The Crepant Resolution Conjecture for 3-dimensional flags modulo an involution

TL;DR

This paper verifies the Crepant Resolution Conjecture for a specific 3D flag manifold with an involution, showing the Gromov-Witten potentials of the orbifold and its crepant resolution match under certain transformations.

Contribution

It provides the first check of the conjecture for a 3D flag manifold with an involution, introducing a new crepant resolution as a hypersurface in a Hilbert scheme.

Findings

Gromov-Witten potentials agree after transformations

Constructs a new crepant resolution as a hypersurface in Hilb^2 P^2

Verifies the conjecture in a specific 3D case

Abstract

After fixing a non-degenerate bilinear form on a vector space V we define an involution of the manifold of flags F in V by taking a flag to its orthogonal complement. When V is of dimension 3 we check that the Crepant Resolution Conjecture of J. Bryan and T. Graber holds: the genus zero (orbifold) Gromov-Witten potential function of [F / Z_2] agrees (up to unstable terms) with the genus zero Gromov-Witten potential function of a crepant resolution Y of the quotient scheme F / Z_2, after setting a quantum parameter to -1, making a linear change of variables, and analytically continuing coefficients. The crepant resolution Y (a hypersurface in the Hilbert scheme Hilb^2 P^2) is the projectivization of a novel rank 2 vector bundle over P^2.