Construction of three-dimensional Euclidean Yang-Mills theory
In plain words
In three dimensions the short-distance problems of Yang-Mills theory are milder, yet even this case has not been built rigorously. It is the natural stepping stone to four dimensions.
Precise statement
Construct the Euclidean Yang-Mills measure for $G = \mathrm{SU}(2)$ on the torus $T^3$ and on $R^3$ as a limit of lattice or smoothed measures, as a gauge-invariant probability measure on distributional connections modulo gauge, satisfying reflection positivity and, on $R^3$, Euclidean invariance. Answer: a construction with proof; a mass gap in 3D is a further question.
What would settle it
A proof that the lattice or stochastically quantized measures converge to a gauge-invariant limit with the Osterwalder-Schrader properties.
Status in the literature
Local-in-time solutions of the 3D Yang-Mills-Higgs Langevin dynamics were constructed by Chandra, Chevyrev, Hairer and Shen around 2022; the invariant measure itself is not yet constructed.