# The small quantum cohomology of the Cayley Grassmannian

## TL;DR

This paper computes the small quantum cohomology ring of the Cayley Grassmannian, verifies positivity of Gromov-Witten invariants, confirms Golyshev's conjecture O, and supports Dubrovin's conjecture regarding semisimplicity and exceptional collections.

## Contribution

It provides the first explicit computation of the quantum cohomology for the Cayley Grassmannian and verifies several conjectures in this context.

## Key findings

- All Gromov-Witten invariants are non-negative.
- Quantum cohomology is semisimple.
- Existence of an exceptional collection of maximal length.

## Abstract

We compute the small cohomology ring of the Cayley Grassmannian, that parametrizes four-dimensional subalgebras of the complexified octonions. We show that all the Gromov-Witten invariants in the multiplication table of the Schubert classes are non negative and deduce Golyshev's conjecture O holds true for this variety. We also check that the quantum cohomology is semisimple and that there exists, as predicted by Dubrovin's conjecture, an exceptional collection of maximal length in the derived category.