Classification of Structure Constants for W-algebras from Highest Weights

TL;DR

This paper classifies the structure constants of W-algebras based on their lowest spin, providing formulas dependent on a functional parameter, and illustrates this with various algebra examples.

Contribution

It introduces a unified classification scheme for W-algebra structure constants using a functional parameter h(c), covering multiple algebra types.

Findings

Structured constants grouped by lowest spin

Derived formulas depend on h(c) for each algebra

Includes examples for various W-algebras and super-Virasoro

Abstract

We show that the structure constants of W-algebras can be grouped according to the lowest (bosonic) spin(s) of the algebra. The structure constants in each group are described by a unique formula, depending on a functional parameter h(c) that is characteristic for each algebra. As examples we give the structure constants C_{33}^4 and C_{44}^4 for the algebras of type W(2,3,4,...) (that include the WA_{n-1}-algebras) and the structure constant C_{44}^4 for the algebras of type W(2,4,...), especially for all the algebras WD_n, WB(0,n), WB_n and WC_n. It also includes the bosonic projection of the super-Virasoro algebra and a yet unexplained algebra of type W(2,4,6) found previously.