EPSRC-SFI - Solving Spins and Strings

近年来，由于对一类特殊的可积系统的研究，人们发现了新的显著的数学结构。这些奇异的代数结构将量子群的标准框架扩展到出现新的意想不到的现象的情况。可积系统具有将其演化方程归结为辅助线性问题即可精确求解的性质。当这些系统与李超代数结合时--也就是存在“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。这在一定程度上是通过申请者的工作建立起来的。例如，描述这些代数的多重(张量)积的所谓的“Hopf”代数获得了非平凡的变形，其结果还没有完全被理解。此外，所讨论的系统表现出从哈密顿公式中看不到的对称性增强。这种“秘密”对称性导致了新奇的更复杂的量在时间演化过程中被守恒。这些效应的完整数学表述尚未形成，人们认为理解数学分支的潜在含义是至关重要的，如代数、几何、纽结和链接不变量的拓扑以及可积系统。本研究项目的目的是了解这种奇异的结构，并利用这种新的理解来解决数学物理和这些连续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泊松结构的“非超局域性”，它支配着可积系统在其半经典近似下的表述。非超局域性使得这些系统的解的代数解释变得非常模糊，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对基本的奇异代数有一个严格的理解。这一领域的任何进展都将对数学界、英国和国际的科学环境产生重大的长期影响。我们计划通过构建一组明确地实现这些奇异代数的作用的不同的“表示”来解决这个问题；特别重要的将是我们所说的“无质量”代数。这些是当参数满足非常特殊的关系时发生的特殊表示，最近被发现在相关的光谱分析中起着关键作用。这将与处理量子超代数和所谓的Bethe ansatz的新技术的开发结合在一起。从这项工作中，我们计划得到关于量子群的新结果，并将它们应用于可积系统中的非超局域问题。该项目的跨学科性质，结合了不同数学领域的思想和技术，将在从群论到几何学(哈密尔顿结构)、拓扑学(纽结不变量、格拉斯曼流形)和组合学(Bethe方程、Baxter算子和Ygians)的广泛主题上产生新的结果。

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).