nonabelian Stokes theorem (changes) in nLab

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Contents

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 A^∈Ω 1(D 2,𝔤)\hat A \in \Omega^1(D^2, \mathfrak{g}) is a Lie algebra valued 1-form on the 2-disk then the parallel transport 𝒫exp(∫ S 1A)\mathcal{P} \exp(\int_{S^1} A) of its restriction A∈Ω 1(S 1,𝔤)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 AF_A over D 2D^2.

In particular if F A^=0F_{\hat A} = 0 then the holonomy of AA 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∈Ω 1(X,𝔤)A \in \Omega^1(X, \mathfrak{g}) is a Lie algebra valued 1-form, then (F A,A)(F_A, A) is a Lie 2-algebra valued 2-form with values in the inner derivation Lie 2-algebra inn(𝔤)inn(\mathfrak{g}) of 𝔤\mathfrak{g} whose curvature 3-form H=d AF AH = \mathbf{d}_A F_A vanishes (which is the Bianchi identity for F AF_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=0F_A = 0 the nonabelian Stokes theorem may be regarded as proving that the Lie integration of 𝔤\mathfrak{g} by “the path method” (see at Lie integration) is indeed the simply connected Lie group corresponding to 𝔤\mathfrak{g} by Lie theory.

References

For instance theorem 3.4 in

• 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](https://arxiv.org/abs/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](https://arxiv.org/abs/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](https://arxiv.org/abs/math-ph/0012035), ISBN:978-0-471-46612-3]

• Urs Schreiber, Konrad Waldorf, Smooth Functors vs. Differential Forms, Homology, Homotopy Appl., 13(1), 143-203 (2011) (arXiv:0802.0663)

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](http://arxiv.org/abs/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]( https://doi.org/10.3390/sym15112000)]