Abstract
We prove that an irreducible representation of the Virasoro algebra can be extracted from an irreducible representation space of theSL(2, ℛ) current algebra by putting a constraint on the latter using the Becchi-Rouet-Stora-Tyutin formalism. Thus there is aSL(2, ℛ) symmetry in the Virasoro algebra, but it is gauged and hidden. This construction of the Virasoro algebra is the quantum analogue of the Hamiltonian reduction. We then are naturally lead to consider a constrainedSL(2, ℛ) Wess-Zumino-Witten model. This system is also related to quantum field theory of coadjoint orbit of the Virasoro group. Based on this result, we present a canonical derivation of theSL(2, ℛ) current algebra in Polyakov's theory of two-dimensional gravity; it is a manifestation of theSL(2, ℛ) symmetry in conformal field theory hidden by the quantum Hamiltonian reduction. We also discuss the quantum Hamiltonian reduction of theSL(2, ℛ) current algebra and its relation to theW n -algebra of Zamolodchikov. This makes it possible to define a natural generalization of the geometric action for theW n -algebra despite its non-Lie-algebraic nature.
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Communicated by A. Jaffe
This paper is dedicated to the memory of Vadik G. Knizhnik
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Bershadsky, M., Ooguri, H. HeiddenSL(n) symmetry in conformal field theories. Commun.Math. Phys. 126, 49–83 (1989). https://doi.org/10.1007/BF02124331
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DOI: https://doi.org/10.1007/BF02124331