Fusion Algebras of Logarithmic Minimal Models

We present explicit conjectures for the chiral fusion algebras of the logarithmic minimal models LM(p,p') considering Virasoro representations with no enlarged or extended symmetry algebra. The generators of fusion are countably infinite in number but the ensuing fusion rules are quasi-rational in the sense that the fusion of a finite number of representations decomposes into a finite direct sum of representations. The fusion rules are commutative, associative and exhibit an sl(2) structure but require so-called Kac representations which are reducible yet indecomposable representations of rank 1. In particular, the identity of the fundamental fusion algebra is in general a reducible yet indecomposable Kac representation of rank 1. We make detailed comparisons of our fusion rules with the results of Gaberdiel and Kausch for p=1 and with Eberle and Flohr for (p,p')=(2,5) corresponding to the logarithmic Yang-Lee model. In the latter case, we confirm the appearance of indecomposable representations of rank 3. We also find that closure of a fundamental fusion algebra is achieved without the introduction of indecomposable representations of rank higher than 3. The conjectured fusion rules are supported, within our lattice approach, by extensive numerical studies of the associated integrable lattice models. Details of our lattice findings and numerical results will be presented elsewhere. The agreement of our fusion rules with the previous fusion rules lends considerable support for the identification of the logarithmic minimal models LM(p,p') with the augmented c_{p,p'} (minimal) models defined algebraically.Comment: 22 pages, v2: comments adde

Grothendieck ring and Verlinde-like formula for the W-extended logarithmic minimal model WLM(1,p)

We consider the Grothendieck ring of the fusion algebra of the W-extended logarithmic minimal model WLM(1,p). Informally, this is the fusion ring of W-irreducible characters so it is blind to the Jordan block structures associated with reducible yet indecomposable representations. As in the rational models, the Grothendieck ring is described by a simple graph fusion algebra. The 2p-dimensional matrices of the regular representation are mutually commuting but not diagonalizable. They are brought simultaneously to Jordan form by the modular data coming from the full (3p-1)-dimensional S-matrix which includes transformations of the p-1 pseudo-characters. The spectral decomposition yields a Verlinde-like formula that is manifestly independent of the modular parameter $\tau$ but is, in fact, equivalent to the Verlinde-like formula recently proposed by Gaberdiel and Runkel involving a $\tau$-dependent S-matrix.Comment: 13 pages, v2: example, comments and references adde

Logarithmic Superconformal Minimal Models

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.Comment: 55 pages, v2: comments and references adde

Polynomial Fusion Rings of Logarithmic Minimal Models

We identify quotient polynomial rings isomorphic to the recently found fundamental fusion algebras of logarithmic minimal models.Comment: 18 page

Solvable Critical Dense Polymers on the Cylinder

A lattice model of critical dense polymers is solved exactly on a cylinder with finite circumference. The model is the first member LM(1,2) of the Yang-Baxter integrable series of logarithmic minimal models. The cylinder topology allows for non-contractible loops with fugacity alpha that wind around the cylinder or for an arbitrary number ell of defects that propagate along the full length of the cylinder. Using an enlarged periodic Temperley-Lieb algebra, we set up commuting transfer matrices acting on states whose links are considered distinct with respect to connectivity around the front or back of the cylinder. These transfer matrices satisfy a functional equation in the form of an inversion identity. For even N, this involves a non-diagonalizable braid operator J and an involution R=-(J^3-12J)/16=(-1)^{F} with eigenvalues R=(-1)^{ell/2}. The number of defects ell separates the theory into sectors. For the case of loop fugacity alpha=2, the inversion identity is solved exactly for the eigenvalues in finite geometry. The eigenvalues are classified by the physical combinatorics of the patterns of zeros in the complex spectral-parameter plane yielding selection rules. The finite-size corrections are obtained from Euler-Maclaurin formulas. In the scaling limit, we obtain the conformal partition functions and confirm the central charge c=-2 and conformal weights Delta_t=(t^2-1)/8. Here t=ell/2 and t=2r-s in the ell even sectors with Kac labels r=1,2,3,...; s=1,2 while t is half-integer in the ell odd sectors. Strikingly, the ell/2 odd sectors exhibit a W-extended symmetry but the ell/2 even sectors do not. Moreover, the naive trace summing over all ell even sectors does not yield a modular invariant.Comment: 44 pages, v3: minor correction

