Symplectic duality pairing of conical symplectic resolutions

Establish that conical symplectic resolutions come in dual pairs under symplectic duality, with specified geometric and representation-theoretic properties interchanged between the duals.

Background

Symplectic duality is a proposed framework interrelating pairs of conical symplectic resolutions, exchanging structures such as categories, fixed-point data, and enumerative invariants.

The conjecture frames the landscape in which quantum cohomology and representation-theoretic symmetries are studied; resolving it would significantly clarify dualities across examples including hypertoric, quiver, and affine Grassmannian slices.

References

Symplectic resolutions were originally studied by Braden, Licata, Proudfoot and Webster in the context of symplectic duality [BPW_Conical_Resolutions_I,BLPW_Conical_Resolutions_II], where they conjectured that these resolutions should come in dual pairs such that certain properties are interchanged.

— Compactifying the Parameter Space for the Quantum Multiplication for Hypertoric Varieties  (2510.15687 - Peters, 17 Oct 2025) in Introduction

Hikita conjectures an isomorphism of graded rings C[XT]\cong \mathsf{H}\bullet_{\mathrm{sing}(\widetilde{X}!,\mathbb{C}), where $\mathsf{H}_{\mathrm{sing}\bullet(-, \mathbb{C})$ denotes the singular cohomology of a topological space with coefficients in $\mathbb{C}$.

— Derived Enhancements of $T$-fixed subschemes  (2608.24226 - Besson et al., 25 Aug 2026) in Section 1, subsection “Symplectic Duality”