# nLab nonabelian Stokes theorem

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

singular cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# 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 $\hat A \in \Omega^1(D^2, \mathfrak{g})$ 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.

For instance theorem 3.4 in