Direct Sum Structure of the Super Virasoro Algebra and a Fermion Algebra Arising from the Quantum Toroidal $\mathfrak{gl}_2$
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866914097385177088 |
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| author | Ohkubo, Yusuke |
| author_facet | Ohkubo, Yusuke |
| contents | 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 $\mathsf{F} \oplus \mathsf{SVir}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_02926 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Direct Sum Structure of the Super Virasoro Algebra and a Fermion Algebra Arising from the Quantum Toroidal $\mathfrak{gl}_2$ Ohkubo, Yusuke Quantum Algebra High Energy Physics - Theory Mathematical Physics Representation Theory 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 $\mathsf{F} \oplus \mathsf{SVir}$. |
| title | Direct Sum Structure of the Super Virasoro Algebra and a Fermion Algebra Arising from the Quantum Toroidal $\mathfrak{gl}_2$ |
| topic | Quantum Algebra High Energy Physics - Theory Mathematical Physics Representation Theory |
| url | https://arxiv.org/abs/2505.02926 |