q-Opers, QQ-Systems, 和 Bethe Ansatz
q-Opers, QQ-Systems, and Bethe Ansatz
AI总结：
本文研究q-Opers与Bethe方程的对应关系，提出QQ系统在量子可积模型与经典几何对象间的联系，揭示了简单李代数与非简单李代数下的不同可积模型。
摘要：
我们引入了(G,q)-opers和Miura (G,q)-opers的概念，并证明了其结构的一些一般结果。随后，我们建立了特定类型的(G,q)-opers集合与非退化Bethe方程解集合之间的一一对应关系。这可视为量子可积模型（IM）谱与经典几何对象（q-微分方程）之间的qDE/IM对应关系。如果g是简单利aced的，所得Bethe方程与XXZ型量子可积模型的方程一致；若g是非简单利aced的，则对应于与U_q {}^L\widehat{\mathfrak{g}}相关的不同可积模型。该qDE/IM对应的关键元素是之前在ODE/IM对应和相关量子affine代数的Grothendieck环研究中出现的QQ系统。
英文摘要：
We introduce the notions of $(G,q)$-opers and Miura $(G,q)$-opers, where $G$ is a simply-connected complex simple Lie group, and prove some general results about their structure. We then establish a one-to-one correspondence between the set of $(G,q)$-opers of a certain kind and the set of nondegenerate solutions of a system of Bethe Ansatz equations. This may be viewed as a $q$DE/IM correspondence between the spectra of a quantum integrable model (IM) and classical geometric objects ($q$-differential equations). If $\mathfrak{g}$ is simply-laced, the Bethe Ansatz equations we obtain coincide with the equations that appear in the quantum integrable model of XXZ-type associated to the quantum affine algebra $U_q \widehat{\mathfrak{g}}$. However, if $\mathfrak{g}$ is non-simply laced, then these equations correspond to a different integrable model, associated to $U_q {}^L\widehat{\mathfrak{g}}$ where $^L\widehat{\mathfrak{g}}$ is the Langlands dual (twisted) affine algebra. A key element in this $q$DE/IM correspondence is the $QQ$-system that has appeared previously in the study of the ODE/IM correspondence and the Grothendieck ring of the category ${\mathcal O}$ of the relevant quantum affine algebra.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
