nLab
nonabelian Stokes theorem

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

tangent cohesion

differential cohesion

graded differential 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& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\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

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

For instance theorem 3.4 in

Revised on March 10, 2017 14:19:16 by Urs Schreiber (46.183.103.17)