On quantum cohomology of Grassmannians of isotropic lines, unfoldings of $A_n$-singularities, and Lefschetz exceptional collections

TL;DR

This paper proves the regularity and generic semisimplicity of the big quantum cohomology rings of symplectic isotropic Grassmannians, linking them to $A_{n-1}$ singularities and suggesting the existence of special exceptional collections in their derived categories.

Contribution

It establishes the regularity and semisimplicity of quantum cohomology rings for $ ext{IG}(2, 2n)$ and connects these rings to $A_{n-1}$ singularities, providing new insights into their structure.

Findings

Quantum cohomology rings of $ ext{IG}(2, 2n)$ are regular.

These rings are generically semisimple.

They correspond to $A_{n-1}$ hypersurface singularities.

Abstract

The subject of this paper is the big quantum cohomology rings of symplectic isotropic Grassmannians $\text{IG}(2, 2n)$. We show that these rings are regular. In particular, by "generic smoothness", we obtain a conceptual proof of generic semisimplicity of the big quantum cohomology for $\text{IG}(2, 2n)$. Further, by a general result of Claus Hertling, the regularity of these rings implies that they have a description in terms of isolated hypersurface singularities, which we show in this case to be of type $A_{n-1}$. By the homological mirror symmetry conjecture, these results suggest the existence of a very special full exceptional collection in the derived category of coherent sheaves on $\text{IG}(2, 2n)$. Such a collection is constructed in the appendix by Alexander Kuznetsov.