Twisted De Rham complex on line and $\widehat{frak{sl}_2}$ singular vectors

Alexey Slinkin, Alexander Varchenko

Published 2018-12-23Version 1

We consider two complexes. The first complex is the twisted De Rham complex of scalar meromorphic differential forms on projective line, holomorphic on the complement to a finite set of points. The second complex is the chain complex of the Lie algebra of $\frak{sl}_2$-valued algebraic functions on the same complement, with coefficients in a tensor product of contragradient dual Verma modules over the affine Lie algebra $\widehat{frak{sl}_2}$. In [SV2] a construction of a monomorphism of the first complex to the second was suggested. It was indicated in [SV2] that under this monomorphism the existence of singular vectors in the Verma modules (the Malikov-Feigin-Fuchs singular vectors) is reflected in the relations between the cohomology classes of the De Rham complex. In this paper we prove the results formulated in [SV2].