Title: Nonabelian Multiplicative Integration on Surfaces
Authors: Amnon Yekutieli
Publication status: Eprint arXiv:1007.1250 [math.DG]

Abstract:
We construct a 2-dimensional twisted nonabelian multiplicative
integral. This is done in the context of a Lie crossed module
(an object composed of two Lie groups interacting), and a
pointed manifold. The integrand is a connection-curvature pair,
that consists of a Lie algebra valued 1-form and a Lie algebra
valued 2-form, satisfying a certain differential equation.
The geometric cycle of the integration is a kite in the pointed
manifold. A kite is made up of a 2-dimensional simplex in the
manifold, together with a path connecting this simplex to the
base point of the manifold. The multiplicative integral is an
element of the second Lie group in the crossed module.

We prove several properties of the multiplicative integral.
Among them is the 2-dimensional nonabelian Stokes Theorem,
which is a generalization of Schlesinger's Theorem. Our main result
is the 3-dimensional nonabelian StokesTheorem. This is a totally
new result.

The methods we used are: the CBH Theorem for the nonabelian
exponential map; piecewise smooth geometry of polyhedra; and
some basic algebraic topology.

The motivation for this work comes from twisted deformation
quantization. In the paper [Ye2] we encountered a problem of
gluing nonabelian gerbes, where the input was certain data in
differential graded algebras. The 2-dimensional multiplicative
integral gives rise, in that situation, to a nonabelian
2-cochain; and the 3-dimensional Stokes Theorem shows that
this cochain is a twisted 2-cocycle.


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updated 20 March 2011