John Baez

2006 Barrett Lectures

April 29-May 1, 2006

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 sort of "higher gauge theory" that describes parallel transport as we move a path through space, tracing out a surface. To find the right mathematical language for this, we must "categorify" concepts from topology and 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. We give an overview of higher gauge theory, with an emphasis on its relation to homotopy theory and the cohomology of groups and Lie algebras.

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

These talks are based on the following 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

• Denis-Charles Cisinski, Batanin higher groupoids and homotopy types

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