Is there a Stokes theorem for covariant derivatives?

A $V$-valued differential $n$-form $\omega$ on a manifold $M$ is a section of the bundle $\Lambda^n (T^*M) \otimes V$. (That is, the restriction $\omega_p$ to any tangent space $T_p M$ for $p \in M$ is a completely antisymmetric map $\omega_p : T_p M \times T_p M \times \cdots \times T_p M \to V$.) $V$ is a vector space here.

One can define a flat covariant derivative $\mathrm{d}\colon \Lambda^n (T^*M) \to \Lambda^{n+1} (T^*M)$ which is just the exterior derivative. It fulfills Stokes' theorem.

Assume now an algebra structure on $V$, and a representation $\rho$ of $V$ on a vector space $W$. For a chosen $V$-valued differential 1-form $\omega$, there is also a covariant derivative (like a principal connection) that acts on all $W$-valued differential forms $\phi$ by the formula $\mathrm{d}_\omega \phi := \mathrm{d} \phi + \omega \wedge_\rho \phi$. The product $\wedge_\rho$ is the composition of $\wedge$, which multiplies a $V$-valued $n$-form and a $W$-valued $m$-form to a $V \otimes W$-valued $(n+m)$-form, and $\rho$.

Is there a generalisation for Stokes' theorem for $\mathrm{d}_\omega$? Maybe something like $\int_M \mathrm{d}_\omega \phi = \int_{\partial M} \phi$ up to terms proportional to the curvature of $\mathrm{d}_\omega$?

Solution 1:

On 1-dimensional manifolds, this is just the holonomy (for circles) or parallel transport (for intervals). One integrates the connection with the path-ordered exponential. It seems that there is a Stokes' theorem for higher gauge theory: There is a notion of 2-holonomy of surfaces for 2-connections, see for example An Invitation to Higher Gauge Theory or Nonabelian Multiplicative Integration on Surfaces.