arXiv subjects
Publications and source records attributed to Rajan Amit Mehta.
We study contact structures on nonnegatively-graded manifolds equipped with homological contact vector fields. In the degree 1 case, we show that there is a one-to-one correspondence between such structures (with fixed contact form) and Jacobi manifolds. This correspondence allows us to reinterpret the Poissonization procedure, taking Jacobi manifolds to Poisson manifolds, as a supergeometric version of symplectization.
We approach Mackenzie's LA-groupoids from a supergeometric point of view by introducing Q-groupoids, which are groupoid objects in the category of Q-manifolds. There is a faithful functor from the category of LA-groupoids to the category of Q-groupoids. We associate to every Q-groupoid a double complex that provides a model for the Q-cohomology of the classifying space. As examples, we obtain models for equivariant Q- and orbifold Q-cohomology, and for equivariant Lie algebroid and orbifold Lie algebroid cohomology. We obtain double complexes associated to Poisson groupoids and groupoid-algebroid "matched pairs".
A VB-algebroid is essentially defined as a Lie algebroid object in the category of vector bundles. There is a one-to-one correspondence between VB-algebroids and certain flat Lie algebroid superconnections, up to a natural notion of equivalence. In this setting, we are able to construct characteristic classes, which in special cases reproduce characteristic classes constructed by Crainic and Fernandes. We give a complete classification of regular VB-algebroids, and in the process we obtain another characteristic class of Lie algebroids that does not appear in the ordinary representation theory of Lie algebroids.
We present a general framework for reduction of symplectic Q-manifolds via graded group actions. In this framework, the homological structure on the acting group is a multiplicative multivector field.
A Q-algebroid is a Lie superalgebroid equipped with a compatible homological vector field and is the infinitesimal object corresponding to a Q-groupoid. We associate to every Q-algebroid a double complex. As a special case, we define the BRST model of a Lie algebroid, which generalizes the BRST model for equivariant cohomology. We extend to this setting the Mathai-Quillen-Kalkman isomorphism of the BRST and Weil models, and we suggest a definition of a basic subcomplex which, however, requires a choice of a connection. Other examples include Roytenberg's homological double of a Lie bialgebroid, Ginzburg's model of equivariant Lie algebroid cohomology, the double of a Lie algebroid matched pair, and Q-algebroids arising from lifted actions on Courant algebroids.
Q-groupoids and Q-algebroids are, respectively, supergroupoids and superalgebroids that are equipped with compatible homological vector fields. These new objects are closely related to the double structures of Mackenzie; in particular, we show that Q-groupoids are intermediary objects between Mackenzie's LA-groupoids and double complexes, which include as a special case the simplicial model of equivariant cohomology. There is also a double complex associated to a Q-algebroid, which in the above special case is the BRST model of equivariant cohomology. Other special cases include models for the Drinfel'd double of a Lie bialgebra and Ginzburg's equivariant Poisson cohomology. Finally, a supergroupoid version of the van Est map is used to give a homomorphism from the double complex of a Q-groupoid to that of a Q-algebroid.