Logarithmic Superconformal Minimal Models

Abstract

The higher fusion level logarithmic minimal models LM(P,P';n) have recently been constructed as the diagonal GKO cosets (A_1^{(1)})_k oplus (A_1^{(1)})_n / (A_1^{(1)})_{k+n} where n>0 is an integer fusion level and k=nP/(P'-P)-2 is a fractional level. For n=1, these are the logarithmic minimal models LM(P,P'). For n>1, we argue that these critical theories are realized on the lattice by n x n fusion of the n=1 models. For n=2, we call them logarithmic superconformal minimal models LSM(p,p') where P=|2p-p'|, P'=p' and p,p' are coprime, and they share the central charges of the rational superconformal minimal models SM(P,P'). Their mathematical description entails the fused planar Temperley-Lieb algebra which is a spin-1 BMW tangle algebra with loop fugacity beta_2=x^2+1+x^{-2} and twist omega=x^4 where x=e^{i(p'-p)pi/p'}. Examples are superconformal dense polymers LSM(2,3) with c=-5/2, beta_2=0 and superconformal percolation LSM(3,4) with c=0, beta_2=1. We calculate the free energies analytically. By numerically studying finite-size spectra on the strip with appropriate boundary conditions in Neveu-Schwarz and Ramond sectors, we argue that, in the continuum scaling limit, these lattice models are associated with the logarithmic superconformal models LM(P,P';2). For system size N, we propose finitized Kac character formulas whose P,P' dependence only enters in the fractional power of q in a prefactor. These characters involve Motzkin and Riordan polynomials defined in terms of q-trinomial coefficients. Using the Hamiltonian limit, we argue that there exist reducible yet indecomposable representations for which the Virasoro dilatation operator L_0 exhibits rank-2 Jordan blocks confirming that these theories are indeed logarithmic. We relate these results to the N=1 superconformal representation theory.

55 pages, v2: comments and references added

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Cited by in corpus (19)

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• A fusion for the periodic Temperley-Lieb algebra and its continuum limit
• Modularity of logarithmic parafermion vertex algebras
• Fusion hierarchies, T-systems and Y-systems of logarithmic minimal models
• Fusion rules for the logarithmic $N=1$ superconformal minimal models II: including the Ramond sector
• Fusion rules for the logarithmic $N=1$ superconformal minimal models I: the Neveu-Schwarz sector
• The action of the Virasoro algebra in quantum spin chains. I. The non-rational case
• Fused RSOS Lattice Models as Higher-Level Nonunitary Minimal Cosets
• Critical behaviour of loop models on causal triangulations
• Kac boundary conditions of the logarithmic minimal models
• Extended T-systems, Q matrices and T-Q relations for $s\ell (2)$ models at roots of unity
• Critical N=(1,1) General Massive Supergravity
• Logarithmic Minimal Models with Robin Boundary Conditions
• Staggered modules of $N=2$ superconformal minimal models
• Integrable models from singly generated planar algebras
• Staggered and affine Kac modules over $A_1^{(1)}$
• Bimodule structure of the mixed tensor product over $U_{q}s\ell (2|1)$ and quantum walled Brauer algebra
• Integrability of planar-algebraic models
• One-Dimensional Sums and Finitized Characters of $2\times 2$ Fused RSOS Models