nLab
nonabelian Stokes theorem

Idea

The nonabelian Stokes theorem is a generalization of the Stokes theorem for Lie algebra valued differential 1-forms and with integration of differential forms refined to parallel transport.

Statement

If $\hat{A}\in {\Omega }^1(D^2,𝔤)$ is a Lie algebra valued 1-form on the 2-disk then the parallel transport $𝒫\exp ({\int }_{S^1}A)$ of its restriction $A\in {\Omega }^1(S^1,𝔤)$ to the boundary circle, hence its holonomy (for a fixed choice of base point) is equal to a certain kind of adjusted 2-dimensional integral of its curvature 2-form $F_A$ over $D^2$.

In particular if $F_{\hat{A}}=0$ then the holonomy of $A$ is trivial.

Properties

Relation to higher parallel transport

In terms of the notion of connection on a 2-bundle the nonabelian Stokes theorem says that if $A\in {\Omega }^1(X,𝔤)$ is a Lie algebra valued 1-form, then $(F_A,A)$ is a Lie 2-algebra valued 2-form with values in the inner derivation Lie 2-algebra $\mathrm{inn}(𝔤)$ of $𝔤$ whose curvature 3-form $H=d_A{F}_A$ vanishes (which is the Bianchi identity for $F_A$) and its higher parallel transport exists. The 2-functorial source-target matching condition in this higher parallel transport is the statement of the nonabelian Stokes theorem.

Relation to Lie integration

For $F_A=0$ the nonabelian Stokes theorem may be regarded as proving that the Lie integration of $𝔤$ by “the path method” (see at Lie integration) is indeed the simply connected Lie group corresponding to $𝔤$ by Lie theory.

References

For instance in

• Urs Schreiber, Konrad Waldorf, Smooth Functors vs. Differential Forms (arXiv:0802.0663)