Twisted de Rham Complex on Line and Singular Vectors in sl₂ˆ Verma Modules

Abstract

We consider two complexes. The first complex is the twisted de Rham complex of scalar meromorphic differential forms on the projective line, holomorphic on the complement to a finite set of points. The second complex is 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 sl₂ˆ. In [Schechtman V., Varchenko A., Mosc. Math. J. 17 (2017), 787-802] a construction of a monomorphism of the first complex to the second was suggested, and it was indicated 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 these results.