On the uniqueness of diffeomorphism symmetry in conformal field theory

A Mobius covariant net of von Neumann algebras on S-1 is diffeomorphism covariant if its Mobius symmetry extends to diffeomorphism symmetry. We prove that in case the net is either a Virasoro net or any at least 4- regular net such an extension is unique: the local algebras together with the Mobius symmetry ( equivalently: the local algebras together with the vacuum vector) completely determine it. We draw the two following conclusions for such theories. ( 1) The value of the central charge c is an invariant and hence the Virasoro nets for different values of c are not isomorphic as Mobius covariant nets. ( 2) A vacuum preserving internal symmetry always commutes with the diffeomorphism symmetries. We further use our result to give a large class of new examples of nets ( even strongly additive ones), which are not diffeomorphism covariant; i. e. which do not admit an extension of the symmetry to Diff(+)( S-1).