Remarks on large N field theories

The nonperturbative aspects of quantum fields related to the large N limit are described. First of all, the nature of the large N limit from the view point of Euclidean path integral is discussed. The large N limit is always dominated by the nontrivial saddle point S = S0. Infinite varieties of different points on the action orbit can be seen, and if the symmetry group is sufficiently large, the action orbit is virtually covered asymptotically in the large N limit. The saddle point action orbit was determined by the Fadeev-Popov method. The saddle-point action orbit cannot be exhausted by the usual internal symmetry transformation. In the case of planar limit, the importance of higher symmetry structure is discussed. The examples of such higher symmetry for general matrix field theory including Yang-Mills theory are shown. This higher symmetry implies the existence of a continuously infinite number of conserved currents which are nonlocal and non-polynomial with respect to the fields. (Kato, T.)