W-Algebras of Negative Rank

TL;DR

This paper explores the extension of W-algebras to negative ranks, demonstrating their finite generation and uncovering connections between different known W-algebras through analytic continuation.

Contribution

It introduces the definition of WA_{-n-1} algebras for negative ranks and reveals new relationships between existing W-algebras.

Findings

WA_{-n-1} algebras are finitely generated.

Connections between CP(k)-models and U(1)-cosets of Polyakov-Bershadsky-algebras.

Extension of W-algebras to negative integer ranks.

Abstract

Recently it has been discovered that the W-algebras (orbifold of) WD_n can be defined even for negative integers n by an analytic continuation of their coupling constants. In this letter we shall argue that also the algebras WA_{-n-1} can be defined and are finitely generated. In addition, we show that a surprising connection exists between already known W-algebras, for example between the CP(k)-models and the U(1)-cosets of the generalized Polyakov-Bershadsky-algebras.