Mirror symmetry for homogeneous varieties

Mirror symmetry for homogeneous varieties - Clelia Pech

Clelia Pech
Kent University
March 15, 2017
Abstract: In this talk reporting on joint work with K. Rietsch and L. Williams, I will explain a new
version of the construction by Rietsch of a mirror for some varieties with a homogeneous Lie
group action. The varieties we study include quadrics and Lagrangian Grassmannians (i.e.,
Grassmannians of Lagrangian vector subspaces of a symplectic vector space). The mirror takes
the shape of a rational function, the superpotential, defined on a Langlands dual homogeneous
variety. I will show that in the mirror manifold has a particular combinatorial structure called a
cluster structure, and that the superpotential is expressed in coordinates dual to the
cohomology classes of the original variety.
I will also explain how these properties lead to new relations in the quantum cohomology, and a
conjectural formula expressing solutions of the quantum differential equation for LG(n) in terms
of the superpotential. If time allows, I will also explain how these results should extend to a
larger family of homogeneous spaces called `cominuscule homogeneous spaces'.