Mass gap in two-dimensional sigma models

3 problems

• Construct the continuum $2\mathrm{D}$ $O(N)$ sigma model as an interacting theory

  Open

  Zooming out at the scale of the correlation length while the temperature goes to zero should give a continuum quantum field theory with massive interacting particles. Building it rigorously would be a two-dimensional model of the Yang-Mills construction problem.

• Asymptotic-freedom growth of the O(N) correlation length

  Open

  Perturbation theory predicts exactly how fast the distance over which arrows stay aligned grows as the temperature goes to zero, an exponential in the inverse temperature. Proving this formula would confirm asymptotic freedom in a lattice model.

• Exponential decay of correlations in the 2D O(N) model at all temperatures

  Open

  Prove that in a flat grid of three-component (or more) arrows, the correlation between two arrows dies off exponentially with their distance at every temperature above zero. For two-component arrows this is false at low temperature, which is why the claim is subtle.