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Title: 3D Mirror Symmetry for Instanton Moduli Spaces

We prove that the Hilbert scheme ofkpoints on$${\mathbb {C}}^2$$C2($$\hbox {Hilb}^k[{\mathbb {C}}^2]$$Hilbk[C2]) is self-dual under three-dimensional mirror symmetry using methods of geometry and integrability. Namely, we demonstrate that the corresponding quantum equivariant K-theory is invariant upon interchanging its Kähler and equivariant parameters as well as inverting the weight of the$${\mathbb {C}}^\times _\hbar $$Cħ×-action. First, we find a two-parameter family$$X_{k,l}$$Xk,lof self-mirror quiver varieties of type A and study their quantum K-theory algebras. The desired quantum K-theory of$$\hbox {Hilb}^k[{\mathbb {C}}^2]$$Hilbk[C2]is obtained via direct limit$$l\longrightarrow \infty $$land by imposing certain periodic boundary conditions on the quiver data. Throughout the proof, we employ the quantum/classical (q-Langlands) correspondence between XXZ Bethe Ansatz equations and spaces of twisted$$\hbar $$ħ-opers. In the end, we propose the 3d mirror dual for the moduli spaces of torsion-free rank-Nsheaves on$${\mathbb {P}}^2$$P2with the help of a different (three-parametric) family of type A quiver varieties with known mirror dual.

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Publisher / Repository:
Springer Science + Business Media
Date Published:
Journal Name:
Communications in Mathematical Physics
Page Range / eLocation ID:
p. 1005-1068
Medium: X
Sponsoring Org:
National Science Foundation
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