John Baez

Higher Gauge Theory, Division Algebras and Superstrings

Classically, superstrings make sense when spacetime has dimension 3, 4, 6, or 10. It is no coincidence that these numbers are two more than 1, 2, 4, and 8, which are the dimensions of the normed division algebras: the real numbers, complex numbers, quaternions and octonions. We sketch an explanation of this already known fact and its relation to "higher gauge theory". Just as gauge theory describes the parallel transport of supersymmetric particles using Lie supergroups, higher gauge theory describes the parallel transport of superstrings using "Lie 2-supergroups". Recently John Huerta has shown that we can use normed division algebras to construct a Lie 2-supergroup extending the Poincaré supergroup when spacetime has dimension 3, 4, 6 or 10. He also used them to construct a Lie 3-supergroup when spacetime has dimension 4, 5, 7 or 11. The 11-dimensional case is related to 11-dimensional supergravity, and thus presumably to "M-theory".

To see the slides click here.
For a version with more math and less physics, try this.

This talk is based on John Huerta's thesis:

• Division Algebras, Supersymmetry and Higher Gauge Theory.

• L_∞-superalgebras for superstrings and 2-branes.
• A categorified supergroup for string theory.

• An Invitation to Higher Gauge Theory.
• Division Algebras and Supersymmetry I.
• Division Algebras and Supersymmetry II.

There are many other relevant papers, such as these:

• 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.
• Leonardo Castellani, Lie Derivatives along Antisymmetric Tensors, and the M-Theory Superalgebra.
• Pietro Fré and Pietro Antonio Grassi, Pure Spinors, Free Differential Algebras, and the Supermembrane.
• Ezra Getzler, Lie Theory for Nilpotent L_∞-Algebras.
• André Henriques, Integrating L_∞-Algebras.
• Hisham Sati, Urs Schreiber and Jim Stasheff, L_∞ Algebra Connections and Applications To String- and Chern-Simons n-Transport.
• Christopher Schommer-Pries, Central Extensions of Smooth 2-Groups and a Finite-Dimensional String 2-Group.
• Jörg Schray, The General Classical Solution of the Superparticle.

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