Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 076, 24 pages      arXiv:2104.06661      https://doi.org/10.3842/SIGMA.2021.076

Quantum Representation of Affine Weyl Groups and Associated Quantum Curves

Sanefumi Moriyama a and Yasuhiko Yamada b
a) Department of Physics/OCAMI/NITEP, Osaka City University, Sugimoto, Osaka 558-8585, Japan
b) Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan

Received May 13, 2021, in final form August 04, 2021; Published online August 15, 2021

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.

Key words: affine Weyl group; quantum curve; Painlevé equation.

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