Lattice realizations of the open descendants of twisted boundary conditions forsl(2) A–D–E models

Abstract

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