权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Nonultralocality and new mathematical structures in quantum integrability

量子可积性中的非超定域性和新数学结构

基本信息

批准号：
EP/H000054/1
负责人：
Niall MacKay
金额：
$ 44.9万
依托单位：
University of York
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2009
资助国家：
英国
起止时间：
2009 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FH000054%2F1
关键词：
Nonultralocality new mathematical structures quantum

项目摘要

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.

在数学物理学中，一个至关重要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，因此问题是我们能否对方程进行积分，问题是否是“可积的”。广义数学的可积性是指使某些问题以某种方式可解而另一些问题不可解的结构。也许令人惊讶的是，基础物理学大量使用可积模型。对现代基础物理学中心问题的数学分析，粒子物理学规范理论与更具投机性的弦理论之间的关系（“规范/弦对应”），近年来已被证明充满了可积模型-但传统的可积性是指时间演化，在规范理论中，重要的可积性是随着能量尺度的演化而变化的。2有些可积模型是“超局域的”--也就是说，它们的相互作用在局域化方面表现得非常好。大多数量子可积性的数学技巧主要适用于这样的模型。行为不太好的“非超本地”（非UL）模型更难处理，但不断重新出现并增加重要性。有时候，他们的问题可以以一种相当特殊的方式得到解决，但对这些问题缺乏系统的了解。这个项目从几个方面着手解决这个问题。一些涉及试图更好地理解已知的非UL模型的代数结构，有或没有边界，特别是那些，或广义的规范/弦对应。然而，一个新的出发点是采取现有的想法处理非UL模型的抽象和尝试“逆向工程”相关联的非UL可积模型。