Title:
Nonabelian
Multiplicative Integration on Surfaces
Authors:
Amnon
Yekutieli
Publication
status:
Appeared
as
a
book
in World
Scientific; eprint arXiv:1007.1250
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 Stokes Theorem. This
is a totally new result.
The methods we use 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
and descent for nonabelian gerbes. Similar questions arise in
nonabelian gauge theory.
Electronic Preprint:
pdf file (139 pages, 38 figures)
Video of lecture:
Univ of Washington 2015
Correspondence with Pierre Deligne (2016)
Here are two letters from Deligne (posted with his permission), and my response.
(updated 16 Feb 2017)