Continuum construction of 2D Yang–Mills–Higgs and 3D pure Yang–Mills theories

Construct the two-dimensional Yang–Mills–Higgs theory and the three-dimensional pure Yang–Mills theory in the continuum by giving a fully rigorous definition and proof of existence that satisfies an appropriate axiomatic framework (e.g., Osterwalder–Schrader or Wightman conditions) without lattice cutoffs.

Background

The authors survey progress on constructive and lattice approaches to Yang–Mills theory, highlighting known constructions (e.g., 2D non-Abelian pure Yang–Mills) and the role of modern stochastic methods.

They emphasize that while some lower-dimensional cases are tractable, constructing certain continuum gauge theories remains unresolved, even in two and three dimensions for specific models.

Resolving these constructions would be a significant step toward a rigorous nonperturbative understanding of Yang–Mills theories beyond exactly solvable cases.