A Quantum Deformation of the Virasoro Algebra and the Macdonald Symmetric Functions
Jun'ichi Shiraishi, Harunobu Kubo, Hidetoshi Awata, Satoru Odake
TL;DR
This work defines a quantum deformation of the Virasoro algebra, $\mathcal{V}ir_{q,t}$, and develops its representation theory and free-boson realization. It conjectures Kac determinants at arbitrary levels and demonstrates that singular vectors can be realized by Macdonald symmetric functions, specifically for rectangular diagrams, via screening currents. A bosonic realization is constructed that expresses the Macdonald operator $D_{q,t}$ through a dressed $q$-Virasoro current, illuminating a deep link between the deformed Virasoro structure, Macdonald theory, and Calogero-Sutherland–type models. The results mirror the classical Virasoro–Jack correspondence and suggest connections to quantum affine algebras and integrable systems, with several limits clarifying relationships to known structures and guiding future investigations.
Abstract
A quantum deformation of the Virasoro algebra is defined. The Kac determinants at arbitrary levels are conjectured. We construct a bosonic realization of the quantum deformed Virasoro algebra. Singular vectors are expressed by the Macdonald symmetric functions. This is proved by constructing screening currents acting on the bosonic Fock space.