Positivity determines the quantum cohomology of the odd symplectic Grassmannian of lines

TL;DR

This paper establishes a positivity condition for the quantum cohomology ring of the odd symplectic Grassmannian of lines, showing it uniquely deforms the classical cohomology ring under certain conditions.

Contribution

It introduces a positivity criterion that ensures the quantum cohomology ring is the unique deformation of the classical cohomology ring for the odd symplectic Grassmannian of lines.

Findings

Positivity condition implies unique quantum deformation

Quantum cohomology ring has negative structure constants

Modification of Fulton's conjecture

Abstract

Let $\mbox{IG}:=\mbox{IG}(2,2n+1)$ denote the odd symplectic Grassmannian of lines which is a horospherical variety of Picard rank 1. The quantum cohomology ring $\mbox{QH}^*(\mbox{IG})$ has negative structure constants. For $n \geq 3$, we give a positivity condition that implies the quantum cohomology ring $\mbox{QH}^*(\mbox{IG})$ is the only quantum deformation of the cohomology ring $\mbox{H}^*(\mbox{IG})$ up to the scaling of the quantum parameter. This is a modification of a conjecture by Fulton.