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 $N=\infty$, in a similar way to the Gross-Witten transition in the $D=1+1$ $SU(N\to \infty )$ 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 $N\ge 5$. Our calculations are not precise enough to determine whether this peak is due to a second-order phase transition at $N=\infty$ 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 $N=\infty$ 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 $N=\infty$ 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.

• Received 17 November 2006

DOI:https://doi.org/10.1103/PhysRevD.74.125010