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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 Hamiltonian of the \(N\)-particle Calogero model can be expressed in terms of generators of a Lie algebra for a definite class of representations. Maintaining this Lie algebra, its representations, and the flatness of the Riemannian metric belonging to the second order differential operator, the set of all possible quadratic Lie algebra forms is investigated. For \(N = 3\) and \(N = 4\) such forms are constructed explicitly and shown to correspond to exactly solvable Sutherland models. The results can be carried over easily to all \(N\).

We present a detailed analysis of a scalar conformal four-point function obtained from AdS/CFT correspondence. We study the scalar exchange graphs in AdS and discuss their analytic properties. Using methods of conformal partial wave analysis, we present a general procedure to study conformal four-point functions in terms of exchanges of scalar and tensor fields. The logarithmic terms in the four-point functions are connected to the anomalous dimensions of the exchanged fields. Comparison of the results from AdS graphs with the conformal partial wave analysis, suggests a possible general form for the operator product expansion of scalar fields in the boundary CFT.

We discuss the analytic properties of AdS scalar exchange graphs in the crossed channel. We show that the possible non-analytic terms drop out by virtue of non-trivial properties of generalized hypergeometric functions. The absence of non-analytic terms is a necessary condition for the existence of an operator product expansion for CFT amplitudes obtained from AdS/CFT correspondence.

Trigonometric invariants are defined for each Weyl group orbit on the root lattice. They are real and periodic on the coroot lattice. Their polynomial algebra is spanned by a basis which is calculated by means of an algorithm. The invariants of the basis can be used as coordinates in any cell of the coroot space and lead to an exactly solvable model of Sutherland type. We apply this construction to the \(F_4\) case.

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.

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.