nonabelian Stokes theorem
∞-Lie theory (higher geometry)
Formal Lie groupoids
The nonabelian Stokes theorem is a generalization of the Stokes theorem for Lie algebra valued differential 1-forms and with integration of differential forms refined to parallel transport.
If is a Lie algebra valued 1-form on the 2-disk then the parallel transport of its restriction 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 over .
In particular if then the holonomy of is trivial.
Relation to higher parallel transport
In terms of the notion of connection on a 2-bundle the nonabelian Stokes theorem says that if is a Lie algebra valued 1-form, then is a Lie 2-algebra valued 2-form with values in the inner derivation Lie 2-algebra of whose curvature 3-form vanishes (which is the Bianchi identity for ) 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 the nonabelian Stokes theorem may be regarded as proving that the Lie integration of by “the path method” (see at Lie integration) is indeed the simply connected Lie group corresponding to by Lie theory.
For instance in
Revised on October 28, 2012 04:29:56
by Eric Forgy