John Baez

2007 Abel Symposium

August 8, 2007

Higher Gauge Theory and Elliptic Cohomology

The concept of elliptic object suggests a relation between elliptic cohomology and "higher gauge theory", a generalization of gauge theory describing the parallel transport of strings. In higher gauge theory, we categorify familiar notions from gauge theory and consider "principal 2-bundles" with a given "structure 2-group". These are a slight generalization of nonabelian gerbes. After a quick introduction to these ideas, we focus on the 2-groups String_k(G) associated to any compact simple Lie group G. We describe how these 2-groups are built using central extensions of the loop group ΩG, and how the classifying space for String_k(G)-2-bundles is related to the "string group" familiar in elliptic cohomology. If there is time, we shall also describe a vector 2-bundle canonically associated to any principal 2-bundle, and how this relates to the von Neumann algebra construction of Stolz and Teichner.

• Higher Gauge Theory and Elliptic Cohomology - in PDF and 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 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, How to Get a Spinning String from Here to There
• Urs Schreiber, On 2-Representations and 2-Vector Bundles

• Stefan Stolz and Peter Teichner, What Is an Elliptic Object?

• Paolo Aschieri, Luigi Cantini, and Branislav Jurco, Nonabelian Bundle Gerbes, their Differential Geometry and Gauge Theory
• Paolo Aschieri and Branislaw Jurco, Gerbes, M5-brane Anomalies and E₈ Gauge Theory
• Nils Baas, Marcel Bökstedt and Tore Kro, Two-Categorical Bundles and Their Classifying Spaces.
• Lawrence Breen and William Messing, Differential Geometry of Gerbes
• Branislaw Jurco, Crossed Module Bundle Gerbes; Classification, String Group and Differential Geometry
• André Henriques, Integrating L_∞-Algebras

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