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Our goal is to develop a more general scheme for constructing integrable lattice regularisations of integrable quantum field theories. Considering the affine Toda theories as examples, we show how to construct such lattice regularisations using the representation theory of quantum affine algebras. This requires us to clarify in particular the relations between the light-cone approach to integrable lattice models and the representation theory of quantum affine algebras. Both are found to be related in a very natural way, suggesting a general scheme for the construction of generalised Baxter Q-operators. One of the main difficulties we need to deal with is coming from the infinite-dimensionality of the relevant families of representations. It is handled by means of suitable renormalisation prescriptions defining what may be called the modular double of quantum affine algebras. This framework allows us to give a representation-theoretic proof of finite-difference equations generalising the Baxter equation., Comment: 155 pages; v2: Refs. added

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 2014

The goal of this note is to show that the Riemann-Hilbert problem to find multivalued analytic functions with $SL(2,\mathbb{C})$-valued monodromy on Riemann surfaces of genus zero with $n$ punctures can be solved by taking suitable linear combinations of the conformal blocks of Liouville theory at $c=1$. This implies a similar representation for the isomonodromic tau-function. In the case $n=4$ we thereby get a proof of the relation between tau-functions and conformal blocks discovered in \cite{GIL}. We briefly discuss a possible application of our results to the study of relations between certain $\mathcal{N}=2$ supersymmetric gauge theories and conformal field theory., Comment: 29 pages, 4 figures; v2: minor changes, added refs

We will propose a derivation of the correspondence between certain gauge theories with N=2 supersymmetry and conformal field theory discovered by Alday, Gaiotto and Tachikawa in the spirit of Seiberg-Witten theory. Based on certain results from the literature we argue that the quantum theory of the moduli spaces of flat SL(2,R)-connections represents a non-perturbative "skeleton" of the gauge theory, protected by supersymmetry. It follows that instanton partition functions can be characterized as solutions to a Riemann-Hilbert type problem. In order to solve it, we describe the quantization of the moduli spaces of flat connections explicitly in terms of two natural sets of Darboux coordinates. The kernel describing the relation between the two pictures represents the solution to the Riemann Hilbert problem, and is naturally identified with the Liouville conformal blocks., Comment: 120 pages, 22 figures; V2: Slightly expanded, some explanations, details, references added, small corrections

We exhibit direct relations between the modular double of U_q(sl(2,R)) and the quantum Teichmueller theory. Explicit representations for the fusion- and braiding operations of the quantum Teichmueller theory are immediate consequences. Our results include a simplified derivation of the Clebsch-Gordan decomposition for the principal series of representation of the modular double of U_q(sl(2,R))., Comment: 40 pages, 30 figures

We revisit the definition of the 6j-symbols from the modular double of U_q(sl(2,R)), referred to as b-6j symbols. Our new results are (i) the identification of particularly natural normalization conditions, and (ii) new integral representations for this object. This is used to briefly discuss possible applications to quantum hyperbolic geometry, and to the study of certain supersymmetric gauge theories. We show, in particular, that the b-6j symbol has leading semiclassical asymptotics given by the volume of a non-ideal tetrahedron. We furthermore observe a close relation with the problem to quantize natural Darboux coordinates for moduli spaces of flat connections on Riemann surfaces related to the Fenchel-Nielsen coordinates. Our new integral representations finally indicate a possible interpretation of the b-6j symbols as partition functions of three-dimensional N=2 supersymmetric gauge theories., Comment: 32 pages, v2: small corrections, v3: another typo

A method is introduced for constructing lattice discretizations of large classes of integrable quantum field theories. The method proceeds in two steps: The quantum algebraic structure underlying the integrability of the model is determined from the algebra of the interaction terms in the light-cone representation. The representation theory of the relevant quantum algebra is then used to construct the basic ingredients of the quantum inverse scattering method, the lattice Lax matrices and R-matrices. This method is illustrated with four examples: The Sinh-Gordon model, the affine sl(3) Toda model, a model called the fermionic sl(2|1) Toda theory, and the N=2 supersymmetric Sine-Gordon model. These models are all related to sigma models in various ways. The N=2 supersymmetric Sine-Gordon model, in particular, describes the Pohlmeyer reduction of string theory on AdS_2 x S^2, and is dual to a supersymmetric non-linear sigma model with a sausage-shaped target space., Comment: 66 pages; V2: minor corrections, references added

We give a direct derivation of a proposal of Nekrasov-Shatashvili concerning the quantization conditions of the Toda chain. The quantization conditions are formulated in terms of solutions to a nonlinear integral equation similar to the ones coming from the thermodynamic Bethe ansatz. This is equivalent to extremizing a certain function called Yang's potential. It is shown that the Nekrasov-Shatashvili formulation of the quantization conditions follows from the solution theory of the Baxter equation, suggesting that this way of formulating the quantization conditions should indeed be applicable to large classes of quantized algebraically integrable models., Comment: 21 pages

