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The structure of theW ∞ algebra

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Abstract

We prove rigorously that the structure constants of the leading (highest spin) linear terms in the commutation relations of the conformal chiral operator algebraW ∞ are identical to those of the Diff +0 ℝ2 algebra generated by area preserving diffeomorphisms of the plane. Moreover, all quadratic terms of theW N algebra are found to be absent in the limitN→∞. In particular we show thatW ∞ is a central extension of Diff +0 ℝ2 with non-trivial cocycles appearing only in the commutation relations of its Virasoro subalgebra. We also propose a representation ofW ∞ in terms of a single scalar field in 2+1 dimensions and discuss its significance in the context of quantum field theory.

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Authors and Affiliations

  1. Center for Theoretical Physics, Department of Physics and Astronomy, University of Maryland, 20742, College Park, MD, USA

    I. Bakas

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  1. I. Bakas
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Communicated by N. Yu. Reshetikhin

Supported in part by the NSF grant PHY-87-17155

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Bakas, I. The structure of theW ∞ algebra. Commun.Math. Phys. 134, 487–508 (1990). https://doi.org/10.1007/BF02098443

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  • DOI: https://doi.org/10.1007/BF02098443

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