We try to develop a coherent picture of Liouville theory as a two-dimensional conformal field theory that takes into account the perspectives of the path-integral approach, bootstrap, canonical quantization and the operator approach. To do this, we need to develop further some of these approaches. This includes in particular the 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.

Liouville theory revisited

We propose a dictionary between geometry of triangulated 3-manifolds and physics of three-dimensional N=2 gauge theories. Under this duality, standard operations on triangulated 3-manifolds and various invariants thereof (classical as well as quantum) find a natural interpretation in field theory. For example, independence of the SL(2) Chern-Simons partition function on the choice of triangulation translates to a statement that S^3_b partition functions of two mirror 3d N=2 gauge theories are equal. Three-dimensional N=2 field theories associated to 3-manifolds can be thought of as theories that describe boundary conditions and duality walls in four-dimensional N=2 SCFTs, thus making the whole construction functorial with respect to cobordisms and gluing.

/pdf/gauge-theories-labelled-by-three-manifolds-1w5hv4p57r.pdf

Gauge Theories Labelled by Three-Manifolds

Liouville field theory is considered with boundary conditions corresponding to a quantization of the classical Lobachevskiy plane (i.e. euclidean version of $AdS_2$). We solve the bootstrap equations for the out-vacuum wave function and find an infinite set of solutions. This solutions are in one to one correspondence with the degenerate representations of the Virasoro algebra. Consistency of these solutions is verified by both boundary and modular bootstrap techniques. Perturbative calculations lead to the conclusion that only the ``basic'' solution corresponding to the identity operator provides a ``natural'' quantization of the Lobachevskiy plane.

Liouville field theory on a pseudosphere

Recent results on the annulus partition function in Liouville field theory are applied to non-critical string theory, both below and above the critical dimension. Liouville gravity coupled to $c\le 1$ matter has a dual formulation as a matrix model. Two well-known matrix model results are reproduced precisely using the worldsheet formulation: (1) the correlation function of two macroscopic loops, and (2) the leading non-perturbative effects. The latter identifies the eigenvalue instanton amplitudes of the matrix approach with disk instantons of the worldsheet approach, thus demonstrating that the matrix model is the effective dynamics of a D-brane realization of $d\le 1$ non-critical string theory. In the context of string theory above the critical dimension, i.e. $d\ge 25$, Liouville field theory realizes two-dimensional de Sitter gravity on the worldsheet. In this case, appropriate D-brane boundary conditions on the annulus realize the S-matrix for two-dimensional de Sitter gravity.

The Annular report on noncritical string theory

We define and study certain integrable lattice models with non-compact quantum group symmetry (the modular double of ) 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 -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 spacetime. 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.

https://www.pdmi.ras.ru/~bytsko/repr/jpa39.pdf

Quantization of models with non-compact quantum group symmetry: Modular XXZ magnet and lattice sinh-Gordon model

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.

B. Ponsot

Papers

Clebsch–Gordan and Racah–Wigner Coefficients for a Continuous Series of Representations of ?q (??(2, ℝ))