Logarithmic Conformal Field Theory: Beyond an Introduction

TL;DR

This paper reviews key topics and examples in logarithmic conformal field theory, emphasizing continuum methods, modular properties, and the structure of bulk state spaces, providing insights beyond introductory material.

Contribution

It offers a comprehensive overview of logarithmic conformal field theories, including new strategies for understanding their modular and fusion properties.

Findings

Analysis of modular properties of characters

Construction of bulk modular invariants

Evaluation of Verlinde formulae in logarithmic theories

Abstract

This article aims to review a selection of central topics and examples in logarithmic conformal field theory. It begins with a pure Virasoro example, critical percolation, then continues with a detailed exposition of symplectic fermions, the fractional level WZW model on SL(2;R) at level -1/2 and the WZW model on the Lie supergroup GL(1|1). It concludes with a general discussion of the so-called staggered modules that give these theories their logarithmic structure, before outlining a proposed strategy to understand more general logarithmic conformal field theories. Throughout, the emphasis is on continuum methods and their generalisation from the familiar rational case. In particular, the modular properties of the characters of the spectrum play a central role and Verlinde formulae are evaluated with the results compared to the known fusion rules. Moreover, bulk modular invariants are constructed, the structures of the corresponding bulk state spaces are elucidated, and a formalism for computing correlation functions is discussed.