On the cohomology of vector fields on parallelizable manifolds

TL;DR

This paper computes the second cohomology groups of the Lie algebra of smooth vector fields on parallelizable compact manifolds with coefficients in certain modules, revealing structures related to universal central extensions.

Contribution

It explicitly determines the cohomology spaces for vector fields on parallelizable manifolds, generalizing known algebraic structures like affine Kac-Moody algebras.

Findings

Cohomology spaces $H^2(V_M, ar ext{Omega}^p_M)$ are explicitly computed.

The case $p=1$ relates to universal central extensions of gauge algebras.

Classifies twists of semidirect products involving these extensions.

Abstract

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