2022/02/20 by Vasily Krylov, Krylov, Vasily, Pavel Shlykov +1
Mathematics · Physics and Astronomy · #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Nonlinear Waves and Solitons #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2202.09934
openalex publication_date 2022/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \mathfrakM0 be an affine Nakajima quiver variety, and M is the corresponding BFN Coulomb branch. Assume that \mathfrakM0 can be resolved by the (smooth) Nakajima quiver variety \mathfrakM. The Hikita-Nakajima conjecture claims that there should be an isomorphism of (graded) algebras H^*S(\mathfrakM,ℂ) ≃ ℂ[M_\mathfraksℂ^×], here S \curvearrowright \mathfrakM0 is a torus acting on \mathfrakM0 preserving the Poisson structure, M_\mathfraks is the (Poisson) deformation of M over \mathfraks=Lie (S), ℂ^× is a generic one-dimensional torus acting on M, and ℂ[M_\mathfraksℂ^×] is the algebra of schematic ℂ^×-fixed points of M_\mathfraks. We prove the Hikita-Nakajima conjecture for \mathfrakM=\mathfrakM(n,r) Gieseker variety (ADHM space). We produce the isomorphism explicitly on generators. We also describe the Hikita-Nakajima isomorphism above using the realization of M_\mathfraks as the spectrum of the center of rational Cherednik algebra corresponding to Sn \ltimes (ℤ/rℤ)n and identify all the algebras that appear in the isomorphism with the center of degenerate cyclotomic Hecke algebra (generalizing some results of Shan, Varagnolo, and Vasserot).