SPDEQFT: Stochastic PDEs meet QFT: Large deviations, Uhlenbeck compactness, and Yang-Mills

SPDEQFT：随机 PDE 满足 QFT：大偏差、Uhlenbeck 紧致性和 Yang-Mills

• 批准号: EP/Y028090/1
• 负责人: Tom Klose
• 金额: $ 23.84万
• 依托单位: University of Edinburgh
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2024
• 资助国家: 英国
• 起止时间: 2024 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FY028090%2F1
• 关键词: SPDEQFT; Stochastic; PDEs; meet; QFT

The overarching goal of this proposal is (a) to develop novel tools in the field of stochastic partial differential equations (SPDEs) and (b) to apply them to mathematical quantum field theory (QFT), particularly quantum Yang-Mills (YM) theory. The YM measure in the physical 4D space-time describes how elementary particles interact at the subatomic level. Its rigorous mathematical construction, however, has so far eluded substantial progress, and accordingly made the list of famously difficult "Millenium problems." On the other hand, SPDE theory has witnessed a number of recent breakthroughs, notably Hairer's theory of regularity structures, which has allowed to make sense of previously ill-posed, singular equations. This proposal aims to develop new tools in SPDEs and regularity structures to analyse the 2D YM measure. The research programme is structured into three projects: 1. We develop a solution theory for singular non-linear elliptic SPDEs. This significantly extends the scope of equations Hairer's theory allows to treat. At the same time, it provides the right framework to extend Uhlenbeck's compactness theorem to distributions. We use that generalisation to give a new gauge-fixed construction of the 2D YM measure, both on the optimal regularity space and with the natural gauge-invariant observables (Wilson loops) well-defined. 2. We show that singular (elliptic and parabolic) SPDEs can be analysed using classical Kusuoka-Stroock theory. This contributes to our theoretical understanding of regularity structures and, in particular, allows to derive precise Laplace asymptotics for these equations. 3. We prove precise Laplace asymptotics of the 2D YM measure in the low temperature limit. This is a novel insight into its qualitative behaviour and generalises a previous large deviation result, which has been obtained by completely different methods.

该建议的总体目标是（a）在随机部分微分方程（SPDE）和（b）领域中开发新的工具，将它们应用于数学量子场理论（QFT），尤其是量子阳米尔（YM）理论。物理4D时空中的YM度量描述了基本粒子在亚原子水平上的相互作用。然而，到目前为止，其严格的数学结构已经取得了实质性的进步，因此使著名的“千年问题”列出了列表。另一方面，SPDE理论见证了许多最近的突破，尤其是Hairer的规律性结构理论，这些理论允许理解以前不适的单数方面。该提案旨在开发SPDES和规律性结构中的新工具，以分析2D YM度量。该研究计划构成了三个项目：1。我们为单数非线性椭圆形SPDE开发了解决方案理论。这大大扩展了方程范围的范围hairer的理论允许治疗。同时，它提供了正确的框架，以将Uhlenbeck的紧凑性定理扩展到分布。我们使用该概括为2D YM度量的新规定结构提供了最佳的规律性空间以及自然规格不变的可观测值（Wilson Loops）的定义明确定义。 2。我们表明，可以使用经典的kusuoka-trocock理论来分析单数（椭圆形和抛物线）SPDE。这有助于我们对规则性结构的理论理解，尤其是允许为这些方程式得出精确的拉普拉斯渐近性​​。 3。我们证明了在低温极限下的2D YM度量的精确拉普拉斯渐近差。这是对其定性行为的新颖见解，并概括了先前的大偏差结果，这是通过完全不同的方法获得的。