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Affine Kac-Moody algebras and semi-infinite flag manifolds

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Abstract

We study representations of affine Kac-Moody algebras from a geometric point of view. It is shown that Wakimoto modules introduced in [18], which are important in conformal field theory, correspond to certain sheaves on a semi-infinite flag manifold with support on its Schhubert cells. This manifold is equipped with a remarkable semi-infinite structure, which is discussed; in particular, the semi-infinite homology of this manifold is computed. The Cousin-Grothendieck resolution of an invertible sheaf on a semi-infinite flag manifold gives a two-sided resolution of an irreducible representation of an affine algebras, consisting of Wakimoto modules. This is just the BRST complex. As a byproduct we compute the homology of an algebra of currents on the real line with values in a nilpotent Lie algebra.

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Authors and Affiliations

  1. Profsoyuznaya, 13/12, apt 91, 117218, Moscow, USSR

    Boris L. Geigin

  2. Suvorova, 100, apt 27, 140410, Kolomna, Moscow district, USSR

    Edward V. Frenkel

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  1. Boris L. Geigin
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  2. Edward V. Frenkel
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Communicated by L. Alvarez-Gaumé

Dedicated to Dmitry Borisovich Fuchs on his 50th birthday

Address after September 15, 1989: Mathematics Department, Harvard University, Cambrdige, MA 02138, USA

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Geigin, B.L., Frenkel, E.V. Affine Kac-Moody algebras and semi-infinite flag manifolds. Commun.Math. Phys. 128, 161–189 (1990). https://doi.org/10.1007/BF02097051

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