1. The Deformation $L_\infty$ algebra of a Dirac--Jacobi structure
- Author
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Tortorella, Alfonso Giuseppe
- Subjects
Mathematics - Differential Geometry ,Mathematics - Quantum Algebra ,Mathematics - Symplectic Geometry ,53D10, 53D17, 58H15, 17B70, 58A50, 17B63 - 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 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)., Comment: 34 pages, comments welcome!
- Published
- 2021