Chiral chains for lattice quantum chromodynamics at N/sub c/=infinity

We study chiral fields (U/sub i/ in the group U(N)) on a periodic lattice (U/sub i/=U/sub i/+L), with action S1/=(g-italic/sup 2/)..sigma../sup L//sub l/=1Tr(U/sub l/U/sup //sub l/+1+ U/sup //sub l/U/sub l/+1), as prototypes for lattice gauge theories (quantum chromodynamics (QCD)) at N/sub c/=infinity. Indeed, these chiral chains are equivalent to gauge theories on the surface of an L-faced polyhedron (e.g., L=4 is a tetrahedron, L=6 is the cube, and L=infinity 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, and infinity, on the weak-coupling side of the Gross-Witten singularity, which occurs at ..beta..=(g-italic/sup 2/N)/sup -1/=1/4, 1/3, ..pi../8, and 1/2, respectively. We carry out weak and strong perturbation expansions at N/sub c/=infinity to estimate the singular part for all L, and to show confinement (as g/sup 2/N..-->..infinity) and asymptotic freedom (g/sup 2/N..-->..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/sub c/=infinity enjoys this singularity in the transition region from strong to weak coupling.