# On the uniqueness of diffeomorphism symmetry in Conformal Field Theory

July 11, 2004

A Moebius covariant net of von Neumann algebras on S^1 is diffeomorphism
covariant if its Moebius 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 Moebius
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 Moebius 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).

Keywords:

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