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Macquarie University  Department of Mathematics

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Workshop on Categorical Methods in Algebra, Geometry and Mathematical Physics

Satellite to the StreetFest conference in honour of Ross Street's sixtieth birthday

July 18-21 2005, Australian National University, Canberra

Wed, 20 July:  9:00 - 10:00

Higher Gauge Theory (II)
Baez, John (University of California, Riverside)

Gauge theory describes how point particles transform as they trace out paths in spacetime using the formalism of connections on bundles. In higher gauge theory, we describe the parallel transport of strings as they sweep out surfaces in spacetime using ``2-connections" on ``2-bundles".

Ordinary gauge theory involves a groupoid in which the objects are points of spacetime and the morphisms are paths. Higher gauge theory goes further and uses a 2-groupoid where the objects are points, the morphisms are paths, and the 2-morphisms are ``paths of paths" which we use to describe surfaces traced out by the motion of strings. So, higher gauge theory is naturally tied to categorification.

Here we show how categorifying the concepts of bundle and connection yields the concepts of ``2-bundle" and ``2-connection". Just as a connection can be locally described as a Lie-algebra-valued 1-form, a 2-connection is locally described by a Lie-algebra-valued 1-form together with a Lie-algebra-valued 2-form. A 2-connection gives well-behaved parallel transport along paths and surfaces if a quantity called its ``fake curvature" vanishes. In this talk we sketch these ideas and discuss how 2-connections on 2-bundles are related to connections on nonabelian gerbes.

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See \url{http://math.ucr.edu/home/baez/street/} for talk transparencies and links to further reading material.

Typeset PDF of this abstract.