Coset Realization of Unifying W-Algebras

TL;DR

This paper constructs and analyzes quantum coset W-algebras, demonstrating their role as unifying structures, their realizations through cosets, and their properties in classical and quantum limits.

Contribution

It introduces new quantum coset W-algebras, shows their unifying role, and explores their realizations and properties, including level-rank duality and orbifolding.

Findings

Constructed several quantum coset W-algebras.

Demonstrated their role as unifying W-algebras of Casimir type.

Explored classical limits indicating potential infinite generation.

Abstract

We construct several quantum coset W-algebras, e.g. sl(2,R)/U(1) and sl(2,R)+sl(2,R) / sl(2,R), and argue that they are finitely nonfreely generated. Furthermore, we discuss in detail their role as unifying W-algebras of Casimir W-algebras. We show that it is possible to give coset realizations of various types of unifying W-algebras, e.g. the diagonal cosets based on the symplectic Lie algebras sp(2n) realize the unifying W-algebras which have previously been introduced as `WD_{-n}'. In addition, minimal models of WD_{-n} are studied. The coset realizations provide a generalization of level-rank-duality of dual coset pairs. As further examples of finitely nonfreely generated quantum W-algebras we discuss orbifolding of W-algebras which on the quantum level has different properties than in the classical case. We demonstrate in some examples that the classical limit according to Bowcock and Watts of these nonfreely finitely generated quantum W-algebras probably yields infinitely nonfreely generated classical W-algebras.