3d Mirror Symmetry and Quantum $K$-theory of Hypertoric Varieties

Abstract

Following the idea of Aganagic--Okounkov \cite{AOelliptic}, we study vertex functions for hypertoric varieties, defined by $K$-theoretic counting of quasimaps from ${\mathbb{P}}^1$. We prove the 3d mirror symmetry statement that the two sets of $q$-difference equations of a 3d hypertoric mirror pairs are equivalent to each other, with Kähler and equivariant parameters exchanged, and the opposite choice of polarization. Vertex functions of a 3d mirror pair, as solutions to the $q$-difference equations, satisfying particular asymptotic conditions, are related by the elliptic stable envelopes. Various notions of quantum $K$-theory for hypertoric varieties are also discussed.

40 pages. v2: minor changes

References in corpus (2)

• The Hirzebruch--Riemann--Roch theorem in true genus-0 quantum K-theory
• Elliptic canonical bases for toric hyper-Kahler manifolds

Cited by in corpus (8)

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• Quantum K theory of symplectic Grassmannians
• 3d $\mathcal{N}=4$ Gauge Theories on an Elliptic Curve
• 3d Mirror Symmetry for Instanton Moduli Spaces
• Bow varieties---geometry, combinatorics, characteristic classes
• Quantum $K$-theory of toric varieties, level structures, and 3d mirror symmetry
• Virtual Coulomb branch and vertex functions
• Quantum differential and difference equations for ${\mathrm{Hilb}}^n({\mathbb{C}}^2)$