On Modular Invariant Partition Functions of Conformal Field Theories with Logarithmic Operators

TL;DR

Flohr extends modular analysis to conformal field theories with logarithmic operators by redefining characters and partition functions to accommodate non-diagonalizable $L_0$ actions. Focusing on the $c_{p,1}$ series at the boundary of the minimal models, he constructs extended $W$-algebras via screening charges and derives both singlet and triplet character theories, including logarithmic terms. He shows that the resulting characters form modular objects with a rich block structure; a modular-invariant logarithmic partition function $Z_{\log}[p]$ is built, and Verlinde's formula is shown to fail globally though a consistent fusion algebra emerges within regular blocks. He also links these theories to two-dimensional polymers, recovering known polymer partition functions and highlighting a one-dimensional moduli space for $c_{\mathrm{eff}}=1$ logarithmic CFTs that is disconnected from the ordinary $c=1$ moduli.

Abstract

We extend the definitions of characters and partition functions to the case of conformal field theories which contain operators with logarithmic correlation functions. As an example we consider the theories with central charge c = c(p,1) = 13-6(p+1/p), the ``border'' of the discrete minimal series. We show that there is a slightly generalized form of the property of rationality for such logarithmic theories. In particular, we obtain a classification of theories with c = c(p,1) which is similar to the A-D-E classification of c = 1 models.