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

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

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

R-operator, co-product and Haar-measure for the modular double of U_q(sl(2,R))

A certain class of unitary representations of U_q(sl(2,R)) has the property of being simultanenously a representation of U_{tilde{q}}(sl(2,R)) for a particular choice of tilde{q}(q). Faddeev has proposed to unify the quantum groups U_q(sl(2,R)) and U_{tilde{q}}(sl(2,R)) into some enlarged object for which he has coined the name ``modular double''. We study the R-operator, the co-product and the Haar-measure for the modular double of U_q(sl(2,R)) and establish their main properties. In particular it is shown that the Clebsch-Gordan maps constructed in [PT2] diagonalize this R-operator.Comment: 27 pages, LaTex (smfart.sty

Liouville bootstrap via harmonic analysis on a noncompact quantum group

The purpose of this short note is to announce results that amount to a verification of the bootstrap for Liouville theory in the generic case under certain assumptions concerning existence and properties of fusion transformations. Under these assumptions one may characterize the fusion and braiding coefficients as solutions of a system of functional equations that follows from the combination of consistency requirements and known results. This system of equations has a unique solution for irrational central charge c>25. The solution is constructed by solving the Clebsch-Gordan problem for a certain continuous series of quantum group representations and constructing the associated Racah-coefficients. This gives an explicit expression for the fusion coefficients. Moreover, the expressions can be continued into the strong coupling region 1<c<25, providing a solution of the bootstrap also for this region.Comment: 16 pages, typos removed incl. important one in (48

Clebsch-Gordan and Racah-Wigner coefficients for a continuous series of representations of U_q(sl(2,R))

The decomposition of tensor products of representations into irreducibles is studied for a continuous family of integrable operator representations of $U_q(sl(2,R)$. It is described by an explicit integral transformation involving a distributional kernel that can be seen as an analogue of the Clebsch-Gordan coefficients. Moreover, we also study the relation between two canonical decompositions of triple tensor products into irreducibles. It can be represented by an integral transformation with a kernel that generalizes the Racah-Wigner coefficients. This kernel is explicitly calculated.Comment: 39 pages, AMS-Latex; V2: Added comments and references concerning relation to Faddeev's modular double, minor corrections, version to be published in CM