We discuss the relation between Liouville theory and the Hitchin integrable system, which can be seen in two ways as a two step process involving quantization and hyperkaehler rotation. The modular duality of Liouville theory and the relation between Liouville theory and the SL(2)-WZNW-model give a new perspective on the geometric Langlands correspondence and on its relation to conformal field theory., Comment: 81 pages; V2: small corrections and improvements, references added; V3: Discussion of Yang's potential moved to 5.5 and extended, small additons and corrections otherwise; V4: a reference added, V5: final version to appear in ATMP

We study integrable lattice regularizations of the sine-Gordon model with the help of the separation of variables method of Sklyanin and the Baxter Q-operators. This leads us to the complete characterization of the spectrum (eigenvalues and eigenstates), in terms of the solutions to the Bethe ansatz equations. The completeness of the set of states that can be constructed from the solutions to the Bethe ansatz equations is proven by our approach., Comment: 44 pages, presentation improved, no result changed

Using the example of Liouville theory, we show how the separation into left- and rightmoving degrees of freedom of a nonrational conformal field theory can be made explicit in terms of its integrable structure. The key observation is that there exist separate Baxter Q-operators for left- and right-moving degrees of freedom. Combining a study of the analytic properties of the Q-operators with Sklyanin's Separation of Variables Method leads to a complete characterization of the spectrum. Taking the continuum limit allows us in particular to rederive the Liouville reflection amplitude using only the integrable structure., Comment: 34 pages; typos corrected, comments added

We discuss the problem to develop a mathematical theory of a certain class of nonrational conformal field theories (CFT) which contain the unitary CFT. A variant of the concept of a modular functor is proposed that appears to be suitable for such CFT., Comment: 44 pages

We derive a characterization of the spectrum of the Sinh-Gordon model in terms of certain nonlinear integral equations. There exists a large class of solutions to these equations which allows a continuation between the infrared and the ultraviolet limits, respectively. We present nontrivial evidence for the claim that the class of solutions in question describes the spectrum of the Sinh-Gordon model completely in both of these limits. The evidence includes some nontrivial relations to Liouville theory., Comment: 33 pages, the new version corrects a mistake in the lattice TBA of the previous versions

We define and study certain integrable lattice models with non-compact quantum group symmetry (the modular double of U_q(sl_2)) including an integrable lattice regularization of the sinh-Gordon model and a non-compact version of the XXZ model. Their fundamental R-matrices are constructed in terms of the non-compact quantum dilogarithm. Our choice of the quantum group representations naturally ensures self-adjointness of the Hamiltonian and the higher integrals of motion. These models are studied with the help of the separation of variables method. We show that the spectral problem for the integrals of motion can be reformulated as the problem to determine a subset among the solutions to certain finite difference equations (Baxter equation and quantum Wronskian equation) which is characterized by suitable analytic and asymptotic properties. A key technical tool is the so-called Q-operator, for which we give an explicit construction. Our results allow us to establish some connections to related results and conjectures on the sinh-Gordon theory in continuous space-time. Our approach also sheds some light on the relations between massive and massless models (in particular, the sinh-Gordon and Liouville theories) from the point of view of their integrable structures., Comment: 60 pages, LaTeX2e, (v.2: minor changes, presentation of section D in Appendix improved, v3: minor last corrections, close to published version)

It is shown that the quantized Teichm"uller spaces have factorization properties like those required in the definition of a modular functor., Comment: Material from math.QA/0405332, substantially reorganized. V2: ref. added; V3: Part III rewritten to make it more accessible, some arguments in Part IV simplified, V4: few last corrections, to appear in "Handbook of Teichmueller 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 corrections

This paper has been withdrawn by the author(s). The material contained in the paper will be published in a subtantially reorganized form, part of it is now included in math.QA/0510174, Comment: withdrawn, large portion of the material now appears in math.QA/0510174, rest will appear elsewhere in reorganized form

The aim of this note is to propose an interpretation for the full (non-chiral) correlation functions of the Liouville conformal field theory within the context of the quantization of spaces of Riemann surfaces., Comment: 16p, Contribution to the Proceedings of the ICMP 2003, Lisbon; v2: A reference added

We review both the construction of conformal blocks in quantum Liouville theory and the quantization of Teichm\"uller spaces as developed by Kashaev, Checkov and Fock. In both cases one assigns to a Riemann surface a Hilbert space acted on by a representation of the mapping class group. According to a conjecture of H. Verlinde, the two are equivalent. We describe some key steps in the verification of this conjecture., Comment: Contribution to the proceedings of the 6th International Conference on CFTs and Integrable Models, Chernogolovka, Russia, September 2002; v2: Typos corrected, typographical changes

We reconsider the construction of exponential fields in the quantized Liouville theory. It is based on a free-field construction of a continuous family or chiral vertex operators. We derive the fusion and braid relations of the chiral vertex operators. This allows us to simplify the verification of locality and crossing symmetry of the exponential fields considerably. The calculation of the matrix elements of the exponential fields leads to a constructive derivation of the formula proposed by Dorn/Otto and the brothers Zamolodchikov., Comment: Contribution to the proceedings of the 6th International Conference on CFTs and Integrable Models, Chernogolovka, Russia, 2002 v2: Remarks added, typos corrected

