A commutant realization of Odake's algebra

TL;DR

This paper constructs a new realization of Odake's algebra within a vertex algebra framework, revealing a Howe pair structure with affine sl_2, and generalizing to other Lie algebra representations.

Contribution

It demonstrates that Odake's algebra and an affine sl_2 algebra form a Howe pair inside a specific vertex algebra, extending the understanding of algebraic structures in conformal field theory.

Findings

Odake's algebra is realized as a commutant in a bcγ-system

Odake's algebra and affine sl_2 form a Howe pair

Generalization to other Lie algebra representations

Abstract

The bc\beta\gamma-system W of rank 3 has an action of the affine vertex algebra V_0(sl_2), and the commutant vertex algebra C =Com(V_0(sl_2), W) contains copies of V_{-3/2}(sl_2) and Odake's algebra O. Odake's algebra is an extension of the N=2 superconformal algebra with c=9, and is generated by eight fields which close nonlinearly under operator product expansions. Our main result is that V_{-3/2}(sl_2) and O form a Howe pair (i.e., a pair of mutual commutants) inside C. More generally, any finite-dimensional representation of a Lie algebra g gives rise to a similar Howe pair, and this example corresponds to the adjoint representation of sl_2.