Deformed Kac-Moody and Virasoro Algebras

TL;DR

The paper presents a general method to twist algebras using charge operators, applies it to Kac-Moody and Virasoro algebras, and explores implications for algebra deformations and statistics.

Contribution

It introduces a novel construction for twisting algebras with charge operators and applies it to deform Kac-Moody and Virasoro algebras.

Findings

Different deformations of Kac-Moody algebra obtained

Constructed Virasoro algebra via Sugawara from deformed Kac-Moody algebra

Implications for particle statistics and algebraic structures

Abstract

Whenever the group $\R^n$ acts on an algebra $\calA$, there is a method to twist $\cal A$ to a new algebra $\calA_\theta$ which depends on an antisymmetric matrix $\theta$ ($\theta^{\mu \nu}=-\theta^{\nu \mu}=\mathrm{constant}$). The Groenewold-Moyal plane $\calA_\theta(\R^{d+1})$ is an example of such a twisted algebra. We give a general construction to realise this twist in terms of $\calA$ itself and certain ``charge'' operators $Q_\mu$. For $\calA_\theta(\R^{d+1})$, $Q_\mu$ are translation generators. This construction is then applied to twist the oscillators realising the Kac-Moody (KM) algebra as well as the KM currents. They give different deformations of the KM algebra. From one of the deformations of the KM algebra, we construct, via the Sugawara construction, the Virasoro algebra. These deformations have implication for statistics as well.