Lattice Realizations of the Open Descendants of Twisted Boundary Conditions for sl(2) A-D-E Models

TL;DR

This paper explores lattice models with twisted boundary conditions on non-orientable surfaces, providing a complete lattice realization of open descendants of sl(2) A-D-E minimal theories, including their Klein bottle and M"obius strip amplitudes.

Contribution

It constructs transfer matrices for A-D-E lattice models that incorporate non-orientable geometries and involutions, advancing the understanding of boundary conditions in conformal field theories.

Findings

Realization of all Klein bottle amplitudes with lattice models

Inclusion of topological flip and involution in transfer matrices

Complete classification of open descendants in lattice models

Abstract

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 M\"obius 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.