Gualtieri, Marco

Research interests

Generalized complex (GC) geometry:

The notion of GC Brane or submanifold developed in my early work, which encompasses the notions of complex submanifold, Lagrangian submanifold, as well as the mysterious Kapustin-Orlov coisotropic A-branes, has posed a major challenge for the field, as it is not yet understood how to define homomorphisms between them. In recent work with my PhD student F. Bischoff, I developed a  method for defining the category of such branes, and implemented it for a large class of toric GC manifolds, obtaining a long-sought direct connection with the subject of noncommutative algebraic geometry.  This method uses the recent Gaiotto-Gukov-Witten theory of Brane Quantization in an essential way. 

Generalized Kahler geometry:

Since 1984 it was conjectured that a GK manifold should have a generalized Kahler potential function, that is, its Riemannian metric should depend on essentially one scalar function.  Several dozen papers over the subsequent decades established this but only in a handful of extreme special cases.  A breakthrough came a few years ago when we adopted and greatly generalized Donaldson's point of view on Kahler geometry, where the Kahler potential takes the global form of a Lagrangian submanifold in the twisted holomorphic cotangent bundle.  Together with my PhD student F. Bischoff and physicist M. Zabzine, we solved an important special case (the case of symplectic type) of the almost 40 year-old conjecture by realizing that the GK potential is also a Lagrangian submanifold, but in the holomorphic symplectic Weinstein groupoid integrating the Hitchin Poisson structure.   This finally made it possible to write down interesting examples of GK metrics just as easily as in the classical Kahler case.

Three other major projects:

Geometric construction of Sklyanin and Feigin-Odesskii Algebras

Isomonodromy and Morita Equivalence for Meromorphic connections 

Double symplectic groupoids and generalized Kahler metrics