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

TL;DR

This paper investigates W-algebras with primary fields of specific dimensions, revealing two solutions for the algebra with fields of dimensions 3, 4, and 5, and identifying unique solutions related to known algebraic structures.

Contribution

It demonstrates the existence of two distinct solutions for W-algebras with certain primary fields and clarifies their relation to known algebras WA_4 and WA_5.

Findings

Two solutions for the W-algebra with fields of dimensions (3,4,5)

Null-fields appear in one solution, indicating exceptional algebra features

Only the WA_5 solution exists when an additional spin 6 field is included

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.