Direct Sum Structure of the Super Virasoro Algebra and a Fermion Algebra Arising from the Quantum Toroidal ${\mathfrak{gl}}_2$

Abstract

It is known that the $q$-deformed Virasoro algebra can be constructed from a certain representation of the quantum toroidal ${\mathfrak{gl}}_1$ algebra. In this paper, we apply the same construction to the quantum toroidal algebra of type ${\mathfrak{gl}}_2$ and study the properties of resulting generators $W_i(z)$ ($i=1,2$). The algebra generated by $W_i(z)$ can be regarded as a $q$-deformation of the direct sum $\mathsf{F}\oplus \mathsf{SVir}$, where $\mathsf{F}$ denotes the free fermion algebra and $\mathsf{SVir}$ stands for the $N=1$ super Virasoro algebra, also referred to as the $N=1$ superconformal algebra or the Neveu-Schwarz-Ramond algebra. Moreover, the generators $W_i(z)$ admit two screening currents, and we show that their degeneration limits coincide with the screening currents of $\mathsf{SVir}$. We also establish quadratic relations satisfied by $W_i(z)$ and show that they generate a pair of commuting $q$-deformed Virasoro algebras, which degenerate into two nontrivial commuting Virasoro algebras included in $\mathsf{F}\oplus \mathsf{SVir}$.

arXiv comment30 pages

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