This article constructs an \(L_\infty\)-algebra governing deformations of a Dirac-Jacobi structure within a fixed Courant-Jacobi algebroid.
Dirac structures are involutive Lagrangian subbundles of Courant algebroids, and are useful in describing and unifying important geometric structures such as (pre)symplectic and Poisson structures, foliations, complex and generalized complex structures.
Dirac-Jacobi structures generalize Dirac structures and unify contact, Jacobi, and generalized contact structures (generalized complex structures in odd-dimensional manifolds). They are involutive Lagrangian subbundles of Courant-Jacobi algebroids, which in turn generalize Courant algebroids. Dirac-Jacobi structures can be thought of as “contact” versions of Dirac structures, via the line bundle approach [L. Vitagliano, J. Symplectic Geom. 16, No. 2, 485–561 (2018; Zbl 1397.53094)] in which one sees contact (respectively Jacobi) geometry as symplectic (respectively Poisson) geometry on the gauge (or Atiyah) algebroid \(DL\) of a line bundle \(L\). This line bundle approach to Courant-Jacobi algebroids and Dirac-Jacobi structures is reviewed in Section 2 of the paper.
The construction of the deformation \(L_\infty\)-algebra of a Dirac-Jacobi structure is inspired by a similar construction of a \(L_\infty\)-algebra for Dirac structures [Y. Frégier and M. Zambon, Compos. Math. 151, No. 9, 1763–1790 (2015; Zbl 1383.17009); D. Roytenberg, Lett. Math. Phys. 61, No. 2, 123–137 (2002; Zbl 1027.53104)]. It uses a graded-geometric interpretation of Courant-Jacobi algebroids and of Dirac-Jacobi structures, and an auxiliary choice of a complementary almost Dirac-Jacobi structure. Sections 3 and 4 of the paper carefully describe the relevant graded (contact) geometry, and use it to construct a curved \(L_\infty\)-algebra associated with a split Courant-Jacobi Lie algebroid, which fully encodes the Courant-Jacobi algebroid structure. This is a cubic curved \(L_\infty\)-algebra, meaning that it only has 0-ary, unary, binary and ternary brackets. In section 5, this description is in turn used to construct, using a complementary almost Dirac-Jacobi structure, the deformation \(L_\infty\)-algebra of a Dirac-Jacobi structure. It is a cubic \(L_\infty\)-algebra, meaning that it only has unary, binary and ternary brackets.
There is one main difference with the case of Dirac structures. To show that the resulting \(L_\infty\)-structure does not depend (up to canonical \(L_\infty\)-isomorphisms) on the choice of complement, the existing proof for the Dirac case used a spinorial description of Dirac structures [M. Gualtieri et al., Int. Math. Res. Not. 2020, No. 14, 4295–4323 (2020; Zbl 1486.53097)], which is not available for Dirac-Jacobi structures. Instead, in Section 5.2 the author proves the independence (up to canonical \(L_\infty\)-isomorphisms) of the deformation \(L_\infty\)-algebra of a Dirac-Jacobi structures via the equivalence of higher derived brackets [A. S. Cattaneo and F. Schätz, J. Pure Appl. Algebra 212, No. 11, 2450–2460 (2008; Zbl 1177.53073)], which is recalled in the appendix of the paper. This provides also an alternative proof for the Dirac case.
Finally, Theorem 5.7 shows that the Maurer-Cartan elements of the deformation \(L_\infty\)-algebra of a Dirac-Jacobi structure \(A\) encode “small” deformations of \(A\) within the fixed Courant-Jacobi algebroid. In Section 5.4, there is a description of infinitesimal deformations of Dirac-Jacobi structures, and of a Kuranishi map \(K\). If \(K(w)\neq 0\) for an infinitesimal deformation \(w\), then \(w\) is obstructed (it cannot be extended to a smooth deformation).