Logarithmic superconformal minimal models

Abstract

The higher fusion level logarithmic minimal models have recently been constructed as the diagonal GKO cosets where n ≥ 1 is an integer fusion level and k = nP/(P′− P) − 2 is a fractional level. For n = 1, these are the well-studied logarithmic minimal models . For n ≥ 2, we argue that these critical theories are realized on the lattice by n × n fusion of the n = 1 models. We study the critical fused lattice models within a lattice approach and focus our study on the n = 2 models. We call these logarithmic superconformal minimal models where P = |2p − p′|, P′ = p′ and p, p′ are coprime. These models share the central charges of the rational superconformal minimal models . Lattice realizations of these theories are constructed by fusing 2 × 2 blocks of the elementary face operators of the n = 1 logarithmic minimal models . Algebraically, this entails the fused planar Temperley–Lieb algebra which is a spin-1 Birman–Murakami–Wenzl tangle algebra with loop fugacity β2 = [x]3 = x2 + 1 + x−2 and twist ω = x4 where x = eiλ and λ = (p′− p)π/p′. The first two members of this n = 2 series are superconformal dense polymers with , β2 = 0 and superconformal percolation with c = 0, β2 = 1. We calculate the bulk and boundary free energies analytically. By numerically studying finite-size conformal spectra on the strip with appropriate boundary conditions, we argue that, in the continuum scaling limit, these lattice models are associated with the logarithmic superconformal models . For system size N, we propose finitized Kac character formulae of the form for s-type boundary conditions with r = 1, s = 1,2,3, …, ℓ = 0,1,2. The P, P′ dependence enters only in the fractional power of q in the prefactor and ℓ = 0,2 label the Neveu–Schwarz sectors (r + s even) and ℓ = 1 labels the Ramond sectors (r + s odd). Combinatorially, the finitized characters involve Motzkin and Riordan polynomials defined in terms of q-trinomial coefficients. Using the Hamiltonian limit and the finitized characters, we argue, from examples of finite-lattice calculations, that there exist reducible yet indecomposable representations for which the Virasoro dilatation operator L0 exhibits rank-2 Jordan cells, confirming that these theories are indeed logarithmic. We relate these results to the N = 1 superconformal representation theory.