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

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.