Strong to weak coupling transitions of SU(N) gauge theories in 2+1 dimensions

We find a strong-to-weak coupling crossover in D=2+1 SU(N) lattice gauge theories that appears to become a third-order phase transition at N=infinity, in a similar way to the Gross-Witten transition in the D=1+1 SU(N ->infinity) lattice gauge theory. There is, in addition, a peak in the specific heat at approximately the same coupling that increases with N, which is connected to Z(N) monopoles (instantons), reminiscent of the first-order bulk transition that occurs in D=3+1 lattice gauge theories for N >= 5. Our calculations are not precise enough to determine whether this peak is due to a second-order phase transition at N=infinity or to the third-order phase transition having a critical behavior different to that of the Gross-Witten transition. We show that as the lattice spacing is reduced, the N=infinity gauge theory on a finite 3-torus appears to undergo a sequence of first-order Z(N) symmetry breaking transitions associated with each of the tori (ordered by size). We discuss how these transitions can be understood in terms of a sequence of deconfining transitions on ever-more dimensionally reduced gauge theories. We investigate whether the trace of the Wilson loop has a nonanalyticity in the coupling at some critical area, but find no evidence for this. However we do find that, just as one can prove occurs in D=1+1, the eigenvalue density of a Wilson loop forms a gap at N=infinity at a critical value of its trace. We show that this gap formation is in fact a corollary of a remarkable similarity between the eigenvalue spectra of Wilson loops in D=1+1 and D=2+1 (and indeed D=3+1): for the same value of the trace, the eigenvalue spectra are nearly identical. This holds for finite as well as infinite N; irrespective of the Wilson loop size in lattice units; and for Polyakov as well as Wilson loops.