Continuum limit of four-dimensional lattice Yang-Mills theory
In plain words
Yang-Mills theory can be defined on a grid of points, and the task is to show that it has a sensible limit as the grid spacing goes to zero. The limit should be interacting, not a free theory.
Precise statement
Take $\mathrm{SU}(N)$ lattice gauge theory with the Wilson action on $(\epsilon \mathrm{Z})^4$, with bare coupling chosen by asymptotic freedom, $g(\epsilon)^2 \sim 1/(2 b_0 \operatorname{log}(1/(\epsilon \Lambda)))$ with $b_0 = 11 N/(48 \pi^2)$, i.e. $1/g^2 \sim (11 N/(24 \pi^2)) \operatorname{log}(1/(\epsilon \Lambda))$. Prove that correlations of gauge-invariant observables (smoothed Wilson loops or smeared $F^2$) converge as $\epsilon \to 0$ in infinite volume to a non-Gaussian limit satisfying the Osterwalder-Schrader axioms. Answer: a proof.
What would settle it
A rigorous renormalization-group or probabilistic proof of the infinite-volume continuum limit with non-Gaussian correlations.
Status in the literature
Balaban's renormalization-group program of the 1980s gave ultraviolet stability bounds in finite volume but did not complete the construction.