Unitary representations of the W_3-algebra with c≥2

Carpi, Sebastiano and Tanimoto, Yoh and Weiner, Mihály (2020) Unitary representations of the W_3-algebra with c≥2. TRANSFORMATION GROUPS. ISSN 1083-4362 (In Press)

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Abstract

We prove unitarity of the vacuum representation of the W_3-algebra for all values ofthe central charge c≥2. We do it by modifying the free field realization of Fateev and Zamolodchikov resulting in a representation which, by a nontrivial argument, can beshown to be unitary on a certain invariant subspace, although it is not unitary on the fullspace of the two currents needed for the construction. These vacuum representationsgive rise to simple unitary vertex operator algebras. We also construct explicitly unitary representations for many positive lowest weight values. Taking into account the knownform of the Kac determinants, we then completely clarify thequestion of unitarity ofthe irreducible lowest weight representations of the W_3-algebra in the 2≤c≤98 region.

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