Solitonic sectors, alpha-induction and symmetry breaking boundaries
  We develop a systematic approach to boundary conditions that break bulk symmetries in a general way such that left and right movers are not necessarily connected by an automorphism. In the context of string compactifications, such boundary conditions typically include non-BPS branes. Our formalism is based on two dual fusion rings, one for the bulk and one for the boundary fields. Only in the Cardy case these two structures coincide. In general they are related by a version of alpha-induction. Symmetry breaking boundary conditions correspond to solitonic sectors. In examples, we compute the annulus amplitudes and boundary states. 
J. Fuchs, C. Schweigert