Gerhard Weigt

Correlation functions and vertex operators of Liouville theory

We calculate correlation functions for vertex operators with negative integer exponentials of a periodic Liouville field, and derive the general case by continuing them as distributions. The path-integral based conjectures of Dorn and Otto prove to be conditionally valid only. We formulate integral representations for the generic vertex operators and indicate structures which are related to the Liouville S-matrix.

Poisson structure and Moyal quantisation of the Liouville theory

Abstract The symplectic and Poisson structures of the Liouville theory are derived from the SL(2, R ) WZNW theory by gauge invariant Hamiltonian reduction. Causal non-equal time Poisson brackets for a Liouville field are presented. Using the symmetries of the Liouville theory, symbols of local fields are constructed and their ∗ -products calculated. Quantum deformations consistent with the canonical quantisation result, and a non-equal time commutator is given.

Analytical solution of the SL(2, R)/U(1) WZNW black hole model

Abstract The gauged SL(2, R )/U(1) Wess-Zumino-Novikov-Witten (WZNW) model is classically an integrable conformal field theory. We have found a Lax pair representation for the non-linear equations of motion, and a Backlund transformation. A second-order differential equation of the Gelfand-Dikii type defines the Poisson bracket structure of the theory, and its fundamental solutions describe the general solution of the WZNW model as well. The physical and free fields are related by non-local transformations. The (anti-)chiral component of the energy-momentum tensor which factorizes into conserved quantities satisfying a non-linear algebra characteristic for parafermions assumes the canonical free filed form. So the black hole model is expected to be prepared for an exact canonical quantization.

The Complete Solution of the Classical SL(2,ℝ/U(1) Gauged WZNW Field Theory

Abstract:We prove that any gauged WZNW model has a Lax pair representation, and give explicitly the general solution of the classical equations of motion of the SL(2,ℝ/U(1) theory. We calculate the symplectic structure of this solution by solving a differential equation of the Gelfand–Dikii type with initial state conditions at infinity, and transform the canonical physical fields non-locally onto canonical free fields. The results will, finally, be collected in a local Bäcklund transformation. These calculations prepare the theory for an exact canonical quantization.

Integration of the SL(2,R)/U(1) gauged WZNW model with periodic boundary conditions

Abstract Gauged WZNW models are integrable conformal field theories. We integrate the classical SL(2, R )/U(1) theory with periodic boundary conditions, which describes closed strings moving in a curved target-space geometry. We calculate its Poisson bracket structure by solving an initial state problem. The results differ from previous field-theoretic calculations due to zero-modes. For a future exact canonical quantization the physical fields are (non-locally) transformed onto canonical free fields.

Causal Poisson brackets of the WZNW model and its coset theories

Abstract From the basic chiral and antichiral Poisson bracket algebra of the SL(2, R ) WZNW model, non equal-time Poisson brackets are derived. Through Hamiltonian reduction we deduce the corresponding brackets for its coset theories.

The Poisson Bracket Structure of the SL(2, R)/U(1) Gauged WZNW Model with Periodic Boundary Conditions

The gauged SL(2, R)/U(1) Wess-Zumino-Novikov-Witten (WZNW) model is classically an integrable conformal field theory. A second-order differential equation of the Gelfand-Dikii type defines the Poisson bracket structure of the theory. For periodic boundary conditions zero modes imply non-local Poisson brackets which, nevertheless, can be represented by canonical free fields.

Exact solution of the SL(2,R)/U(1) WZNW model

Abstract It is shown that the exact solution of the gauged SL(2, R )/U(1) Wess-Zumino-Novikov-Witten (WZNW) black hole model is asymptotically related to the solution of an integrable non-abelian Toda theory. A second-order differential equation characterizes the structure of the theory, and its fundamental solutions describe the general solution of the WZNW model as well. As in Liouville theory, physical and free fields are non-locally related transforming the energy-momentum tensor into a canonical free field form. The black hole model is expected to be canonically quantizable.