Fusion hierarchies, TT-systems and YY-systems for the A2(1)A_2^{(1)} models

The family of $A^{(1)}_2$ models on the square lattice includes a dilute loop model, a $15$-vertex model and, at roots of unity, a family of RSOS models. The fused transfer matrices of the general loop and vertex models are shown to satisfy $s\ell(3)$-type fusion hierarchies. We use these to derive explicit $T$- and $Y$-systems of functional equations. At roots of unity, we further derive closure identities for the functional relations and show that the universal $Y$-system closes finitely. The $A^{(1)}_2$ RSOS models are shown to satisfy the same functional and closure identities but with finite truncation.Comment: 36 page

Fusion hierarchies, T-systems and Y-systems of logarithmic minimal models

A Temperley-Lieb (TL) loop model is a Yang-Baxter integrable lattice model with nonlocal degrees of freedom. On a strip of width N, the evolution operator is the double-row transfer tangle D(u), an element of the TL algebra TL_N(beta) with loop fugacity beta=2cos(lambda). Similarly on a cylinder, the single-row transfer tangle T(u) is an element of the enlarged periodic TL algebra. The logarithmic minimal models LM(p,p') comprise a subfamily of the TL loop models for which the crossing parameter lambda=(p'-p)pi/p' is parameterised by coprime integers 0<p<p'. For these special values, additional symmetries allow for particular degeneracies in the spectra that account for the logarithmic nature of these theories. For critical dense polymers LM(1,2), D(u) and T(u) are known to satisfy inversion identities that allow us to obtain exact eigenvalues in any representation and for all system sizes N. The generalisation for p'>2 takes the form of functional relations for D(u) and T(u) of polynomial degree p'. These derive from fusion hierarchies of commuting transfer tangles D^{m,n}(u) and T^{m,n}(u) where D(u)=D^{1,1}(u) and T(u)=T^{1,1}(u). The fused transfer tangles are constructed from (m,n)-fused face operators involving Wenzl-Jones projectors P_k on k=m or k=n nodes. Some projectors P_k are singular for k>p'-1, but we argue that D^{m,n}(u) and T^{m,n}(u) are well defined for all m,n. For generic lambda, we derive the fusion hierarchies and the associated T- and Y-systems. For the logarithmic theories, the closure of the fusion hierarchies at n=p' translates into functional relations of polynomial degree p' for D^{m,1}(u) and T^{m,1}(u). We also derive the closure of the Y-systems for the logarithmic theories. The T- and Y-systems are the key to exact integrability and we observe that the underlying structure of these functional equations relate to Dynkin diagrams of affine Lie algebras.Comment: 77 page

What is a Good Plan? Cultural Variations in Expert Planners’ Concepts of Plan Quality

This article presents the results of a field research study examining commonalities and differences between American and British operational planners’ mental models of planning. We conducted Cultural Network Analysis (CNA) interviews with 14 experienced operational planners in the US and UK. Our results demonstrate the existence of fundamental differences between the way American and British expert planners conceive of a high quality plan. Our results revealed that the American planners’ model focused on specification of action to achieve synchronization, providing little autonomy at the level of execution, and included the belief that increasing contingencies reduces risk. The British planners’ model stressed the internal coherence of the plan, to support shared situational awareness and thereby flexibility at the level of execution. The British model also emphasized the belief that reducing the number of assumptions decreases risk. Overall, the American ideal plan serves a controlling function, whereas the British ideal plan supports an enabling function. Interestingly, both the US and UK would view the other’s ideal plan as riskier than their own. The implications of cultural models of plans and planning are described for establishing performance measures and designing systems to support multinational planning teams

US/UK Mental Models of Planning: The Relationship Between Plan Detail and Plan Quality

This paper presents the results of a research study applying a new cultural analysis method to capture commonalities and differences between US and UK mental models of operational planning. The results demonstrate the existence of fundamental differences between the way US and UK planners think about what it means to have a high quality plan. Specifically, the present study captures differences in how US and UK planners conceptualize plan quality. Explicit models of cultural differences in conceptions of plan quality are useful for establishing performance metrics for multinational planning teams. This paper discusses the prospects of enabling automatic evaluation of multinational team performance by combining recent advances in cultural modelling with enhanced ontology languages