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

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 $\mathbf{sl}_2$-valued algebraic functions on the same complement with coefficients in a tensor product of contragradient Verma modules over the affine Lie algebra $\widehat{sl}_2$. 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.