Bits and Pieces in Logarithmic Conformal Field Theory

TL;DR

This paper provides an overview of key aspects of logarithmic conformal field theories, including null vectors, correlation functions, modular invariance, and their connections to physical systems like quantum Hall states.

Contribution

It introduces methods for generalizing null vectors and characters in logarithmic CFTs, and proposes a Verlinde formula for fusion rules in this context.

Findings

Generalization of null vectors for logarithmic CFTs

Proposal of a Verlinde formula for fusion rules

Relations between logarithmic CFTs and quantum Hall states

Abstract

These are notes of my lectures held at the first School & Workshop on Logarithmic Conformal Field Theory and its Applications, September 2001 in Tehran, Iran. These notes cover only selected parts of the by now quite extensive knowledge on logarithmic conformal field theories. In particular, I discuss the proper generalization of null vectors towards the logarithmic case, and how these can be used to compute correlation functions. My other main topic is modular invariance, where I discuss the problem of the generalization of characters in the case of indecomposable representations, a proposal for a Verlinde formula for fusion rules and identities relating the partition functions of logarithmic conformal field theories to such of well known ordinary conformal field theories. These two main topics are complemented by some remarks on ghost systems, the Haldane-Rezayi fractional quantum Hall state, and the relation of these two to the logarithmic c=-2 theory.