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

本提案的总体目标是(a)开发随机偏微分方程（SPDEs）领域的新工具，(b)将它们应用于数学量子场论（QFT），特别是量子杨-米尔斯（YM）理论。物理四维时空中的YM测量描述了基本粒子在亚原子水平上的相互作用。然而，它严谨的数学结构至今没有取得实质性进展，因此被列入了著名的“千年难题”名单。另一方面，SPDE理论见证了最近的一些突破，特别是harer的规则结构理论，它使以前的病态奇异方程变得有意义。该提案旨在开发spde和规则结构中的新工具来分析二维YM测量。研究方案分为三个项目：建立了奇异非线性椭圆型SPDEs的解理论。这大大扩展了海勒理论允许处理的方程的范围。同时，它为将Uhlenbeck的紧性定理扩展到分布提供了一个合适的框架。我们利用这一推广给出了二维YM测度的一个新的规范固定结构，既在最优正则性空间上，也在定义良好的自然规范不变可观测值（威尔逊循环）上。2. 我们证明了奇异（椭圆型和抛物型）spde可以用经典Kusuoka-Stroock理论进行分析。这有助于我们对正则结构的理论理解，特别是可以推导出这些方程的精确拉普拉斯渐近性。3. 我们证明了二维YM测度在低温极限下的精确拉普拉斯渐近性。这是对其定性行为的新颖见解，并推广了以前通过完全不同的方法获得的大偏差结果。

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.