Abstract: The classification of quasi - primary fields is outlined. It is proved that the only conserved quasi - primary currents are the energy - momentum tensor and the O(N)-Noether currents. Derivation of all quasi - primary fields and the resolution of degeneracy is sketched. Finally the limits d = 2 and d = 4 of the space dimension are discussed. Whereas the latter is trivial the former is only almost so. (To appear in the Proceedings of the XXII Conference on Differential Geometry Methods in Theoretical Physics, Ixtapa, Mexico, September 20-24, 1993)
The distribution of quasiprimary fields of fixed classes characterized by their O(N) representations Y and the number p of vector fields from which they are composed at N=infty in dependence on their normal dimension delta is shown to obey a Hardy-Ramanujan law at leading order in a 1/N-expansion. We develop a method of collective fusion of the fundamental fields which yields arbitrary qps and resolves any degeneracy.
O(N) vector sigma models possessing catastrophes in their action are studied. Coupling the limit N - > infinity with an appropriate scaling behaviour of the coupling constants, the partition function develops a singular factor. This is a generalized Airy function in the case of spacetime dimension zero and the partition function of a scalar field theory for positive spacetime dimension.
We develop a constructive method to derive exactly solvable quantum mechanical models of rational (Calogero) and trigonometric (Sutherland) type. This method starts from a linear algebra problem: finding eigenvectors of triangular finite matrices. These eigenvectors are transcribed into eigenfunctions of a selfadjoint Schrödinger operator. We prove the feasibility of our method by constructing a new "\(AG_3\) model" of trigonometric type (the rational case was known before from Wolfes 1975). Applying a Coxeter group analysis we prove its equivalence with the \(B_3\) model. In order to better understand features of our construction we exhibit the \(F_4\) rational model with our method.
We transform the two-matrix model, studied by P.Di Francesco and al., into a normal one-matrix model by identifying a 'formal' integral used by these authors as a proper integral. We show also, using their method, that the results obtained for the resolvent and the density are not reliable.
The critical points of the continuous series are characterized by two complex numbers l_1,l_2 (Re(l_1,l_2)< 0), and a natural number n (n>=3) which enters the string susceptibility constant through gamma = -2/(n-1). The critical potentials are analytic functions with a convergence radius depending on l_1 or l_2. We use the orthogonal polynomial method and solve the Schwinger-Dyson equations with a technique borrowed from conformal field theory.