Coisotropic Deformation Problem
Generalizing and unifying analogous notions from the symplectic, Poisson and contact settings, coisotropic submanifolds form a distinguished class of sub-objects of the Jacobi category. More specifically a submanifold S ⊂ M is coisotropic wrt a Jacobi structure on L → M if the submodule of sections of L → M vanishing on S is closed wrt the Jacobi bracket. Coisotropic submanifolds play an important rôle in Jacobi geometry as already in symplectic, Poisson and contact geometry. In particular, in the symplectic/contact setting, Lagrangian/Legendrian submanifolds are special instances of coisotropic submanifolds. They appear naturally as zero level sets of moment maps of Hamiltonian group actions on Jacobi manifolds. Furthermore any coisotropic submanifold is canonically equipped with a singular characteristic foliation which allows us to perform singular reduction of Jacobi manifolds.
The interest towards the coisotropic deformation problem is due to the significance of coisotropic submanifolds of Calabi–Yau manifolds in Homological Mirror Symmetry [Kapustin and Orlov 2003]. Since then it has been intensively investigated by several mathematicians in the symplectic, lcs, and Poisson settings. My PhD research has aimed at studying deformation and moduli theory of coisotropic submanifolds in the extended setting of Jacobi manifolds.
By means of higher derived brackets [Voronov 2005], in [arXiv:1410.8446] we have associated an L∞-algebra (unique up to L∞-isomorphisms) with any coisotropic submanifold of a Jacobi manifold. Our new construction extends, embracing together as its own particular instances, all previous similar constructions in [Oh and Park 2005] (symplectic setting), [Cattaneo and Felder 2007] (Poisson setting), and [Lê and Oh 2012] (lcs setting). In addition our construction applies also to the completely new case of coisotropic submanifolds in contact manifolds.
In [arXiv:1410.8446] we have proved that, at the formal level, the deformation problem of a coisotropic submanifold S is controlled by the associated L∞-algebra, even under Hamiltonian equivalence. This means that there is a one-to-one correspondence between formal coisotropic deformations of S and formal Maurer–Cartan (MC) elements of the L∞-algebra, which moreover intertwines the Hamiltonian equivalence of the former with the gauge equivalence of the latter. Our result extends to the more general setting of Jacobi manifolds similar results previously obtained in the symplectic setting [Oh and Park 2005], lcs setting [Lê and Oh 2012] and Poisson setting [Schätz and Zambon 2012]. So additionally our result also applies to the coisotropic deformation problem in the contact setting.
Since the L∞-algebra only depends on the ∞-jet of the Jacobi structure along S, generically it fails to convey any information about non-formal coisotropic deformations of S. However in [arXiv:1410.8446] we have proved that, if the Jacobi structure is fiberwise entire along S, the L∞-algebra also encodes information about both the space of non-formal coisotropic deformations of S and its moduli space under Hamiltonian equivalence. Notice that such (generically non-trivial) condition holds in the lcs/contact case because of the coisotropic embedding theorems à la Gotay. This result generalizes and sharpens a similar one for the Poisson case [Schätz and Zambon 2012].
In the contact setting, the multibrackets of the L∞-algebra of S have been explicitly expressed, in [arXiv:1410.8446], in terms of the intrinsic pre-contact geometry of S and the geometry transversal to characteristic distribution of S. Our result extends and clarifies similar expressions obtained in [Oh and Park 2005] (symplectic case) and [Lê and Oh 2012] (lcs case). As a by-product I have constructed, in the contact setting, two concrete examples of coisotropic submanifolds whose deformation problem is obstructed (cf. [PhD thesis, Sections 4.8 and 4.9]).
Finally, again by means of higher derived brackets, I have constructed an extension of the L∞-algebra of S. Further I prove that this extended L∞-algebra controls, at the formal level, the simultaneous coisotropic deformation problem of S, i.e. the problem of deforming both the coisotropic submanifold and the ambient Jacobi structure (cf. [PhD thesis, Section 3.9]).
The BRST-BFV formalism was created to handle physical systems with symmetries or constraints, and it is very useful in both classical and quantum physics. In [arXiv:1601.04540], for any coisotropic submanifold S of a Jacobi manifold, we have constructed the associated BFVcomplex. The latter is a certain differential graded Lie algebra (dgLa), unique up to isomorphisms, which can also be seen as a graded Jacobi manifold additionally equipped with a homological Hamiltonian derivation. Our construction is entirely based on standard techniques of homological perturbation theory (which go back to R. Brown and J. Stasheff), so it simplifies and generalizes previous similar constructions restricted to the Poisson setting [Bordemann 2000; Herbig 2007; Schätz 2009].
In [arXiv:1601.04540], extending to the Jacobi setting what first obtained in the Poisson case [Schätz 2009], we construct an L∞ quasi-isomorphism between the L∞-algebra and the BFV-complex of S via homotopy transfer along a certain contraction data. As a consequence L∞-algebra and BFV-complex of S control equally well the formal coisotropic deformation problem of S under Hamiltonian equivalence.
Differently from the L∞-algebra, without any restrictive hypothesis, the BFV-complex of S controls the coisotropic deformation problem of S, at the non-formal level, even under Hamiltonian/Jacobi equivalence. Actually in [arXiv:1601.04540] we have proved that there exists a one-to-one correspondence between (non-formal) coisotropic deformations of S and “geometric” MC-elements of the associated BFV-complex. Moreover such one-to-one correspondence intertwines Hamiltonian/Jacobi equivalence of the former with gauge/extended-gauge equivalence of the latter. This result generalizes and slightly improves what first obtained in the Poisson case [Schätz 2009 and 2011].
Finally, in [PhD thesis, Section 6.4], I have applied this approach via BFV-complex to a concrete example of coisotropic submanifold S in a contact manifold. In this particular example I have got an explicit description of the space of (non-formal) coisotropic deformations of S and a proof that the coisotropic deformation problem of S is obstructed.
Rigidity of integral coisotropic submanifolds of contact manifolds (cf. [arXiv:1605.00411]).
The above examples show that, unlike Legendrian submanifolds, the deformation problem of a coisotropic submanifold in a contact manifold can be obstructed. Starting from this observation, I have single out in the contact setting the special class of integral coisotropic submanifolds as the direct generalization of Legendrian submanifolds for what concerns deformation and moduli theory. Indeed, being integral coisotropic is proved to be a rigid condition, and moreover the integral coisotropic deformation problem is unobstructed with discrete moduli space. These results have been closely inspired by the work [Ruan 2005] about the special class of integral coisotropic submanifolds in the symplectic category.