John Baez, Alissa Crans and Danny Stevenson

Chicago Lectures on Higher Gauge Theory

April 7-11, 2006

Higher Gauge Theory, Higher Categories
John Baez

The work of Eilenberg and Mac Lane marks the beginning of a trend in which mathematics based on sets is generalized to mathematics based on categories and then higher categories. We illustrate this trend towards "categorification" by a detailed introduction to "higher gauge theory".

Gauge theory describes the parallel transport of point particles using the formalism of connections on bundles. In both string theory and loop quantum gravity, point particles are replaced by 1-dimensional extended objects: paths or loops in space. This suggests that we seek some kind of "higher gauge theory" that describes the parallel transport as we move a path through space, tracing out a surface. Surprisingly, this requires that we "categorify" concepts from differential geometry, replacing smooth manifolds by smooth categories, Lie groups by Lie 2-groups, Lie algebras by Lie 2-algebras, bundles by 2-bundles, sheaves by stacks or gerbes, and so on.

To explain how higher gauge theory fits into mathematics as a whole, we start with a lecture reviewing the basic principle of Galois theory and its relation to Klein's Erlangen program, covering spaces and the fundamental group, Eilenberg-Mac Lane spaces, and Grothendieck's ideas on fibrations.

The second lecture treats connections on trivial bundles and 2-connections on trivial 2-bundles, explaining how they can be described either in terms of their holonomies or in terms of Lie-algebra-valued differential forms. For a clean treatment of these concepts, we recall Chen's theory of "smooth spaces", which generalize smooth finite-dimensional manifolds.

The third lecture explains connections on general bundles and 2-connections on general 2-bundles, explaining how they can be described either in terms of holonomies or local data involving differential forms. We also explain how 2-bundles are classified using nonabelian Cech 2-cocycles, and how the theory of 2-connections relates to Breen and Messing's theory of "connections on nonabelian gerbes".

• Higher Gauge Theory, Higher Categories - I - in PDF and Postscript
• Higher Gauge Theory, Higher Categories - II - in PDF and Postscript
• Higher Gauge Theory, Higher Categories - III - in PDF and Postscript

Lie 2-Groups, Lie 2-Algebras and Loop Groups
Alissa Crans

• Lie 2-Groups, Lie 2-Algebras and Loop Groups - in PDF and Postscript

Lie 2-Algebras and the Geometry of Gerbes
Danny Stevenson

• Lie 2-Algebras and the Geometry of Gerbes - in PDF and Postscript

Our talks are based on these papers:

• John Baez and Aaron Lauda, Higher-Dimensional Algebra V: 2-Groups
• John Baez and Alissa Crans, Higher-Dimensional Algebra VI: Lie 2-Algebras
• John Baez and Urs Schreiber, Higher Gauge Theory
• John Baez, Alissa Crans, Urs Schreiber and Danny Stevenson, From Loop Groups to 2-Groups
• Toby Bartels, Higher Gauge Theory I: 2-Bundles
• Urs Schreiber, From Loop Space Mechanics to Nonabelian Strings

• Roger Picken and Marco Mackaay, Holonomy and Parallel Transport for Abelian Gerbes
• Lawrence Breen and William Messing, Differential Geometry of Gerbes
• Marco Mackaay, A Note on the Holonomy of Connections in Twisted Bundles
• Hendryk Pfeiffer, Higher Gauge Theory and a Non-Abelian Generalization of 2-Form Electrodynamics
• Florian Girelli and Hendryk Pfeiffer, Higher Gauge Theory - Differential Versus Integral Formulation
• Paolo Aschieri and Branislaw Jurco, Gerbes, M5-brane Anomalies and E₈ Gauge Theory
• André Henriques, Integrating L-infinity Algebras
• André Henriques, A Model for the String Group

• Felix Klein, Lectures on the Ikosahedron, and Solutions of Equations of the Fifth Degree, trans. George Gavin Morrice, Trübner and Co., London, 1888.
• Jerry Shurman, Geometry of the Quintic, John Wiley and Sons, New York, 1997.

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