Unifying W-Algebras

TL;DR

This paper introduces unifying W-algebras derived from quantum Casimir W-algebras, revealing their truncation at specific central charges, their non-freely generated nature, and their connections via level-rank duality and coset constructions.

Contribution

It demonstrates the unification of W-algebras across different ranks and types, including new examples like WD_{-n}, and explores their structural properties and dualities.

Findings

Quantum Casimir W-algebras truncate at degenerate central charges.

Unifying W-algebras are non-freely generated and finitely presented.

Level-rank duality relates different unifying W-algebras.

Abstract

We show that quantum Casimir W-algebras truncate at degenerate values of the central charge c to a smaller algebra if the rank is high enough: Choosing a suitable parametrization of the central charge in terms of the rank of the underlying simple Lie algebra, the field content does not change with the rank of the Casimir algebra any more. This leads to identifications between the Casimir algebras themselves but also gives rise to new, `unifying' W-algebras. For example, the kth unitary minimal model of WA_n has a unifying W-algebra of type W(2,3,...,k^2 + 3 k + 1). These unifying W-algebras are non-freely generated on the quantum level and belong to a recently discovered class of W-algebras with infinitely, non-freely generated classical counterparts. Some of the identifications are indicated by level-rank-duality leading to a coset realization of these unifying W-algebras. Other unifying W-algebras are new, including e.g. algebras of type WD_{-n}. We point out that all unifying quantum W-algebras are finitely, but non-freely generated.