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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2512.00588 |
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| _version_ | 1866911294617026560 |
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| author | Ghazi, Armin Moradpouri, Ahmad |
| author_facet | Ghazi, Armin Moradpouri, Ahmad |
| contents | In this paper, we study the $U(1)^2$ Kac--Moody algebra and generalize the standard Sugawara construction of the Virasoro algebra to an infinite family of new realizations. In this case, in addition to the standard invariant tensor $δ^{ij}$, there exists another invariant tensor $ε^{ij}$, which enables the construction of genuinely new realizations beyond the conventional one. We show that these new realizations arise from a $\mathbb{Z}_N$--grading of the mode index $n$ of the Virasoro generators $L_n$ and the space of such realizations corresponds to points of a possibly singular algebraic variety. For the $\mathbb{Z}_2$ and $\mathbb{Z}_3$ cases, the space of all such constructions is topologically equivalent to a cylinder, while for $\mathbb{Z}_4$ it forms a non-compact real four-dimensional manifold. We show that the spaces of constructions for $Z_{2N}$ and $Z_{2N+1}$ are closely similar. Furthermore, we reformulate the problem within an action-principle framework by introducing $\mathbb{Z}_N$-equivariant maps, which provide a systematic method for constructing conformal field theories endowed with these generalized Virasoro symmetries. This formulation reproduces the $\mathbb{Z}_2$ case and supports the idea that $\mathbb{Z}_N$-equivariance offers a consistent and unified approach to generating extended conformal algebras. Finally, we analyze the corresponding Virasoro--Kac--Moody-like algebras associated with these constructions and show that they represent nontrivial deformations of the well-known Virasoro-Kac-Moody algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_00588 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The $Z_N$ equivariant Virasoro algebra via alternative Sugawara constructions Ghazi, Armin Moradpouri, Ahmad High Energy Physics - Theory In this paper, we study the $U(1)^2$ Kac--Moody algebra and generalize the standard Sugawara construction of the Virasoro algebra to an infinite family of new realizations. In this case, in addition to the standard invariant tensor $δ^{ij}$, there exists another invariant tensor $ε^{ij}$, which enables the construction of genuinely new realizations beyond the conventional one. We show that these new realizations arise from a $\mathbb{Z}_N$--grading of the mode index $n$ of the Virasoro generators $L_n$ and the space of such realizations corresponds to points of a possibly singular algebraic variety. For the $\mathbb{Z}_2$ and $\mathbb{Z}_3$ cases, the space of all such constructions is topologically equivalent to a cylinder, while for $\mathbb{Z}_4$ it forms a non-compact real four-dimensional manifold. We show that the spaces of constructions for $Z_{2N}$ and $Z_{2N+1}$ are closely similar. Furthermore, we reformulate the problem within an action-principle framework by introducing $\mathbb{Z}_N$-equivariant maps, which provide a systematic method for constructing conformal field theories endowed with these generalized Virasoro symmetries. This formulation reproduces the $\mathbb{Z}_2$ case and supports the idea that $\mathbb{Z}_N$-equivariance offers a consistent and unified approach to generating extended conformal algebras. Finally, we analyze the corresponding Virasoro--Kac--Moody-like algebras associated with these constructions and show that they represent nontrivial deformations of the well-known Virasoro-Kac-Moody algebra. |
| title | The $Z_N$ equivariant Virasoro algebra via alternative Sugawara constructions |
| topic | High Energy Physics - Theory |
| url | https://arxiv.org/abs/2512.00588 |