论文标题
Q-Opers，QQ-Systems和Bethe Ansatz
q-Opers, QQ-Systems, and Bethe Ansatz
论文作者
论文摘要
我们介绍了$（g，q）$ - opers和miura $（g，q）$ - opers的概念，其中$ g $是一个简单连接的复杂简单谎言组，并证明了其结构的一些一般结果。然后，我们在$（g，q）$的集合之间建立一对一的对应关系 - 某种类型的opers和bethe ansatz方程系统的非排效机解决方案集。这可能被视为$ q $ de/im的对应关系之间的量子集成模型（IM）和经典几何对象（$ q $ -Differential方程）之间的光谱之间的对应关系。如果$ \ mathfrak {g} $简单地结束，则您获得的bethe ansatz方程与与量子的xxz-type中出现的方程相吻合，与量子offine offine Algebra $ u_q \ wideHat {\ sathfrak {\ mathfrak {g}} $相关。但是，如果$ \ mathfrak {g} $是非绑扎的，则这些方程对应于与$ u_q {}^l \ wideHat {\ mathfrak {\ mathfrak {g}} $相关的其他可集成模型，其中$^l \ wideHat {\ wideHat {\ mathfrak {\ mathfrak {g}} $是langland的langland（langlands aft and aft and aft and aft bra coft twill（Twiste）。此$ q $ de/im通信中的一个关键要素是$ qq $ - 系统，该系统先前出现在ODE/IM通信研究中，以及类别$ {\ Mathcal O} $的Grothendieck环。
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.
