OAI.MinimalVertex.main

The theorem states that, for a CFT-type vertex operator algebra A on a complex inner product space V (a vertex algebra with vacuum, state-field map and Jacobi identity, together with a conformal vector, central charge, finite-dimensional graded pieces with degree-zero part spanned by the vacuum, conformal vector in degree 2 acting as the grading operator, and the Virasoro relations), equipped with a unitary structure U (an antilinear involution fixing the vacuum and conformal vector, compatible with all modes, together with a unit-norm vacuum and an invariance relation between a mode and the corresponding mode of the transformed adjoint-side vector), if A is simple (nonzero vacuum and no ideals other than 0 and the whole space) and strongly rational (self-contragredient, rational in the sense that every admissible weak module is completely reducible, and C2-cofinite), then three things hold. First, A has polynomial energy bounds: for every a in V there are C>0 and natural numbers p,k with ||a_(n) b|| ≤ C(1+|n|)^p ||(1+L_0)^k b|| for all integers n and all b in V. Second, the CKLW strong locality property holds: all vectors satisfy these bounds, and for every proper circle arc I the von Neumann algebra generated by closed smeared fields supported in I, built on the completion of V, lies in the commutant of the algebra attached to the complementary arc. Third, the assignment of these interval algebras to proper arcs admits the structure of an irreducible conformal net, meaning a separable Hilbert space with isotony, locality, a continuous Mobius representation extended to a continuous projective representation of smooth circle diffeomorphisms with covariance and locality of the action, a unit invariant vacuum that is unique up to scalar and cyclic, a positive self-adjoint Hamiltonian generating the rotation flow, and trivial commutant of all interval algebras apart from scalars.