Liouville Field Theory -- A decade after the revolution

TL;DR

This review summarizes a decade of progress in Liouville field theory, covering bosonic, supersymmetric, and unoriented cases, including dualities, boundary states, and applications up to 2004.

Contribution

It provides a comprehensive overview of recent developments in Liouville theory, including original insights into dual actions and unoriented surfaces.

Findings

Explicit formulas for structure constants (DOZZ formula)

Matrix dual descriptions of supersymmetric Liouville theory

Analysis of crosscap states and tadpole cancellation

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.