A regular normal parabolic geometry of type $G/P$ on a manifold $M$ gives rise to sequences $D_i$ of invariant differential operators, known as the curved version of the BGG resolution. These sequences are constructed from the normal covariant derivative ${\nabla }^{\omega }$ on the corresponding tractor bundle $V$, where $\omega$ is the normal Cartan connection. The first operator $D_0$ in the sequence is overdetermined and it is well known that ${\nabla }^{\omega }$ yields the prolongation of this operator in the homogeneous case $M=G/P$. Our first main result...

In the present paper we determine for each parallelizable smooth compact manifold $M$ the second cohomology spaces of the Lie algebra ${𝒱}_M$ of smooth vector fields on $M$ with values in the module $\overline{\Omega }{}_M^p={\Omega }_M^p/d{\Omega }_M^{p-1}$. The case of $p=1$ is of particular interest since the gauge algebra of functions on $M$ with values in a finite-dimensional simple Lie algebra has the universal central extension with center ${\overline{\Omega }}_M^1$, generalizing affine Kac-Moody algebras. The second cohomology $H^2({𝒱}_M,{\overline{\Omega }}_M^1)$ classifies twists of the semidirect product of ${𝒱}_M$ with the...