The spectra of quantum chains with free boundary conditions and Virasoro algebras

At the critical point the spectra of quantum chains with periodic and twisted boundary conditions are described by irreducible representations of two Virasoro algebras with the same central charge. The authors show that in the case of free boundary conditions, the spectra can be understood in terms of irreducible representations of a single Virasoro algebra. For the Ising and the three-state Potts models, the corresponding irreducible representations are identified.