Okounkov’s conjectural formula for quantum multiplication in equivariant quantum cohomology

Prove Okounkov’s conjecture that for any conical symplectic resolution X equipped with a Hamiltonian torus action and conical C×-action, the quantum multiplication by a divisor u ∈ H^2_G(X) acts on H^•_G(X) as u⋆_q(−) = u∪(−) + Σ_{β∈Φ^+} (q^β/(1−q^β))·ℏ·L_β(−), where L_β(−) are Steinberg correspondence operators in H^{BM}_{2d}(X×_{X_0}X) and Φ^+ is the finite set of positive Kähler roots; the parameter q lies in T^k away from the discriminantal arrangement.

Background

Equivariant quantum cohomology for conical symplectic resolutions is a deformation of equivariant cohomology by effective curve classes. Okounkov proposed a universal, rational-operator formula for the action of divisors in quantum multiplication, expressed via Kähler roots and Steinberg correspondences.

The paper focuses on hypertoric varieties where explicit formulas are known, but the general validity of Okounkov’s formula across conical symplectic resolutions remains conjectural and is a central open question in the area.