John Huerta

An invitation to higher gauge theory

In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2-connections on 2-bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge ‘2-group’. We focus on 6 examples. First, every abelian Lie group gives a Lie 2-group; the case of U(1) yields the theory of U(1) gerbes, which play an important role in string theory and multisymplectic geometry. Second, every group representation gives a Lie 2-group; the representation of the Lorentz group on 4d Minkowski spacetime gives the Poincaré 2-group, which leads to a spin foam model for Minkowski spacetime. Third, taking the adjoint representation of any Lie group on its own Lie algebra gives a ‘tangent 2-group’, which serves as a gauge 2-group in 4d BF theory, which has topological gravity as a special case. Fourth, every Lie group has an ‘inner automorphism 2-group’, which serves as the gauge group in 4d BF theory with cosmological constant term. Fifth, every Lie group has an ‘automorphism 2-group’, which plays an important role in the theory of nonabelian gerbes. And sixth, every compact simple Lie group gives a ‘string 2-group’. We also touch upon higher structures such as the ‘gravity 3-group’, and the Lie 3-superalgebra that governs 11-dimensional supergravity.

The Algebra of Grand Unified Theories

The Standard Model is the best tested and most widely accepted theory of elementary particles we have today. It may seem complicated and arbitrary, but it has hidden patterns that are revealed by the relationship between three ‘grand unied theories’: theories that unify forces and particles by extending the Standard Model symmetry group U(1) SU(2) SU(3) to a larger group. These three are Georgi and Glashow’s SU(5) theory, Georgi’s theory based on the group Spin(10), and the Pati{Salam model based on the group SU(2) SU(2) SU(4). In this expository account for mathematicians, we explain only the portion of these theories that involves nite-dimensional group representations. This allows us to reduce the prerequisites to a bare minimum while still giving a taste of the profound puzzles that physicists are struggling to solve.

and the rolling ball

© 2014 John C. Baez and John Huerta. Understanding the exceptional Lie groups as the symmetry groups of simpler objects is a long-standing program in mathematics. Here, we explore one famous realization of the smallest exceptional Lie group, G2.Its Lie algebra g2acts locally as the symmetries of a ball rolling on a larger ball, but only when the ratio of radii is 1:3. Using the split octonions, we devise a similar, but more global, picture of G2:it acts as the symmetries of a ‘spinorial ball rolling on a projective plane’, again when the ratio of radii is 1:3. We explain this ratio in simple terms, use the dot product and cross product of split octonions to describe the G2incidence geometry, and show how a form of geometric quantization applied to this geometry lets us recover the imaginary split octonions and these operations.

M-theory from the superpoint

The “brane scan” classifies consistent Green–Schwarz strings and membranes in terms of the invariant cocycles on super Minkowski spacetimes. The “brane bouquet” generalizes this by consecutively forming the invariant higher central extensions induced by these cocycles, which yields the complete fundamental brane content of string/M-theory, including the D-branes and the M5-brane, as well as the various duality relations between these. This raises the question whether the super Minkowski spacetimes themselves arise as maximal invariant central extensions. Here, we prove that they do. Starting from the simplest possible super Minkowski spacetime, the superpoint, which has no Lorentz structure and no spinorial structure, we give a systematic process of consecutive “maximal invariant central extensions” and show that it discovers the super Minkowski spacetimes that contain superstrings, culminating in the 10- and 11-dimensional super Minkowski spacetimes of string/M-theory and leading directly to the brane bouquet.

The Magic Square of Lie Groups: The 2 × 2 Case

A unified treatment of the 2 × 2 analog of the Freudenthal–Tits magic square of Lie groups is given, providing an explicit representation in terms of matrix groups over composition algebras.