ar X iv : h ep - t h / 94 12 18 2 v 1 2 0 D ec 1 99 4 BROWN - HET - 974 IFUP - TH 75 / 94 December 1994 SIMPLICIAL CHIRAL MODELS

Principal chiral models on a d-1 dimensional simplex are introduced and studied analytically in the large N limit. The d = 0, 2, 4 and ∞ models are explicitly solved. Relationship with standard lattice models and with few-matrix systems in the double scaling limit are discussed. ⋆ Work supported in part by the Department of Energy under contract DE-FG02-91ER40688Task A The importance of understanding the large N limit of matrix-valued field models cannot be overestimated. Not only is this the basic ingredient of the 1/N expansion in the physically relevant case of QCD, but also the existence of large N criticalities for finite values of the coupling is the starting point for the approach to 2-dimension quantum gravity known as the double scaling limit. Moreover we note that solutions of few-matrix systems may have a direct application to more complex systems in the context of strong-coupling expansion, since they may be reinterpreted as generating functionals for classes of group integrals that are required in strong coupling calculations [1]. Unfortunately our knowledge of exact solutions for the large N limit of unitary matrix models is still limited. After Gross and Witten’s solution [2] of the single-link problem, exact results were obtained only for the external field problem [3,4] and a few toy models (L = 3, 4 chiral chains) [5,6]. In this letter we introduce a new class of lattice chiral models, whose large N behavior can be analyzed by solving an integral equation for the eigenvalue distribution of a single hermitian semi definite positive matrix. The models we are going to study are principal chiral models, with a global U(N)×U(N) symmetry, defined on a (d-1)-dimensional simplex formed by connecting in a fully symmetric way d vertices by (d−1)(d−2) 2 links. The partition function for such a system is defined to be Zd = ∫ d ∏