On the Uniqueness of Diffeomorphism Symmetry in Conformal Field Theory

Abstract

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

Structure and Classification of Superconformal Nets

“…[8,13]. Moreover from the relations [G −r , G r ] = 2L 0 + c 3 (r 2 − 1 4 ), we find the energy bounds…”

“…By the above lemma, the representation of the diffeomorphism group belongs to the Bose subnet, so we may apply the uniqueness result in the local case in [13] and get the following:…”

“…With E = (1 + Γ)/2 the orthogonal projection onto A b Ω, clearly U | EH is the projective unitary representation of Diff(S 1 ) associated with A b , unique by [13,56]. As Ω is separating for the local algebras, U (g)E determines U (g) if g ∈ Diff (2) I (S 1 ) for any interval I ∈ I.…”

Noncommutative Spectral Invariants and Black Hole Entropy

“…By applying a general theorem by Wiesbrock, A is a Möbius covariant net (cf. [50,37,38,47]); moreover A is expected to be diffeomorphism covariant and the diffeomorphism symmetry uniquely determined (see [12]); for example this is the case when the quantum field is free, as A is then isomorphic to the net associated with the U(1)-current algebra, see [22] (this fact has been noticed again in [41]). We thus assume A to be diffeomorphism covariant.…”

“…There A is a local conformal net canonically arising on the horizon. According to the general analysis (by using Wiesbrock's theorem) A is a Möbius covariant net, but A is expected to be diffeomorphism covariant too [12]; for example this is the case when the quantum field is free, as A is then isomorphic to the net associated with the U(1)-current algebra [22] (see also [41]). The incremental free energy dF by adding the charge ρ and removing the charge σ (in the Hartle-Hawking state) in [36] or, more generally, its symmetrization, see [38,Thm.…”