论文标题
可集成的系统，变量的分离和杨巴克斯特方程
Integrable systems, separation of variables and the Yang-Baxter equation
论文作者
论文摘要
本文基于作者的博士学位论文，回顾了量子集成性领域的最新进步，特别是用于高级可整合旋转链的变量（SOV）程序的分离以及解决Yang-Baxter方程的增强机制。我们从量子整合系统的一般概述开始，特别强调了它们在量子代数方面的描述。然后，我们提供$ \ mathfrak {gl}（n）$的Yangian的详细说明，尤其是Bethe代数，Fusion和T-和Q-Systems。然后，我们介绍了可集成系统中变量分离的概念，并基于Sklyanin在等级1模型中的工作，并扩展到更高的等级。通过利用SoV和量子代数表示理论之间的新链接，我们为$ \ Mathfrak {gl}（n）$旋转链构建了分离的变量，用于对称代数的任意紧凑表示，并在途中开发各种新工具。接下来，我们基于上一部分，并开发了一种新技术，用于在SOV框架中计算标量产品，我们称之为功能性SOV或FSOV。与上一个操作员的工作不同，该方法基于Baxter TQ方程。在开发了这项技术之后，我们将其补充新的操作员建设，从而提供了功能和操作型SOV的统一视图。然后，我们将上一部分的结果从紧凑的旋转链到非压缩旋转链中概括。这项工作的最后一部分是基于开发解决杨特克斯方程的工具。我们根据所谓的增强自动形态为此开发了一种自下而上的方法，并将自旋链汉密尔顿作为起点。我们的方法使我们能够对众多解决方案家庭进行分类，尤其是对4 x 4解决方案的完整分类，这些解决方案保留了在ADS/CFT通信中具有应用的费米昂编号。
This article, based on the author's PhD thesis, reviews recent advancements in the field of quantum integrability, in particular the separation of variables (SoV) program for high-rank integrable spin chains and the boost mechanism for solving the Yang-Baxter equation. We begin with a general overview of quantum integrable systems with special emphasis on their description in terms of quantum algebras. We then provide a detailed account of the Yangian of $\mathfrak{gl}(n)$ in particular the Bethe algebra, fusion, and T- and Q-systems. We then introduce the notion of separation of variables in integrable systems and build on Sklyanin's work in rank 1 models and extend to higher rank. By exploiting a novel link between SoV and quantum algebra representation theory we construct the separated variables for $\mathfrak{gl}(n)$ spin chains for arbitrary compact representations of the symmetry algebra and develop various new tools along the way. Next, we build on the previous part and develop a new technique for the computation of scalar products in the SoV framework which we call Functional SoV or FSoV. Unlike the work in the previous, operatorial, part this approach is based on the Baxter TQ equations. After developing this technique we supplement it with a new operator construction providing a unified view of functional and operatorial SoV. Then, we generalise the results of the previous part from compact spin chains to non-compact spin chains. The final part of this work is based on the development of tools for solving the Yang- Baxter equation. We develop a bottom-up approach for this based on the so-called Boost automorphism and uses the spin chain Hamiltonian as a starting point. Our approach allows us to classify numerous families of solutions in particular a complete classification of 4 x 4 solutions which preserve fermion number which have applications in the AdS/CFT correspondence.
