nLab nonabelian Stokes theorem

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

\infty-Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

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 AΩ 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 AΩ 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

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

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]

Last revised on November 2, 2023 at 10:12:42. See the history of this page for a list of all contributions to it.