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

Abstract

Following the idea of Aganagic–Okounkov [2], we study vertex functions for hypertoric varieties, defined by K-theoretic counting of quasimaps from P1. 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.