Melting Instantons, Domain Walls, and Large N

Monte Carlo studies of $CP^{N-1}$ sigma models have shown that the structure of topological charge in these models undergoes a sharp transition at $N=N_c\approx 4$. For $N<N_c$ topological charge is dominated by small instantons, while for $N>N_c$ it is dominated by extended, thin, 1-dimensionally coherent membranes of topological charge, which can be interpreted as domain walls between discrete quasi-stable vacua. These vacua differ by a unit of background electric flux. The transition can be identified as the delocalization of topological charge, or "instanton melting," a phenomenon first suggested by Witten to resolve the conflict between instantons and large $N$ behavior. Implications for $QCD$ are discussed.