nLab nonabelian Stokes theorem

Idea

The nonabelian Stokes theorem (e.g. Schreiber-Waldorf 11, theorem 3.4) is a generalization of the Stokes theorem to Lie algebra valued differential 1-forms and with integration of differential forms refined to parallel transport.

Statement

If $\hat A \in \Omega^1(D^2, \mathfrak{g})$ is a Lie algebra valued 1-form on the 2-disk then the parallel transport $\mathcal{P} \exp(\int_{S^1} A)$ of its restriction $A \in \Omega^1(S^1, \mathfrak{g})$ 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 $A \in \Omega^1(X, \mathfrak{g})$ 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 $inn(\mathfrak{g})$ of $\mathfrak{g}$ whose curvature 3-form $H = \mathbf{d}_A F_A$ 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 $\mathfrak{g}$ by “the path method” (see at Lie integration) is indeed the simply connected Lie group corresponding to $\mathfrak{g}$ by Lie theory.

References

• Robert L. Karp, Freydoon Mansouri, Jung S. Rno, Product Integral Formalism and Non-Abelian Stokes Theorem, J. Math. Phys. 40 (1999) 6033-6043 [arXiv:hep-th/9910173, doi:10.1063/1.533068]

• R. L. Karp, F. Mansouri, J. S. Rno, Product Integral Representations of Wilson Lines and Wilson Loops, and Non-Abelian Stokes Theorem, Turk. J. Phys. 24 (2000) 365-384 [arXiv:hep-th/9903221, journal page]

• Boguslaw Broda, Non-Abelian Stokes theorem in action, in Modern Nonlinear Optics Part 2, Wiley (2001) 429-468 [arXiv:math-ph/0012035, ISBN:978-0-471-46612-3]

In the context of higher parallel transport in principal 2-bundles with connection:

• Urs Schreiber, Konrad Waldorf, Theorem 3.4 in: Smooth Functors vs. Differential Forms, Homology, Homotopy Appl. 13 1 (2011) 143-203 [arXiv:0802.0663, euclid:hha/1311953350]

See also:

• Seramika Ariwahjoedi, Freddy Permana Zen, Alternative Derivation of the Non-Abelian Stokes Theorem in Two Dimensions, Symmetry 2023 15 11 (2000) [doi:10.3390/sym15112000]