John Baez

Lecture at the Union College Mathematics Conference

December 4, 2005

Higher Gauge Theory: 2-Connections

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. To find the right mathematical language for this, it seems we must "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. We give an overview of higher gauge theory, with an emphasis on the concept of "2-connection" for a principal 2-bundle.

• Higher Gauge Theory: 2-Connections - in PDF or Postscript.

The talk is based on this paper:

• John Baez and Urs Schreiber, Higher Gauge Theory - in PDF or Postscript.

• 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, 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

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