W-algebras with set of primary fields of dimensions (3, 4, 5) and (3,4,5,6)

TL;DR

This work classifies W-algebras with primaries of dimensions $3,4,5$ (and extensions to $6$) by enforcing Jacobi-identities on their OPEs. It uncovers two inequivalent solutions for $W(2,3,4,5)$: the known $WA_4$ and a second, generic-$c$ solution that harbors null-fields violating Jacobi-identities, akin to exceptional algebras, while the extension to spin $6$ yields only $WA_5$. The analysis also revisits $W(2,4,6)$, revealing null-field structures that in a Super-Virasoro realization vanish, underscoring how null-fields influence Jacobi-consistency. Overall, the paper raises questions about realizations of nontrivial null-field solutions and the landscape of higher-spin W-algebras.

Abstract

We show that that the Jacobi-identities for a W-algebra with primary fields of dimensions 3, 4 and 5 allow two different solutions. The first solution can be identified with WA_4. The second is special in the sense that, even though associative for general value of the central charge, null-fields appear that violate some of the Jacobi-identities, a fact that is usually linked to exceptional W-algebras. In contrast we find for the algebra that has an additional spin 6 field only the solution WA_5.