The three classical theorems of Lie relating Lie algebras and Lie groups (uniqueness of simply connected Lie group integrating a given Lie algebra, integration of morphisms of Lie algebras to Lie groups,and existence of an integration for any Lie algebra) have been extended to the realm of Lie algebroids and groupoids, the main difference being that Lie's third theorem may fail (obstructions to integrability were described by Crainic and Fernandes). We plan to study the extension of these results further to graded and stacky Lie groupoids and algebroids. Both of this "perpendicular" directions will have several interesting implications for classical objects, such as LA-Courant algebroids and Courant groupoids or 2-groupoids and Lie 2-algebroids.
Many diferent pappers appears in the lasts years trying to explain the integration of Courant algebroids. One of the main difficulties is to match the 2-groupoid maps with the "symplectic" structure. We plan to look into this problem by obtaining the 2-groupoid integrating a Courant algebroid via symplectic reduction. It is expect to find the moment map from the Hamiltonian version of the Courant sigma model. Another open question is to compute the explicit global topological obstructions (as Crainic and Fernandes did for Lie algebroids).
The AKSZ construction defines gauge theories where the space of fields are maps between graded manifolds. The most studied cases are the degree 1, where we obtain the Poisson sigma model, and the Chern-Simons gauge theory, that is a particular case of degree 2. The general degree 2, the Courant sigma model, offers a lot to be explored. Other case to study is when the target is a general graded cotangent bundle, that are non-linear analogs of BF theory as defined by Cattaneo and Mnev. The computations fit in the framework of Batalin-Vilkovisky and the long term objective is to compute new topological invariants for manifolds. The programe include: Gauge fixing, renormalization of the BV-laplacians, perturbative invariants, correlator functions...
Some singular symplectic spaces appear as symplectic reduction of degree preserving maps between graded Q-manifolds. Examples of that include the symplectic groupoid integrating a Poisson manifold and the moduli space of flat connections over a surface. It would be interesting to compute other explicit examples of these singular spaces, including their descriptions as stacks modeled on Lie groupoids.Articles
My PhD thesis, available here.
Here you can find some of my recorded talks:
at Friday fish seminar (Utrech) 09/10/2020
at Junior global Poisson workshop 16/09/2020
at Higher structures and field theory programme (Vienna) 09/09/2020
at International Conference on Poisson Geometry (Rio de Janeiro) 09/12/2019
at V Workshop on Poisson Geometry and Related Topics (Rio de Janeiro) 23/11/2017