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

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

Boundary Liouville Field Theory: Boundary Three Point Function

Liouville field theory is considered on domains with conformally invariant boundary conditions. We present an explicit expression for the three point function of boundary fields in terms of the fusion coefficients which determine the monodromy properties of the conformal blocks.Comment: 18 pages; v2: minor change

Are there really any AdS_2 branes in the euclidean (or not) AdS_3?

We do not find any AdS_2 branes, neither in the H_3^+ WZNW model nor the SL(2,R) WZNW model. We then reexamine the case of the branes that possess a su(2) symmetry: we speculate that they would have to live on the boundary of AdS_3. This cannot be realized in an euclidean spacetime, but in the SL(2,R) WZNW model by analytical continuation

Branes in the Euclidean AdS_3

In this work we propose an exact microscopic description of maximally symmetric branes in a Euclidean $AdS_3$ background. As shown by Bachas and Petropoulos, the most important such branes are localized along a Euclidean $AdS_2 \subset AdS_3$. We provide explicit formulas for the coupling of closed strings to such branes (boundary states) and for the spectral density of open strings. The latter is computed in two different ways first in terms of the open string reflection amplitude and then also from the boundary states by world-sheet duality. This gives rise to an important Cardy type consistency check. All the results are compared in detail with the geometrical picture. We also discuss a second class of branes with spherical symmetry and finally comment on some implications for D-branes in a 2D back hole geometry

Liouville Field Theory on an Unoriented Surface

Liouville field theory on an unoriented surface is investigated, in particular, the one point function on a RP^2 is calculated. The constraint of the one point function is obtained by using the crossing symmetry of the two point function. There are many solutions of the constraint and we can choose one of them by considering the modular bootstrap.Comment: 13 pages, no figures, LaTeX, minor changes, equations in section 4 are correcte

Continuously Crossing u=z in the H3+ Boundary CFT

For AdS boundary conditions, we give a solution of the H3+ two point function involving degenerate field with SL(2)-label b^{-2}/2, which is defined on the full (u,z) unit square. It consists of two patches, one for z<u and one for u<z. Along the u=z "singularity", the solutions from both patches are shown to have finite limits and are merged continuously as suggested by the work of Hosomichi and Ribault. From this two point function, we can derive b^{-2}/2-shift equations for AdS_2 D-branes. We show that discrete as well as continuous AdS_2 branes are consistent with our novel shift equations without any new restrictions.Comment: version to appear in JHEP - 12 pages now; sign error with impact on some parts of the interpretation fixed; material added to become more self-contained; role of bulk-boundary OPE in section 4 more carefully discussed; 3 references adde

New AdS3AdS_3 Branes and Boundary States

We examine D-branes on $AdS_3$, and find a three-brane wrapping the entire $AdS_3$, in addition to 1-branes and instantonic 2-branes previously discussed in the literature. The three-brane is found using a construction of Maldacena, Moore, and Seiberg. We show that all these branes satisfy Cardy's condition and extract the open string spectrum on them.Comment: 18 pages, late

Open String Creation by S-Branes

An sp-brane can be viewed as the creation and decay of an unstable D(p+1)-brane. It is argued that the decaying half of an sp-brane can be described by a variant of boundary Liouville theory. The pair creation of open strings by a decaying s-brane is studied in the minisuperspace approximation to the Liouville theory. In this approximation a Hagedorn-like divergence is found in the pair creation rate, suggesting the s-brane energy is rapidly transferred into closed string radiation.Comment: Talk presented at the Hangzhou String 2002 Conference, August 12-1

On the crossing relation in the presence of defects

The OPE of local operators in the presence of defect lines is considered both in the rational CFT and the $c>25$ Virasoro (Liouville) theory. The duality transformation of the 4-point function with inserted defect operators is explicitly computed. The two channels of the correlator reproduce the expectation values of the Wilson and 't Hooft operators, recently discussed in Liouville theory in relation to the AGT conjecture.Comment: TEX file with harvmac; v3: JHEP versio