Liouville Field Theory -- A decade after the revolution

TL;DR

This review surveys Liouville field theory and its matrix-model dual, focusing on developments up to 2004 across bosonic, supersymmetric, and unoriented formulations. It highlights the DOZZ three-point function, exact boundary states (FZZT/ZZ branes), and boundary-bulk couplings, all interconnected through open/closed dualities and nonperturbative string physics. The matrix-model perspective is developed for $c<1$ and $c=1$, linking to Painlevé and integrable hierarchies, while Teschner’s trick and higher equations of motion underpin the Liouville bootstrap and boundary data. The work emphasizes noncritical string theory, tachyon dynamics, and the deep ties between Liouville theory, topological strings, and CY geometries, illustrating a coherent framework for exact, nonperturbative descriptions in low dimensions with broad implications for higher-dimensional string theory.

Abstract

We review recent developments (up to January 2004) of the Liouville field theory and its matrix model dual. This review consists of three parts. In part I, we review the bosonic Liouville theory. After briefly reviewing the necessary background, we discuss the bulk structure constants (the DOZZ formula) and the boundary states (the FZZT brane and the ZZ brane). Various applications are also presented. In part II, we review the supersymmetric extension of the Liouville theory. We first discuss the bulk structure constants and the branes as in the bosonic Liouville theory, and then we present the matrix dual descriptions with some applications. In part III, the Liouville theory on unoriented surfaces is reviewed. After introducing the crosscap state, we discuss the matrix model dual description and the tadpole cancellation condition. This review also includes some original material such as the derivation of the conjectured dual action for the N = 2 Liouville theory from other known dualities and the comparison of the Liouville crosscap state with the c = 0 unoriented matrix model. This is based on my master's thesis submitted to Department of Physics, Faculty of Science, University of Tokyo on January 2004.

Figures (32)

• Figure 1: An incoming wave is reflected from the Liouville wall and becomes an outgoing wave with a reflection coefficient $R(p)$.
• Figure 2: The summation over $m$ can be effectively replaced with the integration over the horizontally hatched region.
• Figure 3: Sen's conjecture: the closed string vacuum is the global minimum of the effective tachyon potential and the open string vacuum is the extremum of the potential. The potential difference corresponds to the D-brane tension.
• Figure 4: (a) the boundary effect can be regarded as an insertion of the Ishibashi states in the closed string language. (b) in a certain limit, we expect that the existence of the boundaries can be replaced with a closed string theory on a nontrivial background. This picture is taken from Gopakumar:1999.
• Figure 5: The geometric transition: the topological open A-model on the deformed conifold with $N$ A-branes is dual to the closed topological A-model on the resolved conifold.