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 $q\mapsto q^{-1}$.

arXiv commentComments welcome

References in corpus (10)

• Quantum Groups and Quantum Cohomology
• Mirror Symmetry and Toric Geometry in Three-Dimensional Gauge Theories
• On the $K$-theory of smooth toric DM stacks
• 3d Mirror Symmetry and Elliptic Stable Envelopes
• qKZ/tRS Duality via Quantum K-Theoretic Counts
• 3d Mirror Symmetry and Quantum $K$-theory of Hypertoric Varieties
• Permutation-equivariant quantum K-theory V. Toric $q$-hypergeometric functions
• Permutation-equivariant quantum K-theory II. Fixed point localization
• On the WDVV-equation in quantum K-theory
• K-theoretic quasimap invariants and their wall-crossing

Cited by in corpus (3)

• 3d Mirror Symmetry and Elliptic Stable Envelopes
• Quantum $K$-theory and $q$-Difference equations
• $K$-theoretic quasimap wall-crossing