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

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 Sch\"atz'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).