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Semi-infinite Weil complex and the Virasoro algebra

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Abstract

We define a semi-infinite analogue of the Weil algebra associated an infinite-dimensional Lie algebra. It can be used for the definition of semi-infinite characteristic classes by analogy with the Chern-Weil construction. The second term of a spectral sequence of this Weil complex consists of the semi-infinite cohomology of the Lie algebra with coefficients in its “adjoint semi-infinite symmetric powers.” We compute this cohomology for the Virasoro algebra. This is just the BRST cohomology of the bosonic βγ-system with central charge 26. We give a complete description of the Fock representations of this bosonic system as modules over the virasoro algebra, using Friedan-Martinec-Shenker bosonization. We derive a combinatorial identity from this result.

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Authors and Affiliations

  1. Landau Institute for Theoretical Physics, Moscow, USSR

    Boris Feigin

  2. Department of Mathematics, Harvard University, 02138, Cambridge, MA, USA

    Edward Frenkel

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  1. Boris Feigin
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  2. Edward Frenkel
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Communicated by N.Yu. Reshetikhin

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Feigin, B., Frenkel, E. Semi-infinite Weil complex and the Virasoro algebra. Commun.Math. Phys. 137, 617–639 (1991). https://doi.org/10.1007/BF02100281

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