Quantum K-theory of toric varieties, level structures, and 3d mirror symmetry

Abstract

We introduce a new version of 3d mirror symmetry for the toric quotient stack [Cn/K] where K=(C⁎)k and the torus action is determined by the given charge matrix. It is inspired by a 3d N=2 abelian mirror symmetry construction in physics. Given some toric data, we introduce the K-theoretic I-function with the effective level structure for the associated quotient stack. When a particular stability condition is chosen, it restricts to the I-function for the particular toric variety. The mirror of a GIT quotient stack is defined by the Gale dual of the original toric data. We then prove the mirror conjecture that the I-functions of a mirror pair coincide, under the mirror map, which switches Kähler and equivariant parameters, and changes q to q−1.