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

Yongbin Ruan, Yaoxiong Wen+1 more|arXiv (Cornell University)|Nov 15, 2020

Algebraic structures and combinatorial models23 references4 citations

TL;DR

This paper establishes a 3d $υ=2$ mirror symmetry for toric stacks using a novel $K$-theoretic $I$-function with effective level structures. It proves that the $I$-functions of mirror pairs coincide under a mirror map that exchanges Kähler and equivariant parameters and inverts the quantum parameter $q$, resolving a conjecture in 3d physics and extending quantum $K$-theory to toric stacks via fixed-point localization and $q$-difference equations.

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

Motivation & Objective

• To formulate and prove a 3d $υ=2$ mirror symmetry conjecture for toric stacks, incorporating effective level structures from physics.
• To define a global $K$-theoretic $I$-function for the entire toric stack, rather than individual GIT quotients, to resolve chamber dependence issues.
• To establish the mirror map that exchanges Kähler and equivariant parameters and inverts $q$, proving the $I$-functions of mirror pairs are equivalent.
• To extend quantum $K$-theory to toric stacks by introducing a modified $I$-function with $q$-difference equation structure.
• To resolve the non-uniqueness of $I$-functions across chambers by proving fixed-point contributions are chamber-independent.

Proposed method

• Introduces a $K$-theoretic $I$-function for toric stacks using quasimap graph moduli spaces, with contributions localized at $\mathbb{C}^*$-fixed points.
• Defines an effective level structure via the quadratic form $\frac{1}{2}\sum_{j,l} \iota_{ij}\iota_{il}$, which encodes the level structure from 3d $υ=2$ physics.
• Constructs a modified $I$-function $\widetilde{I}^{\text{eff}}(\mathbf{p})$ with exponential prefactors involving $\ln z_i \ln U_i|_{\mathbf{p}} / \ln q$ to handle asymptotic behavior.
• Uses a mirror map $\tau$ that exchanges Kähler and equivariant parameters and sends $q \mapsto q^{-1}$, mapping $I$-functions between mirror pairs.
• Applies $q$-binomial identities and asymptotic analysis to show that $\tau(\widetilde{I}^{\text{eff}}(\mathbf{p}^!))$ matches the asymptotic form of the original $I$-function.
• Employs uniqueness of solutions to $q$-difference equations to conclude that the $I$-functions of mirror pairs are equivalent under the mirror map.

Experimental results

Research questions

• RQ1 Does the $K$-theoretic $I$-function for a toric stack remain invariant across different GIT chambers, despite fixed points appearing in multiple chambers?
• RQ2 Can a global $I$-function for the entire toric stack be defined that unifies contributions from all chambers and captures the correct mirror symmetry?
• RQ3 How does the effective level structure from 3d $υ=2$ physics modify the $K$-theoretic $I$-function and ensure mirror symmetry under parameter duality?
• RQ4 What is the precise form of the mirror map that exchanges Kähler and equivariant parameters and inverts $q$, and does it preserve the $I$-function structure?
• RQ5 Do the $I$-functions of mirror toric stacks satisfy the same $q$-difference equations after applying the mirror map, implying equivalence?

Key findings

• The fixed-point contributions $I_{\mathbf{p},\theta}$ to the $I$-function are independent of the GIT chamber $\theta$, allowing the definition of a global $I$-function $I^{\text{eff}}(\mathfrak{X}) = \sum_{\mathbf{p}} I^{\text{eff}}(\mathbf{p})$ for the toric stack.
• The modified $I$-function $\widetilde{I}^{\text{eff}}(\mathbf{p})$ incorporates exponential prefactors and $q$-Pochhammer symbols to handle asymptotic behavior and poles at $U_i|_{\mathbf{p}} = q^{\mathbb{Z}}$, ensuring analytic control.
• Under the mirror map $\tau$, the $I$-function of the mirror toric stack $\mathfrak{X}^!$ transforms such that $\tau(\widetilde{I}^{\text{eff}}(\mathbf{p}^!))$ matches the asymptotic form of the original $I$-function, up to a prefactor.
• The $I$-functions of mirror pairs are shown to satisfy the same $q$-difference equations, and by uniqueness of solutions, they are equivalent under the mirror map.
• The $I$-function $I^{\text{eff}}(\mathfrak{X})$ is invariant under the mirror map, which exchanges Kähler and equivariant parameters and sends $q \mapsto q^{-1}$, proving the main mirror conjecture.
• The construction resolves the issue of chamber dependence in 2d mirror symmetry by working with the full stack, not individual GIT quotients, and proves the conjecture for toric stacks with effective level structures.