arXiv 2201.120572022-01-28math-phhep-thmath.MPnlin.SI

可积系统、分离变量与杨-巴克斯特方程

Integrable systems, separation of variables and the Yang-Baxter equation

Paul Ryan

备注：: 213 pages. Based on the author's PhD thesis submitted to Trinity College Dublin in June 2021. Based on the author's publications arxiv:1810.10996, arxiv:1904.12005, arxiv:1910.13442, arxiv:1911.01439, arxiv:2002.12341, arxiv:2003.04332, arxiv:2010.11231, arXiv:2011.08229

摘要：

本文以作者的博士论文为基础，综述了近年来量子可积性领域的最新进展，特别是高阶可积自旋链的分离变量（SoV）程序和求解Yang-Baxter方程的boost机制。我们首先概述了量子可积系统，特别强调其描述的量子代数。然后，我们提供了一个详细的帐户Yangian的$\mathfrak{gl}(n)$特别是贝特代数，融合，T-和Q-系统。然后，我们引入分离变量的概念，在可积系统和Sklyanin的工作秩1模型和扩展到更高的秩。通过利用SoV和量子代数表示理论之间的一种新的联系，我们构造了$\mathfrak{gl}(n)$自旋链的分离变量，用于对称代数的任意紧致表示，并开发了各种新的工具。接下来，我们在前一部分的基础上开发了一种新的技术，用于在SoV框架中计算标量积，我们称之为Functional SoV或FSoV。与前一部分的工作不同，这一方法是基于Baxter TQ方程的。开发这种技术后，我们补充它与一个新的运营商建设提供一个统一的功能和操作的SoV的视图。然后，我们将前一部分的结果从紧自旋链推广到非紧自旋链。这项工作的最后一部分是基于解决杨-巴克斯特方程的工具的发展。我们开发了一个自底向上的方法，这是基于所谓的Boost自同构，并使用自旋链哈密顿量作为起点。我们的方法使我们能够对许多家庭的解决方案进行分类，特别是一个完整的分类4 × 4的解决方案，保留费米子数，在AdS/CFT对应的应用。

英文摘要：

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.

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。

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