The $Z_N$ equivariant Virasoro algebra via alternative Sugawara constructions

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 ${\delta }^{ij}$, there exists another invariant tensor ${\epsilon }^{ij}$, 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.