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'.

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'.