The $Z_N$ equivariant Virasoro algebra via alternative Sugawara constructions

TL;DR

This work develops a comprehensive framework to realize infinite families of Virasoro-like algebras from the abelian $oldsymbol{U}(1)^2$ current system by introducing a $oldsymbol{ m Z}_N$ grading of Virasoro modes and leveraging a second invariant tensor $oldsymbol{ m oldsymbol{\epsilon}}^{ij}$. It characterizes the moduli spaces of these realizations for $N=2,3,4$ as cylinder-like and higher-dimensional manifolds, shows a universal pattern for even/odd $N$, and establishes a $oldsymbol{ m Z}_N$-equivariant action principle that reproduces and extends the algebraic constructions into conformal field theories. The paper also analyzes deformations of the Virasoro–Kac–Moody algebra, derives modified OPEs, and defines primary fields under these deformations, revealing new two-parameter families of algebras that are not equivalent to the conventional Sugawara realization. Collectively, these results provide a unified, symmetry-guided route to extended conformal algebras with potential applications to non-abelian generalizations and higher-dimensional generalizations of current algebras.

Abstract

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.