The Logarithmic Conformal Field Theories

TL;DR

This paper analyzes the correlation functions and operator product expansions in logarithmic conformal field theories, providing explicit calculations and a method to derive logarithmic correlators from ordinary conformal theory.

Contribution

It introduces a formal derivative approach to relate logarithmic fields to ordinary fields, enabling systematic calculation of correlation functions and OPE coefficients.

Findings

Explicit formulas for two- and three-point functions.

Logarithmic fields as derivatives of ordinary fields.

OPE coefficients derived from ordinary conformal theory.

Abstract

We study the correlation functions of logarithmic conformal field theories. First, assuming conformal invariance, we explicitly calculate two-- and three-- point functions. This calculation is done for the general case of more than one logarithmic field in a block, and more than one set of logarithmic fields. Then we show that one can regard the logarithmic field as a formal derivative of the ordinary field with respect to its conformal weight. This enables one to calculate any $n$-- point function containing the logarithmic field in terms of ordinary $n$--point functions. At last, we calculate the operator product expansion (OPE) coefficients of a logarithmic conformal field theory, and show that these can be obtained from the corresponding coefficients of ordinary conformal theory by a simple derivation.