Quantum Representation of Affine Weyl Groups and Associated Quantum Curves
Quantum Representation of Affine Weyl Groups and Associated Quantum Curves
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DOI：
10.3842/sigma.2021.076
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发表时间：
2021-04
期刊：
影响因子：
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通讯作者：
Sanefumi Moriyama;Y. Yamada
中科院分区：
文献类型：
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作者：
Sanefumi Moriyama;Y. Yamada
本文研究了主要类型为$E_8^{(1)}$的仿射Weyl群的量子（非交换）表示，其中该表示是由两个变量$x$, $y$上具有$q$对易关系的双生作用给出的。使用tau变量，我们还构建了量子“基本”多项式$F(x,y)$，它完全控制Weyl群的行为。交换情况下多项式$F(x,y)$的几何性质在量子情况下作为$q$差分算子被明显地提升到某些奇异结构。这一性质被进一步用作量子多项式$F(x,y)$的表征。作为一种应用，本文利用Weyl群对称重新推导了第一作者最近提出的拓扑弦相关量子曲线。类型的情况下美元D_5 ^ {(1)} $, $ E_6 ^ {(1)} $, $ E_7 ^{(1)} $也进行了讨论。
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.