Twisted de Rham Complex on Line and Singular Vectors in $\hat{{\mathfrak{sl}_2}}$ Verma Modules
Abstract
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 -valued algebraic functions on the same complement, with coefficients in a tensor product of contragradient Verma modules over the affine Lie algebra . 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.
Cite
@article{arxiv.1812.09791,
title = {Twisted de Rham Complex on Line and Singular Vectors in $\hat{{\mathfrak{sl}_2}}$ Verma Modules},
author = {Alexey Slinkin and Alexander Varchenko},
journal= {arXiv preprint arXiv:1812.09791},
year = {2019}
}