3d mirror symmetry and quantum K-theory of hypertoric varieties

Following the idea of Aganagic-Okounkov [2], we study vertex functions for hypertoric varieties, defined by K theoretic counting of quasimaps from 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 Kahler 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. (c) 2021 Elsevier Inc. All rights reserved.

Automated summary

In this study, we investigate the vertex functions for hypertoric varieties and their application in 3D mirror pairs. By examining the equivalence of two sets of q-difference equations, we establish a relationship between the exchanged Kahler and equivariant parameters and the opposite choice of polarization. Additionally, we discuss various notions of quantum K-theory for hypertoric varieties.