Use of Nilpotent weights in Logarithmic Conformal Field Theories

TL;DR

This paper introduces a nilpotent weight approach to derive properties of logarithmic conformal field theories (LCFTs), including correlation functions, singular vectors, and boundary effects, providing new symmetry-based insights.

Contribution

It presents a novel method using nilpotent weights to systematically derive LCFT properties and their correlation functions from symmetry principles.

Findings

Derived two and three-point correlation functions using symmetry arguments.

Obtained singular vectors and Kac determinant via nilpotent variables.

Explored boundary effects and the AdS/LCFT correspondence.

Abstract

We show that logarithmic conformal field theories may be derived using nilpotent scale transformation. Using such nilpotent weights we derive properties of LCFT's, such as two and three point correlation functions solely from symmetry arguments. Singular vectors and the Kac determinant may also be obtained using these nilpotent variables, hence the structure of the four point functions can also be derived. This leads to non homogeneous hypergeometric functions. Also we consider LCFT's near a boundary. Constructing "superfields" using a nilpotent variable, we show that the superfield of conformal weight zero, composed of the identity and the pseudo identity is related to a superfield of conformal dimension two, which comprises of energy momentum tensor and its logarithmic partner. This device also allows us to derive the operator product expansion for logarithmic operators. Finally we discuss the AdS/LCFT correspondence and derive some correlation functions and a BRST symmetry.