Torus structure on graphs and twisted partition functions for minimal and affine models

Using the Ocneanu quantum geometry of ADE diagrams (and of other diagrams belonging to higher Coxeter–Dynkin systems), we discuss the classification of twisted partition functions for affine and minimal models in conformal field theory and study several examples associated with the WZW, Virasoro and cases.     

The Level Two and Three Modular Invariants of SU (n)

In this Letter, we explicitly classify all modular invariant partition functions for at levels 2 and 3. Previously, these were known only for level 1. Level 2 exceptions exist at r=9, 15, and 27;level 3 exceptions exist at r=4, 8, and 20. One of these is new, but the others were all anticipated by the rank-level duality relating level k and level r+1. The main recent result which this Letter rests on is the classification of -type invariants.     

On the Lattice of Subalgebras Associated with the Principle of Half-sided Modular Inclusion

Given a von Neumann algebra M with cyclic and separating vector and with the modular group it which is associated with the pair (M, ) we will investigate the von Neumann subalgebras N M which fulfil the principle of half-sided modular inclusion. We show that this set is almost a lattice with respect to intersection and union. Furthermore, in this set we can introduce an equivalence relation respecting the lattice structure. To every von Neumann subalgebra fulfilling the condition of half-sided modular inclusion is associated a unique one-parametric translation group which fulfils the spectrum condition. Within this setting, one deals with two orders, the order of inclusion of subalgebras, and the order of positive operators between the generators of the translations. The first order implies the reverse of the second but the converse holds only if the corresponding translations commute.     

The spin-1/2XXZ Heisenberg chain,the quantum algebra Uq[sl(2)], and duality transformations for minimal models

The finite-size scaling spectra of the spin-1/2XXZ Heisenberg chain with toroidal boundary conditions and an even number of sites provide a projection mechanism yielding the spectra of models with a central chargec < 1, including the unitary and nonunitary minimal series. Taking into account the half-integer angular momentum sectors—which correspond to chains with an odd number of sites—in many cases leads to new spinor operators appearing in the projected systems. These new sectors in theXXZ chain correspond to new types of frustration lines in the projected minimal models. The corresponding new boundary conditions in the Hamiltonian limit are investigated for the Ising model and the 3-state Potts model and are shown to be related to duality transformations which are an additional symmetry at their self-dual critical point. By different ways of projecting systems we find models with the same central charge sharing the same operator content and modular invariant partition function which, however, differ in the distribution of operators into sectors and hence in the physical meaning of the operators involved. Related to the projection mechanism in the continuum there are remarkable symmetry properties of the finiteXXZ chain. The observed degeneracies in the energy and momentum spectra are shown to be the consequence of intertwining relations involvingU _q [sl(2)] quantum algebra transformations.     

Exact Results for the Universal Area Distribution of Clusters in Percolation, Ising, and Potts Models

At the critical point in two dimensions, the number of percolation clusters of enclosed area greater than A is proportional to A ^–1, with a proportionality constant C that is universal. We show theoretically (based upon Coulomb gas methods), and verify numerically to high precision, that . We also derive, and verify to varying precision, the corresponding constant for Ising spin clusters, and for Fortuin–Kasteleyn clusters of the Q = 2, 3 and 4-state Potts models.     

Loop Observables for BF Theories in Any Dimension and the Cohomology of Knots

A generalization of Wilson loop observables for BF theories in any dimension is introduced within the Batalin–Vilkovisky framework. The expectation values of these observables are cohomology classes of the space of imbeddings of a circle. One of the resulting theories discussed in the Letter has only trivalent interactions and, irrespective of the actual dimension, looks like a three-dimensional Chern–Simons theory.