Toward logarithmic extensions of ^sl(2)_k conformal field models

TL;DR

This paper constructs a logarithmic extension of the ^sl(2)_k conformal field theory using fermionic screening operators, explores its W-algebra representations, and analyzes their modular properties, linking to the logarithmic (p,1) models.

Contribution

It introduces a new logarithmic extension of ^sl(2)_k models via fermionic screenings and characterizes its W-algebra representations and modular structure.

Findings

Constructed 2p W-algebra representations and their characters.

Demonstrated the modular group representation as a deformation of known structures.

Linked the W-algebra currents to the triplet W-algebra of the logarithmic (p,1) model.

Abstract

For positive integer p=k+2, we construct a logarithmic extension of the ^sl(2)_k conformal field theory of integrable representations by taking the kernel of two fermionic screening operators in a three-boson realization of ^sl(2)_k. The currents W^-(z) and W^+(z) of a W-algebra acting in the kernel are determined by a highest-weight state of dimension 4p-2 and charge 2p-1, and a (theta=1)-twisted highest-weight state of the same dimension 4p-2 and charge -2p+1. We construct 2p W-algebra representations, evaluate their characters, and show that together with the p-1 integrable representation characters they generate a modular group representation whose structure is described as a deformation of the (9p-3)-dimensional representation $R_{p-1} \oplus C^2 \tensor R_{p-1} \oplus R_{p-1} \oplus C^2 \tensor R_{p-1} \oplus C^3 \tensor R_{p-1}$, where R_{p-1} is the SL(2,Z)-representation on integrable representation characters and R_{p-1} is a (p+1)-dimensional SL(2,Z)-representation known from the logarithmic (p,1) model. The dimension 9p-3 is conjecturally the dimension of the space of torus amplitudes, and the C^n with n=2 and 3 suggest the Jordan cell sizes in indecomposable W-algebra modules. Under Hamiltonian reduction, the W-algebra currents map into the currents of the triplet W-algebra of the logarithmic (p,1) model.