Nonultralocality and new mathematical structures in quantum integrability

对于数学物理问题，最关键的自然问题是我们能否解决它。因为大多数这样的问题都是用微分方程的语言表达的，所以问题是我们能否对方程积分，问题是否“可积”。可积性的广义数学是一种结构，它使一些问题可以求解，而另一些问题则不能。也许令人惊讶的是，基础物理学大量使用可积模型。现代基础物理学的核心问题，即粒子物理学的规范理论与更具有推测性的弦理论（“规范/弦对应”）之间的关系的数学分析，近年来已被证明充满了可积模型——但传统的可积性指的是时间演化，而规范理论中关键的可积性是与能量尺度的演化有关。一些可积模型是“超局部的”——也就是说，它们在相互作用的局部化方面表现得非常好。大多数量子可积性的数学技术主要适用于这样的模型。表现不佳的“非超局部”（non- ultra - local，非ul）模型更难处理，但不断出现，重要性也越来越高。有时他们的问题可以以一种相当特别的方式解决，但缺乏对它们的系统理解。这个项目从多个方向着手解决这个问题。其中一些涉及尝试更好地理解已知非ul模型（有或没有边界）的代数结构，特别是规范/弦对应的代数结构或由其推广的代数结构。然而，一个新的出发点是采用现有的抽象处理非ul模型的思想，并尝试对相关的非ul可积模型进行“逆向工程”。

The crucial natural question to ask of a problem in mathematical physics is whether we can solve it. Since most such problems are couched in the language of differential equations, the question is therefore of whether we can integrate the equations, of whether the problem is 'integrable'. The generalized mathematics of integrability is that of the structures which make some problems solvable in a way that others are not. Perhaps surprisingly, it turns out that fundamental physics makes much use of integrable models. The mathematical analysis of the central issue in modern fundamental physics, the relationship between the gauge theories of particle physics and the much more speculative string theory (the 'gauge/string correspondence'), has proved to be full of integrable models in recent years - but whereas conventional integrability refers to time-evolution, the crucial integrability in gauge theory is with the evolution of the energy scale.Some integrable models are 'ultralocal' - that is, very well-behaved in terms of the localization of their interactions. Most of the mathematical techniques of quantum integrability apply principally to such models. Less well-behaved, 'nonultralocal' (non-UL) models are harder to handle, but keep re-appearing and increasing in importance. Sometimes their problems can be accommodated in a rather ad-hoc way, but a systematic understanding of them is lacking. This project attacks the problem from a number of directions. Some involve attempting to understand better the algebraic structures underlying known non-UL models, with or without a boundary, especially those of, or generalized from, the gauge/string correspondence. However, a new departure is to take existing ideas for handling non-UL models in the abstract and attempt 'reverse-engineer' the associated non-UL integrable models.