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ZN equivariant Virasoro algebra via alternative Sugawara constructions

Armin Ghazi* and Ahmad Moradpouri†

  • *Contact author: armin.gh.kh@gmail.com
  • †Contact author: Ahmadreza.Moradpour@gmail.com
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Phys. Rev. D 114, 066013 – Published 21 September, 2026

DOI: https://doi.org/10.1103/tmss-5n5q

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 δij, there exists another invariant tensor εij, which enables the construction of genuinely new realizations beyond the conventional one. We show that these new realizations arise from a ZN grading of the mode index n of the Virasoro generators Ln and the space of such realizations corresponds to points of a possibly singular algebraic variety. For the Z2 and Z3 cases, the space of all such constructions is topologically equivalent to a cylinder, while for Z4 it forms a noncompact real four-dimensional manifold. We show that the spaces of constructions for Z2N and Z2N+1 are closely similar. Furthermore, we reformulate the problem within an action-principle framework by introducing ZN-equivariant maps, which provide a systematic method for constructing conformal field theories endowed with these generalized Virasoro symmetries. This formulation reproduces the Z2 case and supports the idea that ZN 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 (up to local field transformations) of the well-known Virasoro-Kac-Moody algebra.

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Phys. Rev. D 114, 066013– Published 21 September, 2026
  • Received 1 July 2026
  • Accepted 24 August 2026
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