We construct integrable lattice 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 A-D-E lattice models with positive spectral parameter. The integrable seam boundary conditions are labelled by (r, s, ζ) ∈ (A g−2 , A g−1 , Γ) where Γ is the group of automorphisms of the graph G and g is the Coxeter number of G = A, D, E. Taking symmetries into account, these are identified with conformal twisted boundary conditions of Petkova and Zuber labelled by (a, b, γ) ∈ (A g−2 ⊗G, A g−2 ⊗G, Z 2 ) and associated with nodes of the minimal analog of the Ocneanu quantum graph. Our results are illustrated using the Ising (A 2 , A 3 ) and 3-state Potts (A 4 , D 4 ) models.

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 → ±i∞ associated with the graph fusion algebra, and labelled by nodes a, b ∈ G respectively. When combined with automorphisms, they lead to general integrable seams labelled by x = (r, a, b, κ) ∈ (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| ≤ 6. We also show, in the case of D 2ℓ , that the noncommutativity of the Ocneanu algebra of seams arises because the automorphisms do not commute with the fusions. 1

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The twisted boundary conditions and associated partition functions of the conformal sl(2) A-D-E models are studied on the Klein bottle and the Möbius strip. The A-D-E minimal lattice models give realization to the complete classification of the open descendants of the sl(2) minimal theories. We construct the transfer matrices of these lattice models that are consistent with non-orientable geometries. In particular, we show that in order to realize all the Klein bottle amplitudes of different crosscap states, not only the topological flip on the lattice but also the involution in the spin configuration space must be taken into account. This involution is the Z 2 symmetry of the Dynkin diagrams which corresponds to the simple current of the Ocneanu algebra.