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

2020/11/15 by Yongbin Ruan, Ruan, Yongbin, Yaoxiong Wen +3
Mathematics · #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2011.07519

openalex publication_date 2020/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce a new version of 3d mirror symmetry for toric stacks, inspired by a 3d 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↦ q^-1.

Citations

• On the K-theory of smooth toric DM stacks
• Quantum Groups and Quantum Cohomology
• Permutation-equivariant quantum K-theory V. Toric q-hypergeometric\n functions
• K-theoretic quasimap invariants and their wall-crossing
• 3d Mirror Symmetry and Elliptic Stable Envelopes

Related

• 3d Mirror Symmetry and Quantum K-theory of Hypertoric Varieties
• Permutation-equivariant quantum K-theory VI. Mirrors
• 3D mirror symmetry in positive characteristic
• Pursuing quantum difference equations II: 3D-mirror symmetry
• The level structure in quantum K-theory and mock theta functions