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arXiv · 1812.09791

Twisted de Rham Complex on Line and Singular Vectors in $\hat{{\mathfrak{sl}_2}}$ Verma Modules

Alexey Slinkin·Alexander Varchenko
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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 $\mathfrak{sl}_2$-valued algebraic functions on the same complement, with coefficients in a tensor product of contragradient Verma modules over the affine Lie algebra $\hat{{\mathfrak{sl}_2}}$. 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.

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Alexey Slinkin, Alexander Varchenko. 2018-12-23. Twisted de Rham Complex on Line and Singular Vectors in $\hat{{\mathfrak{sl}_2}}$ Verma Modules. https://doi.org/10.3842/sigma.2019.075

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