On unitary vertex operator algebras
Abstract
We prove that a conformal vertex algebra with a non-zero invariant Hermitian form is unitary as soon as it admits a non-zero positive energy module with a positive definite invariant Hermitian form. Using free field realizations, we deduce that the principal simple W-algebra Wk(g) of a simple Lie algebra g is unitary for central charge c≥ rank(g)+24((ρ|ρ∨)+|ρ| |ρ∨|), and, if g is simply laced, for all c≥ rank(g). Among basic Lie superalgebras which are not Lie algebras, the free field argument applies to $osp(1|2n)$ and $osp(2|2n)$. It shows that Wk(osp(1|2n)) is unitary for c≥ n+1/2 and gives an explicit bound for $osp(2|2n)$. Next, we prove that if s is an even sl2-subalgebra of a simple Lie algebra g whose centralizer is not abelian, then the simple W-algebra Wk(g,s) is unitary for at most finitely many k. We illustrate this on the W-algebras of slqm attached to the nilpotents of Jordan type qm: unitarity is possible only at two levels, and for q=2 it does occur at both. For instance, the W-algebra of sl4 attached to the partition 2+2 is unitary only for k=−2 and k=−3/2. This fails for Lie superalgebras, as minimal W-algebras show, but still holds under stronger assumptions. Finally, we prove that a simple W-algebra Wk(g,s) can be unitary for only finitely many k if g is a simple Lie algebra and s is not a distinguished sl2 subalgebra.
Related
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- Classification of Strongly Rational Vertex Operator Algebras with Central and Effective Central Charge One
- Reverse Module Categories of Vertex Operator Algebras
- Classification and C2-cofiniteness of rational vertex operator algebras generated in low conformal weights
- Vector Bundles of Coinvariants for Admissible Affine Vertex Operator Algebras
BibTeX
@misc{kac2026unitary,
title = {{On unitary vertex operator algebras}},
author = {Victor G. Kac and Pierluigi Möseneder Frajria and Paolo Papi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29878},
eprint = {2610.29878},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29878},
}
RIS
TY - UNPB
TI - On unitary vertex operator algebras
AU - Victor G. Kac
AU - Pierluigi Möseneder Frajria
AU - Paolo Papi
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29878
UR - https://arxiv.org/abs/2610.29878
AB - We prove that a conformal vertex algebra with a non-zero invariant Hermitian form is unitary as soon as it admits a non-zero positive energy module with a positive definite invariant Hermitian form. Using free field realizations, we deduce that the principal simple W-algebra W_k(g) of a simple Lie algebra g is unitary for central charge c≥ rank(g)+24((ρ|ρ^∨)+|ρ| |ρ^∨|), and, if g is simply laced, for all c≥ rank(g). Among basic Lie superalgebras which are not Lie algebras, the free field argument applies to $osp(1|2n)$ and $osp(2|2n)$. It shows that W_k(osp(1|2n)) is unitary for c≥ n+1/2 and gives an explicit bound for $osp(2|2n)$. Next, we prove that if s is an even sl_2-subalgebra of a simple Lie algebra g whose centralizer is not abelian, then the simple W-algebra W_k(g,s) is unitary for at most finitely many k. We illustrate this on the W-algebras of sl_qm attached to the nilpotents of Jordan type q^m: unitarity is possible only at two levels, and for q=2 it does occur at both. For instance, the W-algebra of sl_4 attached to the partition 2+2 is unitary only for k=−2 and k=−3/2. This fails for Lie superalgebras, as minimal W-algebras show, but still holds under stronger assumptions. Finally, we prove that a simple W-algebra W_k(g,s) can be unitary for only finitely many k if g is a simple Lie algebra and s is not a distinguished sl_2 subalgebra.
ER -