arXiv 2301.051472023-01hep-thmath-phmath.MP

论2维共形场论的Bethe方程

On Bethe equations of 2d conformal field theory

Tomáš Procházka, Akimi Watanabe

AI总结：

本文从量子可积性角度研究二维共形场论的高自旋代数，利用代数Bethe ansatz获得ILW hierarchy的无穷交换Hamiltonian族，验证并猜想其联合谱，分析了退化极限与局部Hamiltonian，并揭示了Bethe根的解析结构。

摘要：

我们从量子可积性的角度研究了二维共形场论的高自旋代数。从Maulik-Okounkov瞬子R-matrix出发，并应用代数Bethe ansatz程序，我们获得了由辅助环面的形状参数化的量子ILW hierarchy的无穷交换Hamiltonian族。我们显式计算了这些Hamiltonian的前五个。然后，我们数值验证了它们的联合谱可以由Litvinov的Bethe ansatz方程的解参数化，并且基于Feigin、Jimbo、Miwa和Mukhin的结果，我们猜想了一个适用于所有ILW Hamiltonian联合谱的一般公式。存在两个有趣的退化极限，即无限厚和无限薄的辅助环面。在其中一个极限中，ILW hierarchy退化为Yangian或Benjamin-Ono hierarchy，且Bethe方程可以容易地求解。另一个极限是奇异的，但我们仍然可以提取对应于量子KdV或KP hierarchy的局部Hamiltonian。在这个局部极限下，Litvinov的Bethe方程提供了Bazhanov、Lukyanov和Zamolodchikov的Bethe ansatz方程的一种替代方案，但更为透明，适用于任意秩，并且在$\mathcal{W}_{1+\infty}$的triality对称性下具有明显的对称性。最后，我们阐明了在极小模型中，特别是对于Lee-Yang CFT，Bethe方程解的解析性质。Bethe根的解析结构非常丰富，因为它捕获了$\mathcal{W}_N$极小模型的表示论。

英文摘要：

We study the higher spin algebras of two-dimensional conformal field theory from the perspective of quantum integrability. Starting from Maulik-Okounkov instanton R-matrix and applying the procedure of algebraic Bethe ansatz, we obtain infinite commuting families of Hamiltonians of quantum ILW hierarchy parametrized by the shape of the auxiliary torus. We calculate explicitly the first five of these Hamiltonians. Then, we numerically verify that their joint spectrum can be parametrized by solutions of Litvinov's Bethe ansatz equations and we conjecture a general formula for the joint spectrum of all ILW Hamiltonians, based on results of Feigin, Jimbo, Miwa and Mukhin. There are two interesting degeneration limits, the infinitely thick and the infinitely thin auxiliary torus. In one of these limits, the ILW hierarchy degenerates to Yangian or Benjamin-Ono hierarchy and the Bethe equations can be easily solved. The other limit is singular but we can nevertheless extract local Hamiltonians corresponding to quantum KdV or KP hierarchy. Litvinov's Bethe equations in this local limit provide an alternative to Bethe ansatz equations of Bazhanov, Lukyanov and Zamolodchikov, but are more transparent, work at any rank and are manifestly symmetric under triality symmetry of $\mathcal{W}_{1+\infty}$. Finally, we illustrate analytic properties of the solutions of Bethe equations in minimal models, in particular for Lee-Yang CFT. The analytic structure of Bethe roots is very rich as it captures the representation theory of $\mathcal{W}_N$ minimal models.

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