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Moscow Mathematical Journal

Volume 17, Issue 4, October–December 2017  pp. 787–802.

Rational Differential Forms on the Line and Singular Vectors in Verma Modules over \widehat{sl}2

Authors:  Vadim Schechtman (1) and Alexander Varchenko (2)
Author institution:(1) Institut de Mathématiques de Toulouse – Université Paul Sabatier, 118 Route de Narbonne, 31062 Toulouse, France
(2) Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA


We construct a monomorphism of the De Rham complex of scalar multivalued meromorphic forms on the projective line, holomorphic on the complement to a finite set of points, to the chain complex of the Lie algebra of sl2-valued algebraic functions on the same complement with coefficients in a tensor product of contragradient Verma modules over the affine Lie algebra \widehat{sl}2. We show that the existence of singular vectors in the Verma modules (the Malikov–Feigin–Fuchs singular vectors) is reflected in the new relations between the cohomology classes of logarithmic differential forms.

2010 Math. Subj. Class. Primary: 17B56; Secondary: 17B67, 33C80, 52B30.

Keywords: Gauss–Manin connection, Malikov–Feigin–Fuchs singular vectors, conformal blocks.

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