The quantum Hikita conjecture

TL;DR

This paper proposes a quantum version of the Hikita conjecture linking quantized coordinate rings to quantum cohomology, and proves it for hypertoric varieties and Springer resolutions.

Contribution

It formulates and proves a quantum Hikita conjecture for specific classes of symplectic varieties, extending the classical conjecture to the quantum setting.

Findings

Proved the quantum Hikita conjecture for hypertoric varieties.

Proved the quantum Hikita conjecture for Springer resolutions.

Included an analysis of highest weights for quantizations with torus actions.

Abstract

The Hikita conjecture relates the coordinate ring of a conical symplectic singularity to the cohomology ring of a symplectic resolution of the dual conical symplectic singularity. We formulate a quantum version of this conjecture, which relates the quantized coordinate ring of the first variety to the quantum cohomology of a symplectic resolution of the dual variety. We prove this conjecture for hypertoric varieties and for the Springer resolution. Our paper includes an appendix, written by Ben Webster, which studies highest weights for quantizations of symplectic resolutions with isolated torus actions.