Logarithmic Operators in Conformal Field Theory and The $\W_\infty$-algebra

TL;DR

This paper demonstrates that correlation functions in logarithmic conformal field theories are invariant under a differential realization of the Borel subalgebra of the _-algebra, providing explicit differential equations and OPE coefficients.

Contribution

It introduces a method to derive differential equations for logarithmic CFT correlation functions using the _-algebra's differential representation, including explicit OPE coefficients.

Findings

Correlation functions are invariant under _-algebra transformations.

Derived differential equations for three and four-point functions with logarithmic operators.

Computed OPE coefficients up to third level for logarithmic CFTs.

Abstract

It is shown explicitly that the correlation functions of Conformal Field Theories (CFT) with the logarithmic operators are invariant under the differential realization of Borel subalgebra of $\W_\infty$-algebra. This algebra is constructed by tensor-operator algebra of differential representation of ordinary $sl(2,C)$. This method allows us to write differential equations which can be used to find general expression for three and four-point correlation functions possessing logarithmic operators. The operator product expansion (OPE) coefficients of general logarithmic CFT are given up to third level.