalized Contact Bundles
• Practical Introduction to Action-Dependent Field Theories
• Kirillov structures up to homotopy
• Brackets in multicontact geometry and multisymplectization
• The $L_{\infty }$-algebra of a symplectic manifold