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

Abstract

We introduce a new version of 3d mirror symmetry for toric stacks, inspired by a 3d $\mathcal{N}=2$ abelian mirror symmetry construction in physics. Given some toric data, we introduce the $K$-theoretic $I$-function with effective level structure for the associated toric stack. When a particular stability condition is chosen, it restricts to the $I$-function for the particular toric GIT quotient. The mirror of a toric stack is defined by the Gale dual of the original toric data. We then proved the mirror conjecture that the $I$-functions of a mirror pair coincide, under the mirror map, which switches Kähler and equivariant parameters, and maps $q\mapsto q^{-1}$.

Comments welcome

References in corpus (1)

• On the $K$-theory of smooth toric DM stacks

Cited by in corpus (2)

• Quantum $K$-theory and $q$-Difference equations
• $K$-theoretic quasimap wall-crossing