The three point gauge algebra $\mathcal V\ltimes \mathfrak{sl}(2, \mathcal R) \oplus\left(\Omega_{\mathcal R}/d{\mathcal R}\right)$ and an action on a Fock space

TL;DR

This paper demonstrates that the semi-direct product of a three-point current algebra and a three-point Virasoro algebra acts on a Fock space, extending previous results on algebra actions.

Contribution

It proves that the gauge algebra, formed as a semi-direct product, acts on the Fock space, which was not previously established.

Findings

The gauge algebra acts on the Fock space.

Extension of algebra actions to the gauge algebra.

Validation of the semi-direct product action.

Abstract

The three point current algebra $\mathfrak{sl}(2,\mathcal R)$ where $\mathcal R=\mathbb C[t,t^{-1},u\,|\,u^2=t^2+4t ]$ and three-point Virasoro algebra both act on a previously constructed Fock space. In this paper we prove that the semi-direct product, i.e. the gauge algebra acts on the Fock space as well.