Hikita-Nakajima conjecture for the Gieseker variety
Year: 2025
Type: article
Abstract: Let $${\mathfrak {M}}_0$$ M 0 be an affine Nakajima quiver variety, and let $${\mathcal {M}}$$ M be the corresponding BFN Coulomb branch. Assume that $${\mathfrak {M}}_0$$ M 0 can be resolved by the (smooth) Nakajima quiver variety $${\mathfrak {M}}$$ M . The Hikita-Nakajima conjecture claims that there should be an isomorphism of (graded) algebras $$H^*_{S}({\mathfrak {M}},{\mathbb {C}}) \simeq {\mathbb {C}}[{\mathcal {M}}_{{\mathfrak {s}}}^{{\mathbb {C}}^\times }]$$ H S ∗ ( M , C ) ≃ C [ M s C × ] , where $$S \curvearrowright ~{\mathfrak {M}}_0$$ S ↷ M 0 is a torus acting on $${\mathfrak {M}}_0$$ M 0 preserving the Poisson structure, $${\mathcal {M}}_{{\mathfrak {s}}}$$ M s is the (Poisson) deformation of $${\mathcal {M}}$$ M over $${\mathfrak {s}}=\operatorname {Lie}S$$ s = Lie S , $${\mathbb {C}}^\times $$ C × is a generic one-dimensional torus acting on $${\mathcal {M}}$$ M , and $${\mathbb {C}}[{\mathcal {M}}_{{\mathfrak {s}}}^{{\mathbb {C}}^\times }]$$ C [ M s C × ] is the algebra of schematic $${\mathbb {C}}^\times $$ C × -fixed points of $${\mathcal {M}}_{{\mathfrak {s}}}$$ M s . We prove the Hikita-Nakajima conjecture for $${\mathfrak {M}}={\mathfrak {M}}(n,r)$$ M = M ( 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 $${\mathcal {M}}_{{\mathfrak {s}}}$$ M
Source: Selecta Mathematica
Authors: Vasily Krylov, Pavel Shlykov
Institutions: Harvard University, University of Glasgow
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
Award: G8534380803
OpenAlex ID: W4408963194