Chiral chains for lattice quantum chromodynamics atNc=∞

We study chiral fields [$U_i$ in the group $\mathrm{U}\mathrm{}\left(N\right)\mathrm{}$] on a periodic lattice ($U_i=U_{i+L}$), with action $S=\left(\frac{1}{g^2}\right)\Sigma {l=1\mathrm{}}^L\mathrm{Tr}(U_l{U}_{l+1\mathrm{}}^{†}+U_l^{†}{U}_{l+1\mathrm{}})$, as prototypes for lattice gauge theories [quantum chromodynamics (QCD)] at $N_c=\infty$. Indeed, these chiral chains are equivalent to gauge theories on the surface of an $L$-faced polyhedron (e.g., $L=4\mathrm{}$ is a tetrahedron, $L=6\mathrm{}$ is the cube, and $L=\infty$ is two-dimensional QCD). The one-link Schwinger-Dyson equation of Brower and Nauenberg, which gives the square of the transfer matrix, is solved exactly for all $N$. From the large-$N$ solution, we solve exactly the finite chains for $L=2, 3, 4, \mathrm{and} \infty$, on the weak-coupling side of the Gross-Witten singularity, which occurs at $\beta ={\left(g^{2}N\right)}^{-1\mathrm{}}=\frac{1}{4}, \frac{1}{3}, \frac{\pi }{8}, \mathrm{and} \frac{1}{2}$, respectively. We carry out weak and strong perturbation expansions at $N_c=\infty$ to estimate the singular part for all $L$, and to show confinement (as $g^{2}N\to \infty$) and asymptotic freedom ($g^{2}N\to 0$) in the Migdal $\beta$ function for QCD. The stability of the location of the Gross-Witten singularity for different-size lattices ($L$) suggests that QCD at $N_c=\infty$ enjoys this singularity in the transition region from strong to weak coupling.