Unitarity and strong graded locality of holomorphic vertex operator superalgebras with central charge at most 24
Tiziano Gaudio
TL;DR
The paper addresses unitarity and strong graded locality for nice holomorphic VOSAs with central charge $c\le 24$, clarifying exceptions involving fake copies such as $VB^\natural\hat{\otimes}F$ and $VO^{\natural}$. It develops a two-step strategy: (i) prove unitarity and strong locality for the $c=24$ case via dual pairs, lattice/moonshine structures, and Gui's unitary theory, and (ii) extend to $c<24$ using the Free-Fermion Splitting to reduce to the $c=24$ situation. A key outcome is the unitarity of all such VOSAs (excluding the specified fake copies) and the strong graded locality of 910 out of 969 central-charge-$24$ examples with nontrivial odd part, yielding holomorphic graded-local conformal nets. The work strengthens the link between algebraic VOSA methods and operator-algebraic CFT, providing a robust framework for constructing and classifying holomorphic graded-local nets from VOSAs, including connections to lattice and moonshine theories. Overall, it advances the program of understanding how unitarity and locality properties propagate through VOSA extensions and their associated conformal nets in the holomorphic, low-central-charge regime.
Abstract
We prove that all nice holomorphic vertex operator superalgebras (VOSAs) with central charge at most 24 and with non-trivial odd part are unitary, apart from the hypothetical ones arising as fake copies of the shorter moonshine VOSA or of the latter tensorized with a real free fermion VOSA. Furthermore, excluding the ones with central charge 24 of glueing type III and with no real free fermion, we show that they are all strongly graded-local. In particular, they naturally give rise to holomorphic graded-local conformal nets. In total, we are able to prove that 910 of the 969 nice holomorphic VOSAs with central charge 24 and with non-trivial odd part are strongly graded-local, without counting hypothetical fake copies of the shorter moonshine VOSA tensorized with a real free fermion VOSA.