Origin of Logarithmic Operators in Conformal Field Theories

TL;DR

This paper investigates the emergence of logarithmic operators in various conformal field theories, including Coulomb gas, Liouville, and WZNW models, revealing their algebraic structures and conditions for appearance.

Contribution

It demonstrates that logarithmic operators arise when the puncture operator is included in Liouville theory and identifies their algebraic properties in WZNW models.

Findings

Logarithmic operators appear with the puncture operator in Liouville theory.

Logarithmic operators form Jordan blocks for currents and Virasoro algebra in WZNW models.

Logarithmic operators are found at specific levels in WZNW models.

Abstract

We study logarithmic operators in Coulomb gas models, and show that they occur when the ``puncture'' operator of the Liouville theory is included in the model. We also consider WZNW models for $SL(2,R)$, and for SU(2) at level 0, in which we find logarithmic operators which form Jordan blocks for the current as well as the Virasoro algebra.