Integrable and conformal twisted boundary conditions forsl(2)A-D-Elattice 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 $|G| \leq 6$. 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.