Integrable and Conformal Twisted Boundary Conditions for sl(2) A-D-E Lattice Models

Abstract

We study integrable realizations of conformal twisted boundary conditions for sl(2) unitary minimal models on a torus. These conformal field theories are realized as the continuum scaling limit of critical G = A,D,E lattice models with positive spectral parameter u > 0 and Coxeter number g. Integrable seams are constructed by fusing blocks of elementary local face weights. The usual A-type fusions are labelled by the Kac labels (r, s) and are associated with the Verlinde fusion algebra. We introduce a new type of fusion in the two braid limits u->\pm i\infty associated with the graph fusion algebra, and labelled by nodes $a,b\in G$ respectively. When combined with automorphisms, they lead to general integrable seams labelled by $x=(r,a,b,\kappa )\in (A_{g-2},H,H,Z_2)$ where H is the graph G itself for Type I theories and its parent for Type II theories. Identifying our construction labels with the conformal labels of Petkova and Zuber, we find that the integrable seams are in one-to-one correspondence with the conformal seams. The distinct seams are thus associated with the nodes of the Ocneanu quantum graph. The quantum symmetries and twisted partition functions are checked numerically for $|G|\le 6$. We also show, in the case of $D_{2l}$, that the non-commutativity of the Ocneanu algebra of seams arises because the automorphisms do not commute with the fusions.

43 pages. Typos fixed

References in corpus (13)

• Generalised twisted partition functions
• Interaction-Round-a-Face Models with Fixed Boundary Conditions: The ABF Fusion Hierarchy
• On the Classification of Bulk and Boundary Conformal Field Theories
• The many faces of Ocneanu cells
• Integrable and Conformal Boundary Conditions for sl(2) A-D-E Lattice Models and Unitary Minimal Conformal Field Theories
• Twisted partition functions for ADE boundary conformal field theories and Ocneanu algebras of quantum symmetries
• Integrable Lattice Realizations of Conformal Twisted Boundary Conditions
• Integrable Boundaries, Conformal Boundary Conditions and A-D-E Fusion Rules
• Spectrum of a duality-twisted Ising quantum chain
• Lattice Approach to Excited TBA Boundary Flows: Tricritical Ising Model
• Integrable Lattice Realizations of N=1 Superconformal Boundary Conditions
• On Discrete Symmetries in su(2) and su(3) Affine Theories and Related Graphs
• Integrable and Conformal Boundary Conditions for Z_k Parafermions on a Cylinder

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• Comments about quantum symmetries of SU(3) graphs
• Landau-Ginzburg Description of Boundary Critical Phenomena in Two Dimensions
• The critical Ising model on a torus with a defect line
• Shaping Lattice through irrelevant perturbation: Ising model
• Lattice Realizations of the Open Descendants of Twisted Boundary Conditions for sl(2) A-D-E Models
• Exact solution of the critical Ising model with special toroidal boundary conditions