Covariant exterior derivative on ad(P)-valued forms

Let $\pi:P\to M$ be a principal G-bundle with Lie algebra $\mathfrak{g}$ . The adjoint bundle is the vector bundle

\mathrm{ad}(P)\coloneqq P\times_{\mathrm{Ad}} \mathfrak{g}\;\to\;M,

associated to the adjoint action of $G$ on the Lie algebra $\mathfrak{g}$ .

A principal connection on $P$ induces a connection $\nabla$ on $\mathrm{ad}(P)$ (equivalently, it induces the operator on tensorial forms described by the exterior covariant derivative). The covariant exterior derivative

d_\nabla:\Omega^k(M;\mathrm{ad}(P))\to \Omega^{k+1}(M;\mathrm{ad}(P))

is the unique graded derivation extending $\nabla$ and satisfying the usual Koszul formula: for $\alpha\in \Omega^k(M;\mathrm{ad}(P))$ and vector fields $X_0,\dots,X_k$ ,

\begin{aligned} (d_\nabla \alpha)(X_0,\dots,X_k) &= \sum_{i=0}^k (-1)^i\,\nabla_{X_i}\bigl(\alpha(X_0,\dots,\widehat{X_i},\dots,X_k)\bigr) \\ &\quad + \sum_{0\le i<j\le k} (-1)^{i+j}\,\alpha([X_i,X_j],X_0,\dots,\widehat{X_i},\dots,\widehat{X_j},\dots,X_k), \end{aligned}

where the bracket is the Lie bracket of vector fields on $M$ .

Locally, if $A\in \Omega^1(U;\mathfrak{g})$ is a local connection form and we identify $\mathrm{ad}(P)|_U\cong U\times \mathfrak{g}$ , then for $\alpha\in \Omega^k(U;\mathfrak{g})$ ,

d_\nabla \alpha = d\alpha + [A\wedge \alpha],

with $d$ the exterior derivative.

Examples

Sections of $\mathrm{ad}(P)$ (degree 0). For $\phi\in \Omega^0(M;\mathrm{ad}(P))$ , locally $\phi:U\to\mathfrak{g}$ , one has $d_\nabla \phi = d\phi + [A,\phi],$ which is the usual covariant derivative in the adjoint representation.

Bianchi identity on the base. Let $F\in \Omega^2(U;\mathfrak{g})$ be the local curvature 2-form. Then $d_\nabla F = 0$ is the Bianchi identity written as a covariant closure condition.

Abelian structure group. If $G$ is abelian, then $[A\wedge \alpha]=0$ for all $\alpha$ , so $d_\nabla$ reduces to the ordinary exterior derivative on $\mathfrak{g}$ -valued forms.