Toward logarithmic extensions of $\widehat{s\ell}(2)_k$ conformal field models

For positive integers $p=k+2$, we construct a logarithmic extension of the $\widehat{s\ell}_k$ conformal field theory of integrable representations by taking the kernel of two fermionic screening operators in a butterfly resolution of a three-boson realization of $\widehat{s\ell}_k$. 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 by a $(\theta=1)$-twisted highest-weight state of the same dimension $4p-2$ and opposite 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 $\mathscr{R}_{p+1}\oplus\mathbb{C}^2{\otimes}\mathscr{R}_{p+1}\oplus \mathscr{R}_{p-1}\oplus\mathbb{C}^2\otimes \mathscr{R}_{p-1}\oplus\mathbb{C}^3\otimes\mathscr{R}_{p-1}$, where $\mathscr{R}_{p-1}$ is the $SL(2,\mathbb{Z})$-representation on $\widehat{s\ell}_k$ integrable-representation characters and $\mathscr{R}_{p+1}$ is a $(p+1)$-dimensional $SL(2,\mathbb{Z})$-representation known from the logarithmic $(p,1)$ model. The dimension $9p-3$ is conjecturally the dimension of the space of torus amplitudes, and the $\mathbb{C}^n$ with $n=2$ and $3$ suggest the Jordan cell sizes in indecomposable $W$-algebra modules. We show that under Hamiltonian reduction, the $W$-algebra currents map into the currents of the triplet $W$-algebra of the logarithmic $(p,1)$ model.