Stochastic quantum gauge theories

This proposal aims to solve central open problems in the mathematical foundation of quantum gauge theories (QGTs), an important challenge comprising the Yang-Mills (YM) Millennium Prize Problem.

Introduction

A key outcome of the proposal will be the first constructions in finite volume of 2- and 3-dimensional non-exactly solvable QGTs, with a view towards the physical case of 4 dimensions.

Tools and Methods

The principal tools that will be developed and used to address these problems are in the field of rough analysis, in particular singular stochastic partial differential equations (SPDEs).

Importance of SPDEs

Singular SPDEs appear widely in the study of dynamics with randomness and have seen revolutionary progress in the past decade. By developing new rough analytic methods applicable to QGTs, the proposal will push the frontiers of rough analysis, particularly by:

• Studying discrete approximations of SPDEs
• Introducing novel geometric solution theories
• Linking SPDEs with random matrix theory

Research Background

My research has shown that the stochastic quantisation equations of YM (SYM) can be renormalised in a geometrically faithful way, which has already revealed new properties of the exactly solvable 2D YM measure. This is strong evidence that rough analytic techniques can bring new light to the study of QGTs and render their construction in 2D and 3D finally within reach.

Long-term Projects

The proposal is split into the following three long-term projects:

1. Two-dimensional theories:

   • Solve and identify the invariant measure of SYM for non-trivial principal bundles
   • Prove large N convergence of SYM
   • Construct the non-Abelian YM-Higgs measure in finite volume

2. Three-dimensional theories:

   • Give the first construction of the 3-dimensional YM measure in finite volume
   • Prove a discrete version of the BPHZ renormalisation theorem in regularity structures

3. Axiomatic quantum gauge theory:

   • Formulate and prove the Osterwalder-Schrader reconstruction theorem applicable to QGTs
   • Prove Uhlenbeck's regularity theorem for distributions