The $Z_N$ equivariant Virasoro algebra via alternative Sugawara constructions

TL;DR

This paper introduces a new family of Virasoro algebra realizations derived from a $ ext{U}(1)^2$ Kac--Moody algebra using alternative Sugawara constructions, revealing rich geometric structures and systematic methods for extended conformal symmetries.

Contribution

It generalizes the Sugawara construction to include $ ext{Z}_N$-graded realizations, connecting algebraic structures with geometric varieties and formulating a systematic approach via $ ext{Z}_N$-equivariant maps.

Findings

New realizations arise from $ ext{Z}_N$-grading of Virasoro generators.

The space of constructions forms algebraic varieties with specific topologies.

Deformed Virasoro--Kac--Moody algebras are identified as nontrivial extensions.

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 $\delta^{ij}$, 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.