Operator Product Expansion in Logarithmic Conformal Field Theory

TL;DR

This paper investigates operator product expansions in logarithmic conformal field theory, focusing on cases with primary fields and pre-logarithmic operators, and explores algebraic structures related to zero modes.

Contribution

It provides explicit OPE formulas for specific cases in logarithmic CFT and discusses the algebraic structures and assumptions involved.

Findings

Derived OPE formulas for primary-primary and pre-logarithmic cases

Identified algebraic structures generated by zero modes

Discussed potential generalizations to more complex cases

Abstract

In logarithmic conformal field theory, primary fields come together with logarithmic partner fields on which the stress-energy tensor acts non-diagonally. Exploiting this fact and global conformal invariance of two- and three-point functions, operator product expansions of logarithmic operators in arbitrary rank logarithmic conformal field theory are investigated. Since the precise relationship between logarithmic operators and their primary partners is not yet sufficiently understood in all cases, the derivation of operator product expansion formulae is only possible under certain assumptions. The easiest cases are studied in this paper: firstly, where operator product expansions of two primaries only contain primary fields, secondly, where the primary fields are pre-logarithmic operators. Some comments on generalization towards more relaxed assumptions are made, in particular towards the case where logarithmic fields are not quasi-primary. We identify an algebraic structure generated by the zero modes of the fields, which proves useful in determining settings in which our approach can be successfully applied.