Abstract
We prove unitarity of the vacuum representation of the \( \mathcal{W} \)3-algebra for all values of the 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 be shown to be unitary on a certain invariant subspace, although it is not unitary on the full space of the two currents needed for the construction. These vacuum representations give rise to simple unitary vertex operator algebras. We also construct explicitly unitary representations for many positive lowest weight values. Taking into account the known form of the Kac determinants, we then completely clarify the question of unitarity of the irreducible lowest weight representations of the \( \mathcal{W} \)3-algebra in the 2 ≤ c ≤ 98 region.
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15 July 2022
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Sebastiano Carpi is supported in part by ERC advanced grant 669240 QUEST “Quantum Algebraic Structures and Models” and GNAMPA-INDAM.
Yoh Tanimoto is supported by Programma per giovani ricercatori, anno 2014 “Rita Levi Montalcini” of the Italian Ministry of Education, University and Research.
Mihály Weiner is supported in part by the Bolyai János and Bolyai+ scholarships, by the NRDI grants K 124152, KH 129601, K 132097 and by the ERC advanced grant 669240 QUEST “Quantum Algebraic Structures and Models”.
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CARPI, S., TANIMOTO, Y. & WEINER, M. UNITARY REPRESENTATIONS OF THE \( \mathcal{W} \)3-ALGEBRA WITH c ≥ 2. Transformation Groups 28, 561–590 (2023). https://doi.org/10.1007/s00031-022-09699-8
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DOI: https://doi.org/10.1007/s00031-022-09699-8