synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
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(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality $\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
∞-Lie theory (higher geometry)
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.
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.
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.
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
Last revised on March 10, 2017 at 14:19:16. See the history of this page for a list of all contributions to it.