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

## TL;DR

This paper establishes a connection between twisted de Rham complexes on the projective line and singular vectors in Verma modules of affine TA4A4A4A4A4A4A4A4A4A4A4A4A4A4A4A4A4A4A4A4A4A4A4A4A4A4A4A4A4A4 complex and proves the reflection of singular vectors in cohomology relations.

## Contribution

The paper proves the conjectured relationship between the de Rham complex and singular vectors in Verma modules, confirming a construction suggested earlier.

## Key findings

- Established a monomorphism from the de Rham complex to the Lie algebra chain complex.
- Proved that singular vectors correspond to relations in the cohomology classes.
- Confirmed the reflection of Malikov-Feigin-Fuchs singular vectors in the cohomology structure.

## 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.