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 Symposium

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)

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., Comment: 58 pages, 2 figures

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 changes

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 pages

We try to develop a coherent picture on Liouville theory as a two-dimensional conformal field theory that takes into account the perspectives of path-integral approach, bootstrap, canonical quantization and operator approach. To do this, we need to develop further some of these approaches. This includes in particular a construction of general exponential field operators from a set of covariant chiral operators. The latter are shown to satisfy braid relations that allow one to prove the locality of the former., Comment: 78 pages, uses bourbaki.cls; some remarks in sec. 9 altered, minor corrs

The bootstrap for Liouville theory with conformally invariant boundary conditions will be discussed. After reviewing some results on one- and boundary two-point functions we discuss some analogue of the Cardy condition linking these data. This allows to determine the spectrum of the theory on the strip, and illustrates in what respects the bootstrap for noncompact conformal field theories with boundary is richer than in RCFT. We briefly indicate some connections with $U_q(sl(2,R))$ that should help completing the bootstrap., Comment: Talk presented at TMR-conference "Nonperturbative Quantum Effects 2000", 7 pages, LATEX with JHEP proceedings style

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 CMP

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

Precise descriptions are given for the operator product expansion of generic primary fields as well as the factorization of four point functions as sum over intermediate states. The conjecture underlying the recent derivation of the space-time current algebra for string theory on $ADS_3$ by Kutasov and Seiberg is thereby verified. The roles of microscopic and macroscopic states are further clarified. The present work provides the conformal field theory prerequisites for a future study of factorization of amplitudes for string theory on $ADS_3$ as well as operator product expansion in the corresponding conformal field theory on the boundary., Comment: 28 pages, references added

A deformation of the wave equation on a two-dimensional black hole is considered as a toy-model for possible gravitational or stringy nonlocal effects. The deformed wave-equation allows for an initial-value problem despite being nonlocal. The classical singularity present in the classical geometry is resolved by the deformation, so that propagation of a wave-packet can be continued through the classically singular region, ultimately reaching another asymptotically ``flat'' region., Comment: 12 pager, Latex with eepic macros, minor changes

A closed formula for the structure constants in the SL(2,C)/SU(2) WZNW model is derived by a method previously used in Liouville theory. With the help of a reflection amplitude that follows from the structure constants one obtains a proposal for the fusion rules from canonical quantization. Taken together these pieces of information allow an unambigous definition of any genus zero n-point function., Comment: 25 pages, AMSLATEX2e, discussion of canonical quantization and fusion rules clarified, a few corrections

Many qualitatively new features of WZNW models associated to noncompact cosets are due to zero modes with continuous spectrum. Insight may be gained by reducing the theory to its zero-mode sector, the mini-superspace limit. This will be discussed in some detail for the example of SL(2,C)/SU(2)-WZNW model. The mini-superspace limit of this model can be formulated as baby-CFT. Spectrum, structure constants and fusion rules as well as factorization of four point functions are obtained from the harmonic analysis on SL(2,C)/SU(2). The issues of operator-state correspondence or the appearance of non-normalizable intermediate states in correlation functions can be discussed transparently in this context., Comment: 17 pages, AMS-LATEX2e, minor corrections

This paper has been withdrawn. A more satisfactory account of the results it was supposed to present will appear in a series of papers starting with hep-th/9712256 and hep-th/9712258, Comment: 40 pages, amssymb. Withdrawn

The chiral vertex operators for the minimal models are constructed and used to define a fusion product of representations. The existence of commutativity and associativity operations is proved. The matrix elements of the associativity operations are shown to be given in terms of the 6-j symbols of the weak quasitriangular quasi-Hopf algebra obtained by truncating $\usl$ at roots of unity., Comment: 18 pages, amslatex, proof of prop. 6.2 corrected

The recently proposed expression for the general three point function of exponential fields in quantum Liouville theory on the sphere is considered. By exploiting locality or crossing symmetry in the case of those four-point functions, which may be expressed in terms of hypergeometric functions, a set of functional equations is found for the general three point function. It is shown that the expression proposed by the Zamolodchikovs solves these functional equations and that under certain assumptions the solution is unique., Comment: 8 Pages, LATEX

Borcherds algebras represent a new class of Lie algebras which have almost all the properties that ordinary Kac-Moody algebras have, and the only major difference is that these generalized Kac-Moody algebras are allowed to have imaginary simple roots. The simplest nontrivial examples one can think of are those where one adds ``by hand'' one imaginary simple root to an ordinary Kac-Moody algebra. We study the fundamental representation of this class of examples and prove that an irreducible module is given by the full tensor algebra over some integrable highest weight module of the underlying Kac-Moody algebra. We also comment on possible realizations of these Lie algebras in physics as symmetry algebras in quantum field theory., Comment: 8 pages