Bits and Pieces in Logarithmic Conformal Field Theory

TL;DR

This work surveys the emergence and structure of logarithmic conformal field theories (LCFTs), focusing on how indecomposable (Jordan-cell) representations modify standard CFT tools. It develops a formalism for logarithmic null vectors using nilpotent variables, derives differential equations for correlation functions, and analyzes the Shapovalov form within LCFTs. The modular properties of LCFTs are explored through the c_{p,1} series and their W-algebras, with generalized characters involving Theta, derivative-Theta, and log-Theta functions; a Verlinde-like framework is proposed using a limiting procedure. The discussion highlights concrete realizations (notably c = -2) and physical applications such as the Haldane-Rezayi state, illustrating how LCFTs extend rational CFTs and challenge conventional notions of locality, unitarity, and modular invariance. Overall, the notes sketch a coherent program for incorporating indecomposable structures into the LCFT toolkit, with implications for statistical physics and string-inspired contexts, while acknowledging ongoing mathematical development and open questions.

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.