The three point gauge algebra \(\mathcal{V} \ltimes \mathfrak{sl}(2, \mathcal{R}) \oplus(\Omega_{\mathcal{R}} / d \mathcal{R})\) and its action on a Fock space (Q1755549)

In a previous paper two of these three authors [Cox and Jurisich, Pac. J. Math. 270, No. 1, 27--48 (2014; Zbl 1355.17030)] studied the universal central extension \(\hat{\mathfrak{g}}\) of the three-point current algebra \(\mathfrak{sl}(2,\mathcal{R})\) where \(\mathcal{R}=\mathbb{C}[t,t^{-1},u|u^2=t^2=4t]\) obtaining a Wakimoto-type realization of \(\hat{\mathfrak{g}}\). This is one of the simplest nontrivial examples of a Krichever-Novikov algebra beyond an affine Kac-Moody algebra. Then again the three authors gave a natural action of the three-point Virasoro algebra on \(\hat{\mathfrak{g}}\). In the present paper they put together these representations to get conditions so that the semidirect product of these two algebras acts on the Fock space.