Quantum Representation of Affine Weyl Groups and Associated Quantum Curves

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.

References in corpus (8)

• Webs of five-branes and N=2 superconformal field theories
• Operators and higher genus mirror curves
• Superconformal Chern-Simons Theories from del Pezzo Geometries
• E-string Quantum Curve
• Quantum spectral problems and isomonodromic deformations
• CFT approach to the $q$-Painlevé VI equation
• An Elliptic Garnier System from Interpolation
• Twisted reduction of quiver W-algebras

Cited by in corpus (5)

• Quantum spectral problems and isomonodromic deformations
• New Chandrasekhar transformation in Kerr spacetime
• Duality Cascades and Affine Weyl Groups
• Duality Cascades and Parallelotopes
• Non-Stationary Difference Equation and Affine Laumon Space: Quantization of Discrete Painlevé Equation