[Paper Review] Representation theory of a W-algebra from generalised DS reduction

Peter Bowcock|ArXiv.org|Mar 25, 1994

Algebraic structures and combinatorial models2 references3 citations

TL;DR

This paper constructs the W-algebra arising from generalized Drinfeld-Sokolov reduction of a B₂ WZW model using a short root grading, explicitly deriving its quantum generators and a free field realization. It establishes a screening charge that commutes with zero-grade currents, enabling a complete representation theory via fusion rules, character formulas, and a Kac determinant formula for degenerate primary fields.

ABSTRACT

We investigate the W-algebra resulting from Drinfel'd-Sokolov reduction of a $B_2$ WZW model with respect to the grading induced by a short root. The quantum algebra, which is generated by three fields of spin-2 and a field of spin-1, is explicitly constructed. A `free field' realisation of the algebra in terms of the zero-grade currents is given, and it is shown that these commute with a screening charge. We investigate the representation theory of the algebra using a combination of the explicit fusion method of Bauer et al. and free field methods. We discuss the fusion rules of degenerate primary fields, and give various character formulae and a Kac determinant formula for the algebra

Motivation & Objective

• To construct the quantum W-algebra obtained from Drinfeld-Sokolov reduction of a B₂ WZW model under a short root grading.
• To provide an explicit free field realization of the algebra in terms of zero-grade currents.
• To develop a representation theory framework using fusion rules and screening charges.
• To derive character formulae and a Kac determinant formula for degenerate primary fields in the algebra.
• To establish the commutativity of the screening charge with the zero-grade currents, enabling representation-theoretic analysis.

Proposed method

• Generalized Drinfeld-Sokolov reduction is applied to a B₂ WZW model using a grading induced by a short root.
• The resulting W-algebra is shown to be generated by three spin-2 fields and one spin-1 field.
• A free field realization is constructed using the zero-grade currents of the underlying affine algebra.
• The screening charge is explicitly identified and proven to commute with the zero-grade currents.
• The fusion rules of degenerate primary fields are computed using the explicit fusion method of Bauer et al.
• Character formulae and a Kac determinant formula are derived for the algebra's highest weight representations.

Experimental results

Research questions

• RQ1 What is the explicit structure of the W-algebra obtained from generalized DS reduction of a B₂ WZW model with short root grading?
• RQ2 How can the W-algebra be realized in terms of free fields using the zero-grade currents?
• RQ3 What is the role of the screening charge in the representation theory of this W-algebra?
• RQ4 What are the fusion rules for degenerate primary fields in this algebra?
• RQ5 What character formulae and Kac determinant formula can be derived for highest weight modules of this W-algebra?

Key findings

• The W-algebra is explicitly constructed with three spin-2 generators and one spin-1 generator, confirming its non-linear structure.
• A free field realization is achieved using the zero-grade currents of the B₂ affine algebra.
• The screening charge commutes with all zero-grade currents, a crucial condition for constructing representations.
• Fusion rules for degenerate primary fields are computed explicitly using the method of Bauer et al.
• Character formulae are derived for highest weight modules, providing a tool for analyzing the algebra's spectrum.
• A Kac determinant formula is obtained, enabling the identification of singular vectors and unitarity conditions in the representation theory.