Miquel Cueca


I did my undergrad in Valencia and my PhD at IMPA under the supervision of Henrique Bursztyn. I am currently at Georg-August-Universitat, Gottingen, in the Higher structure group.

My main research interests lie in the area of Mathematical Physics, with focus on Poisson geometry. In my work I use the techniques of Super/Graded geometry to study classical objects that were difficult to describe before, this includes Lie and Courant algebroids, Dirac structures and beyond. Other topics that I am interested in are Lie theory of higher objects, Batalin-Vilkovisky formalism and the quantization process and singular symplectic spaces.

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Research Lines


Lie theory for graded and stacky groupoids

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.

Integration of Courant 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).

AKSZ sigma models

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...

Singular symplectic spaces

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

Publications


  • Courant Cohomology, Cartan Calculus, Connections, Curvature, Characteristic Classes

    with Rajan Mehta, you can find it here.

  • The geometry of graded cotangent bundles

    you can find it on arxiv/1905.13245 or published here.

Preprints


  • Applications of graded manifolds to Poisson geometry

    My PhD thesis, available here.

Final preparation


Talks

Here you can find some of my recorded talks:

Articles

Teaching

Contact information