Logarithmic superconformal minimal models

Pearce, Paul A., Rasmussen, Jørgen and Tartaglia, Elena (2014) Logarithmic superconformal minimal models. Journal of Statistical Mechanics: Theory and Experiment, 2014 5: P05001.1-P05001.61. doi:10.1088/1742-5468/2014/05/P05001

Author Pearce, Paul A.
Rasmussen, Jørgen
Tartaglia, Elena
Title Logarithmic superconformal minimal models
Journal name Journal of Statistical Mechanics: Theory and Experiment   Check publisher's open access policy
ISSN 1742-5468
Publication date 2014-05-09
Sub-type Article (original research)
DOI 10.1088/1742-5468/2014/05/P05001
Open Access Status
Volume 2014
Issue 5
Start page P05001.1
End page P05001.61
Total pages 61
Place of publication Bristol, United Kingdom
Publisher Institute of Physics Publishing
Language eng
Formatted 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 (A(1) 1)n/(A (1) 1)k+n 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 LM(P, P') ≡ LM(P; P'; 1). 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 LM(p, p')n×n within a lattice approach and focus our study on the n = 2 models. We call these logarithmic superconformal minimal models LSM(p, p') ≡ LM(P, P'; 2) where P = |2p - p'|, P' = p' and p' p' are coprime. These models share the central charges c = cP;P';2 = 3/2 (1 - 2(P' - P)2=PP') of the rational superconformal minimal models SM(P, P'). Lattice realizations of these theories are constructed by fusing 2 × 2 blocks of the elementary face operators of the n = 1 logarithmic minimal models LM(p, p'). 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 = e iλ and λ = (p' - p)π/p'. The first two members of this n = 2 series are superconformal dense polymers LSM(2, 3) with c = -5/2, β2 = 0 and superconformal percolation LSM(3, 4) 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 LM(P, P'; 2). For system size N, we propose finitized Kac character formulae of the form q -cPP';2/24+δP,P';2 r;s;ℓ Χ̃ (N) r;s;ℓ(q) 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 coeficients. 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. 
Keyword Conformal field theory
Loop models and polymers
Q-Index Code C1
Q-Index Status Confirmed Code
Institutional Status UQ

Document type: Journal Article
Sub-type: Article (original research)
Collections: School of Mathematics and Physics
Official 2015 Collection
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Created: Thu, 22 May 2014, 02:00:49 EST by Jorgen Rasmussen on behalf of School of Mathematics & Physics