The Deformation $L_{\infty }$ algebra of a Dirac--Jacobi structure

Abstract

We develop the deformations theory of a Dirac--Jacobi structure within a fixed Courant--Jacobi algebroid. Using the description of split Courant--Jacobi algebroids as degree $2$ contact $\mathbb{N}Q$ manifolds and Voronov's higher derived brackets, each Dirac--Jacobi structure is associated with a cubic $L_{\infty }$ algebra for any choice of a complementary almost Dirac--Jacobi structure. This $L_{\infty }$ algebra governs the deformations of the Dirac--Jacobi structure: there is a one-to-one correspondence between the MC elements of this $L_{\infty }$ algebra and the small deformations of the Dirac-Jacobi structure. Further, by Cattaneo and Schätz's equivalence of higher derived brackets, this $L_{\infty }$ algebra does not depend (up to $L_{\infty }$-isomorphisms) on the choice of the complementary almost Dirac--Jacobi structure. These same ideas apply to get a new proof of the independence of the $L_{\infty }$ algebra of Dirac structure from the choice of a complementary almost Dirac structure (a result proved using other techniques by Gualtieri, Matviichuk and Scott).

34 pages, comments welcome!

References in corpus (3)

• Formal Deformations of Dirac Structures
• Deformations of Pre-Symplectic Structures: a Dirac Geometry Approach
• Contact Dual Pairs