Quantum Representation of Affine Weyl Groups and Associated Quantum Curves

TL;DR

This paper develops a quantum non-commutative representation of affine Weyl groups, especially of type E8^{(1)}, using birational actions and quantum polynomials, and applies it to derive quantum curves related to topological strings.

Contribution

It introduces a novel quantum representation of affine Weyl groups via birational actions and quantum polynomials, extending classical geometric properties to the quantum setting.

Findings

Constructed quantum Weyl group actions with q-commutation relations.

Defined quantum fundamental polynomials controlling Weyl group actions.

Rederived the quantum curve associated with topological strings using Weyl group symmetry.

Abstract

We study a quantum (non-commutative) representation of the affine Weyl group mainly of type $E_8^{(1)}$, where the representation is given by birational actions on two variables $x$, $y$ with $q$-commutation relations. Using the tau variables, we also construct quantum "fundamental" polynomials $F(x,y)$ which completely control the Weyl group actions. The geometric properties of the polynomials $F(x,y)$ for the commutative case is lifted distinctively in the quantum case to certain singularity structures as the $q$-difference operators. This property is further utilized as the characterization of the quantum polynomials $F(x,y)$. As an application, the quantum curve associated with topological strings proposed recently by the first named author is rederived by the Weyl group symmetry. The cases of type $D_5^{(1)}$, $E_6^{(1)}$, $E_7^{(1)}$ are also discussed.