Homological Mirror Symmetry for Homogeneous Spaces

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

The principal investigator's main research interests lie in symplectic topology. Symplectic topology is concerned with global structures of symplectic manifolds, a class of spaces that appeared first as the phase space in classical mechanics. Gromov developed a theory of pseudoholomorphic curves, which has led to a number of advancements in symplectic topology, including a class of symplectic invariants now known as Gromov-Witten invariants. More sophisticated invariants, namely Fukaya categories, are the subject of mainstream research. Much of this research arises from predictions made by physicists under the name of mirror symmetry, an area of both mathematics and physics that remains largely conjectural. These conjectures are striking as they have been understood to predict a rather general correspondence between symplectic geometry and algebraic geometry - a huge field of mathematics whose roots go back to ancient times. Various questions one may ask about the geometry of a symplectic manifold can be answered by studying instead the algebraic geometry of a different manifold (its "mirror"), and vice-versa. This project will be concerned with a study of a mirror theory to the classical Bott-Borel-Weil construction in algebraic geometry. Bott-Borel-Weil gives a geometric description of all finite dimensional, irreducible representations of semi-simple Lie groups in terms of equivariant vector bundles on the corresponding homogeneous spaces. The latter are Fano varieties, for which Kontsevich's homological mirror symmetry has been studied extensively over the past decade. In particular, the predicted mirror partner to such varieties is an explicitly known Landau-Ginzburg model. Homological mirror symmetry suggests that there should be Lagrangian submanifolds, mirroring the equivariant vector bundles, and Floer cohomology of pairs of such Lagrangians should form the underlying vector space of a representation of the Lie group. In a recent work (jointly with J. Pascaleff), PI defined the notion of an equivariant Lagrangian brane where equivariance is to be understood with respect to an algebraic action of a Lie algebra on the mirror variety. In addition, the simplest non-trivial example of the aforementioned mirror theory to Bott-Borel-Weil construction was worked out. The current project will extend these constructions to the case of an arbitrary semisimple Lie algebra with the main motivation being the identification of a canonical basis of representations coming from intersections of Lagrangian submanifolds.