Rational differential forms on line and singular vectors in Verma modules over $\widehat {sl}_2$

TL;DR

This paper establishes a connection between rational differential forms on the projective line and singular vectors in Verma modules over affine , revealing new relations in their cohomology classes.

Contribution

It constructs a monomorphism linking the De Rham complex of multivalued forms to the chain complex of -valued functions with Verma module coefficients, highlighting the role of singular vectors.

Findings

Existence of singular vectors corresponds to new relations among cohomology classes.

The monomorphism provides a bridge between differential forms and representation theory.

Singular vectors influence the structure of the cohomology classes in the complex.

Abstract

We construct a monomorphism of the De Rham complex of scalar multivalued meromorphic forms on the projective line, holomorphic on the complement to a finite set of points, to the chain complex of the Lie algebra of $sl_2$-valued algebraic functions on the same complement with coefficients in a tensor product of contragradient Verma modules over the affine Lie algebra $\hat{sl}_2$. We show that the existence of singular vectors in the Verma modules (the Malikov-Feigin-Fuchs singular vectors) is reflected in the new relations between the cohomology classes of logarithmic differential forms.