Crossing Symmetry in the H3+H_3^+ WZNW model

We show that crossing symmetry of four point functions in the $H_3^+$ WZNW model follows from similar properties of certain five point correlation functions in Liouville theory that have already been proven previously.Comment: 7 page

A guide to two-dimensional conformal field theory

This is a review of two-dimensional conformal field theory including some of the relations to integrable models. An effort is made to develop the basic formalism in a way which is as elementary and flexible as possible at the same time. Some advanced topics like conformal field theory on higher genus surfaces and relations to the isomonodromic deformation problem are discussed, for other topics we offer a first guide to the literature.Comment: 57 Pages; v2: refs. added, minor correction

Quantum Liouville theory versus quantized Teichm\"uller spaces

This note announces the proof of a conjecture of H. Verlinde, according to which the spaces of Liouville conformal blocks and the Hilbert spaces from the quantization of the Teichm\"uller spaces of Riemann surfaces carry equivalent representations of the mapping class group. This provides a basis for the geometrical interpretation of quantum Liouville theory in its relation to quantized spaces of Riemann surfaces.Comment: Contribution to the proceedings of the 35th Ahrenshoop Symposiu

On Tachyon condensation and open-closed duality in the c=1 string theory

We present an exact representation for decaying ZZ-branes within the dual matrix model formulation of c=1 string theory. It is shown how to trade the insertion of decaying ZZ-branes for a shift of the closed string background. Our formlaism allows us to demonstrate that the conjectured world-sheet mechanism behind the open-closed dualities (summing over disc insertions) is realized here in a clear way. On the way we need to clarify certain infrared issues - insertion of ZZ-branes creates solitonic superselection sectors.Comment: 37 pages; v2: Minor correction

Quantisation conditions of the quantum Hitchin system and the real geometric Langlands correspondence

Single-valuedness of the eigenfunctions of the quantised Hitchin Hamiltonians is proposed as a natural quantisation condition. Separation of Variables can be used to relate the classification of eigenstates to the classification of projective structures with real holonomy. Using complex Fenchel-Nielsen coordinates one may reformulate the quantisation conditions in terms of the generating function for the variety of opers. These results are interpreted as a variant of the geometric Langlands correspondence.Comment: 30 pages; v2: relevant corrections, close to fina

Quantization of moduli spaces of flat connections and Liouville theory

We review known results on the relations between conformal field theory, the quantization of moduli spaces of flat PSL(2,R)-connections on Riemann surfaces, and the quantum Teichmueller theory.Comment: 25 pages, contribution to the proceedings of the ICM 201

Classical conformal blocks and isomonodromic deformations

The leading classical asymptotics of Virasoro conformal blocks on the Riemann sphere with n generic and n-3 "heavy" degenerate field insertions can be described in terms of the geometry of Garnier system describing the monodromy preserving deformations of second order Fuchsian differential equations on an n-punctured sphere. This allows us to characterise the leading classical asymptotics of Virasoro conformal blocks completely, and to clarify in which sense conformal field theory represents a quantisation of the isomonodromic deformation problem.Comment: 14 page

Liouville theory without an action

We show that the crossing symmetry of the four-point function in the Liouville conformal field theory on the sphere contains more information than what was hitherto considered. Under certain assumptions, it provides the special structure constants that were previously computed perturbatively and allows to solve the theory without using the Liouville interaction.Comment: 9 page