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

该提案的总体目标是：（a）在随机偏微分方程（SPDE）领域开发新的工具;（B）将其应用于数学量子场论（QFT），特别是量子Yang-Mills（YM）理论。物理4D时空中的YM测度描述了基本粒子如何在亚原子水平上相互作用。然而，它严格的数学构造迄今为止还没有取得实质性的进展，因此被列入了著名的“千年难题”名单。另一方面，SPDE理论最近取得了一些突破，特别是Hairer的正则性结构理论，它使以前不适定的奇异方程变得有意义。该建议旨在开发SPDE和规则性结构中的新工具来分析2D YM措施。该研究计划分为三个项目：1。我们发展了奇异非线性椭圆型随机微分方程的解理论。这大大扩展了Hairer理论所能处理的方程的范围，同时，它也为将Uhlenbeck紧性定理扩展到分布提供了正确的框架。我们使用的推广给一个新的规范固定建设的二维YM措施，无论是在最佳的正则性空间和自然规范不变的可观测量（威尔逊循环）定义良好。2.我们发现，奇异（椭圆和抛物）SPDE可以使用经典Kusuoka-Stroock理论进行分析。这有助于我们的理论理解的规律性结构，特别是，可以得到精确的拉普拉斯渐近这些方程。3.我们证明了精确的拉普拉斯渐近的二维YM措施在低温限制。这是一个新的洞察其定性行为，并概括了以前的大偏差的结果，这是通过完全不同的方法获得的。

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.