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Macquarie University  Department of Mathematics

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Categories in Algebra, Geometry and Mathematical Physics

Conference in honour of Ross Street's sixtieth birthday

July 11-16 2005, Macquarie University, Sydney

Tue, 12 July:  9:00 - 10:00

Noncommutative Fourier transform, Chen's iterated integrals and higher-dimensional holonomy~(I)
Kapranov, Michael (Yale University)

We set up a framework for a noncommutative version of
the Fourier transform which relates functions of noncommuting variables
and ordinary functions on the space of unparametrized paths. It is
based on Chen's analogs of exponential functions that are
generating functions of his iterated integrals. Then we explain how to
extend this correspondence to represent higher-dimensional membranes
by elements of a certain differential graded algebra $A$. This is
related to the concept of holonomy of gerbes that attracted
a lot of attention recently. We will also give the interpretation
of higher gerbe holonomy in terms of Chen's iterated integrals
of forms of higher degree with coefficients in Lie-algebraic
analogs of crossed modules and crossed complexes. If one views higher
holonomy as a ``pasting integral" then 2-dimensional associativities
translate into vanishing of some brackets in the structure dg-Lie
algebra which is automatic in the crossed module case but has to be
imposed in general.

Typeset PDF of this abstract.