Quantum Cohomology of a Fano Quiver Moduli Space

TL;DR

This paper computes the quantum cohomology ring of a specific Fano 6-fold quiver moduli space, verifies Dubrovin's Conjecture, and provides Quantum Chevalley formulas for certain subvarieties.

Contribution

It explicitly calculates the quantum cohomology of a Fano quiver moduli space and confirms Dubrovin's Conjecture for this case, advancing understanding of quantum cohomology and derived categories.

Findings

Quantum cohomology ring of the Fano 6-fold $Y$ is explicitly computed.

Quantum Chevalley formulas for Schubert type subvarieties are derived.

Dubrovin's Conjecture is verified for the space $Y$.

Abstract

We consider a prime Fano 6-fold $Y$ of index 3, which is a fine quiver moduli space and a blow down of $\mathrm{Hilb}^3(\mathds{P}^2)$. We calculate the quantum cohomology ring of $Y$ and obtain Quantum Chevalley formulas for the Schubert type subvarieties. The famous Dubrovin's Conjecture relating the quantum cohomology and the derived category is verified for $Y$.