We present a comprehensive analysis of branes in the Euclidean 2D black hole (cigar). In particular, exact boundary states and annulus amplitudes are provided for D0-branes which are localized at the tip of the cigar as well as for two families of extended D1 and D2-branes. Our results are based on closely related studies for the Euclidean AdS 3 model [1] and, as predicted by the conjectured duality between the 2D black hole and the sine-Liouville model, they share many features with branes in Liouville theory. New features arise here due to the presence of closed string modes which are localized near the tip of the cigar. The paper concludes with some remarks on possible applications.

We prove that arbitrary correlation functions of the H + 3 -WZNW model on a sphere have a simple expression in terms of Liouville theory correlation functions. This is based on the correspondence between the KZ and BPZ equations, and on relations between the structure constants of Liouville theory and the H + 3 -WZNW model. In the critical level limit, these results imply a direct link between eigenvectors of the Gaudin Hamiltonians and the problem of uniformization of Riemann surfaces. We also present an expression for correlation functions of the SL(2)/U (1) gauged WZNW model in terms of correlation functions in Liouville theory. 93.

I generalize the Knizhnik-Zamolodchikov equations to correlators of spectral flowed fields in AdS 3 string theory. If spectral flow is preserved or violated by one unit, the resulting equations are equivalent to the KZ equations. If spectral flow is violated by two units or more, only some linear combinations of the KZ equations hold, but extra equations appear. Then I explicitly show how these correlators and the associated conformal blocks are related to Liouville theory correlators and conformal blocks with degenerate field insertions, where each unit of spectral flow violation removes one degenerate field. A similar relation to Liouville theory holds for noncompact parafermions.

We study four-point functions of critical percolation in two dimensions, and more generally of the Potts model. We propose an exact ansatz for the spectrum: an infinite, discrete and non-diagonal combination of representations of the Virasoro algebra. Based on this ansatz, we compute four-point functions using a numerical conformal bootstrap approach. The results agree with Monte-Carlo computations of connectivities of random clusters.

Abstract:We determine the spectrum and correlation functions of Liouville theory with a central charge less than (or equal) one. This completes the definition of Liouville theory for all complex values of the central charge. The spectrum is always spacelike, and there is no consistent timelike Liouville theory. We also study the non-analytic conformal field theories that exist at rational values of the central charge. Our claims are supported by numerical checks of crossing symmetry. We provide Python code for computing Virasoro conformal blocks, and correlation functions in Liouville theory and (generalized) minimal models.

We provide an introduction to conformal field theory on the plane in the conformal bootstrap approach. We introduce the main ideas of the bootstrap approach to quantum field theory, and how they apply to two-dimensional theories with local conformal symmetry. We describe the mathematical structures that appear in such theories, from the Virasoro algebra and its representations, to the BPZ equations and their solutions. As examples, we study a number of models: Liouville theory, (generalized) minimal models, free bosonic theories, the H + 3 model, and the SU 2 and SL 2 (R) WZW models.

We perform Monte-Carlo computations of four-point cluster connectivities in the critical 2d Potts model, for numbers of states Q ∈ (0, 4) that are not necessarily integer. We compare these connectivities to four-point functions in a CFT that interpolates between D-series minimal models. We find that 3 combinations of the 4 independent connectivities agree with CFT four-point functions, down to the 2 to 4 significant digits of our Monte-Carlo computations. However, we argue that the agreement is exact only in the special cases Q = 0, 3, 4. We conjecture that the Potts model can be analytically continued to a double cover of the half-plane {ℜc < 13}, where c is the central charge of the Virasoro symmetry algebra.

Under the assumption that degenerate fields exist, diagonal CFTs such as Liouville theory can be solved analytically using the conformal bootstrap method. Here we generalize this approach to non-diagonal CFTs, i.e. CFTs whose primary fields have nonzero conformal spins. Assuming generic values of the central charge, we find that the non-diagonal sector of the spectrum must be parametrized by two integer numbers. We then derive and solve the equations that determine how three-and four-point structure constants depend on these numbers. In order to test these results, we numerically check crossing symmetry of a class of four-point functions in a non-rational limit of D-series minimal models. The simplest four-point functions in this class were previously argued to describe connectivities of clusters in the critical Potts model.