## Fachbereich Physik

- Aspects of the conformal operator product expansion in AdS/CFT correspondence (2000)
- 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.

- A note on the analyticity of AdS scalar exchange graphs in the crossed channel (2000)
- 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.

- The construction of trigonometric invariants for Weyl groups and the derivation of corresponding exactly solvable Sutherland models (1999)
- 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.

- On the critical behaviour of hermitean f-matrix models in the double scaling limit with f >= 3 (1999)
- An algorithm for the isolation of any singularity of f-matrix models in the double scaling limit is presented. In particular it is proved by construction that only those universality classes exist that are known from 2-matrix models.

- Exactly solvable dynamical systems in the neighborhood of the Calogero model (1999)
- 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 \(N\).

- An exactly solvable model of the Calogero type for the icosahedral group (1998)
- We construct a quantum mechanical model of the Calogero type for the icosahedral group as the structural group. Exact solvability is proved and the spectrum is derived explicitly.

- Is it possible to construct exactly solvable models? (1998)
- 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.

- Remarks on 'Coloring Random Triangulation' (1998)
- 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 Continuous Series of Critical Points of the Two-Matrix Model at N -> infinity in the Double Scaling Limit (1998)
- 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.

- Construction of exactly solvable quantum models of Calogero and Sutherland type with translation invariant four-particle interactions (1998)
- We construct exactly solvable models for four particles moving on a real line or on a circle with translation invariant two- and four-particle interactions.