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EPSRC-SFI - Solving Spins and Strings

EPSRC-SFI - 解决旋转和字符串

基本信息

批准号：
EP/S020888/1
负责人：
Alessandro Torrielli
金额：
$ 49.7万
依托单位：
University of Surrey
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2019
资助国家：
英国
起止时间：
2019 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FS020888%2F1
关键词：
EPSRC SFI Solving Spins Strings

项目摘要

In recent years, stemming from the study of a particular class of integrable systems, new remarkable mathematical structures have been discovered. These exotic algebraic constructions extend the standard framework of quantum groups to situations where novel unexpected phenomena are seen to emerge. Integrable systems have the property that their evolution equations can be exactly solved via reduction to an auxiliary linear problem. When these systems are combined with Lie superalgebras - that is, Lie algebras for which there exists a notion of "even" (commuting) and "odd" (anti-commuting) generators - new exciting facts occur. This has been partly established through the work of the applicants. The so called "Hopf" algebra describing multiple (tensorial) products of these algebras, for instance, acquires non-trivial deformations, whose consequences have not yet been fully understood. Furthermore, the systems in question exhibit a symmetry-enhancement which is not manifest from the Hamiltonian formulation. This "secret" symmetry results in novel more complicated quantities being conserved during the time evolution. A complete mathematical formulation of these effects has yet to be developed, and it is believed to be crucial to understand potential implications for branches of mathematics such as algebra, geometry, the topology of knots and link-invariants, and integrable systems.The aim of this research project is to understand such exotic structures, and use this new understanding to attack challenging problems at the interface between Mathematical Physics and these contiguous areas. One such problem is the so-called "non-ultralocality" of Poisson structures, governing the formulation of integrable systems in their semi-classical approximation. Non-ultralocality makes the algebraic interpretation of the solution to these systems dramatically more obscure, and it is a difficult problem which has challenged mathematicians for years. We believe that the key to significant progress in this direction is a rigorous understanding of the underlying exotic algebras. Any progress in this area will have a major long-term impact on the mathematical community, and on the scientific environment in the UK and internationally.We plan to attack the problem by constructing a diverse set of "representations" which explicitly realise the action of these exotic algebras; especially important will be what we call the "massless" ones. These are special representations occurring when the parameters satisfy very particular relations, and have recently been found to play a crucial role in the associated spectral analysis. This will be combined with the development of new techniques to treat quantum superalgebras and the so-called Bethe ansatz. From this work, we plan to derive new results on quantum groups and apply them to the problem of non-ultralocality in integrable systems. The intradisciplinary character of the project, combining ideas and techniques from different areas of mathematics, will lead to new results across a broad range of topics, from group theory to geometry (Hamiltonian structures), topology (knot invariants, Grassmannian manifolds) and combinatorics (Bethe equations, Baxter operators and Yangians).

近年来，通过对一类特殊的可积系统的研究，人们发现了一些新的数学结构。这些奇异的代数结构将量子群的标准框架扩展到了新的意想不到的现象出现的情况。可积系统的演化方程可以通过化为辅助线性问题来精确求解。当这些系统与李超代数相结合时--也就是说，存在“偶”（交换）和“奇”（反交换）生成元概念的李代数--新的令人兴奋的事实出现了。这部分是通过申请人的工作确定的。例如，描述这些代数的多重（张量）乘积的所谓“Hopf”代数获得了非平凡的变形，其后果尚未完全理解。此外，所讨论的系统表现出一种从哈密顿公式中没有表现出来的能量增强。这种“秘密”对称性导致了在时间演化过程中保持的新的更复杂的量。一个完整的数学公式，这些影响还有待开发，它被认为是至关重要的，以了解潜在的影响，数学分支，如代数，几何，拓扑结构的结和链接不变量，和可积系统。并利用这种新的理解来攻击数学物理和这些相邻领域之间的界面上的挑战性问题。一个这样的问题是所谓的“非超局部性”的泊松结构，管理制定的可积系统在其半经典近似。非超局部性使得这些系统的解的代数解释变得更加模糊，这是一个多年来一直挑战数学家的难题。我们认为，在这个方向上取得重大进展的关键是严格理解的基础奇异代数。在这一领域的任何进展将有一个重大的长期影响的数学界，并在英国和国际上的科学环境。我们计划攻击的问题，通过构建一套不同的“表示”，明确实现这些奇异代数的行动，特别重要的是我们所谓的“无质量”的。这些都是特殊的表示时发生的参数满足非常特殊的关系，最近被发现在相关的频谱分析中发挥着至关重要的作用。这将与新技术的发展相结合，以处理量子超代数和所谓的贝特代数。从这项工作中，我们计划推出新的结果量子群，并将其应用到可积系统的非超局部性问题。该项目的跨学科性质，结合了来自不同数学领域的思想和技术，将导致新的结果跨越广泛的主题，从群论几何（哈密顿结构），拓扑学（结不变量，格拉斯曼流形）和组合学（贝特方程，巴克斯特运营商和Yangians）。