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.

Abstract. 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 unified 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 finite-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.

Supersymmetry is deeply related to division algebras. For example, nonabelian Yang-Mills fields minimally coupled to massless spinors are supersymmetric if and only if the dimension of spacetime is 3, 4, 6 or 10. The same is true for the Green-Schwarz superstring. In both cases, supersymmetry relies on the vanishing of a certain trilinear expression involving a spinor field. The reason for this, in turn, is the existence of normed division algebras in dimensions two less, namely 1, 2, 4 and 8: the real numbers, complex numbers, quaternions and octonions. Here we provide a self-contained account of how this works.For any square matrix A with entries in K, we define its trace tr(A) to be the sum of its diagonal entries. This trace lacks the usual cyclic property, because K is noncommutative, so in general tr(AB) = tr(BA). Luckily, taking the real part restores this property:Proposition 3. Let a, b, and c be elements of K. Then Re((ab)c) = Re(a(bc)) and this quantity is invariant under cyclic permutations of a, b, and c.Proof. Proposition 2 implies that Re((ab)c) = Re(a(bc)). For the cyclic property, it then suffices to prove Re(ab) = Re(ba). Since (a, b) = (b, a) and the inner product is defined by (a, b) = Re(ab * ) = Re(a * b), we see:Re(ab * ) = Re(b * a).The desired result follows upon substituting b * for b. Proposition 4. Let A, B, and C be k × ℓ, ℓ × m and m × k matrices with entries in K. Then Re tr((AB)C) = Re tr(A(BC))and this quantity is invariant under cyclic permutations of A, B, and C. We call this quantity the real trace Re tr(ABC).Proof. This follows from the previous proposition and the definition of the trace.The reader will have noticed three trilinears in this section: the associator [a, b, c], the real part Re((ab)c), and the real trace Re tr(ABC). This is no coincidence, as they all relate to the star of the show, tri ψ. In fact: tri ψ = Re tr(ψ † (ǫ · ψ)ψ).for some suitable matrices ψ † , ǫ · ψ and ψ. Of course, we have not yet said how to construct these. We turn to this now.

Starting from the four normed division algebras -the real numbers, complex numbers, quaternions and octonions -a systematic procedure gives a 3-cocycle on the Poincaré Lie superalgebra in dimensions 3, 4, 6 and 10. A related procedure gives a 4-cocycle on the Poincaré Lie superalgebra in dimensions 4, 5, 7 and 11. In general, an (n + 1)-cocycle on a Lie superalgebra yields a "Lie n-superalgebra": that is, roughly speaking, an n-term chain complex equipped with a bracket satisfying the axioms of a Lie superalgebra up to chain homotopy. We thus obtain Lie 2-superalgebras extending the Poincaré superalgebra in dimensions 3, 4, 6 and 10, and Lie 3-superalgebras extending the Poincaré superalgebra in dimensions 4, 5, 7 and 11. As shown in Sati, Schreiber and Stasheff's work on higher gauge theory, Lie 2-superalgebra connections describe the parallel transport of strings, while Lie 3-superalgebra connections describe the parallel transport of 2-branes. Moreover, in the octonionic case, these connections concisely summarize the fields appearing in 10-and 11-dimensional supergravity.e-print archive: http://lanl.arXiv.org/abs/1003.3436v2

Understanding the exceptional Lie groups as the symmetry groups of simpler objects is a longstanding program in mathematics. Here, we explore one famous realization of the smallest exceptional Lie group, G2. Its Lie algebra g2 acts 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 G2 incidence geometry, and show how a form of geometric quantization applied to this geometry lets us recover the imaginary split octonions and these operations.On the other hand, we can recover the imaginary split octonions from this variant of the 1:3 rolling ball via geometric quantization.Using a spinorial variant of the rolling ball system may seem odd, but it is essential if we want to see the hidden G 2 symmetry. In fact, we must consider three variants of the rolling ball system. The first is the ordinary rolling ball, which has configuration space S 2 × SO(3). This never has G 2 symmetry. We thus pass to the double cover, S 2 × SU(2), where such symmetry is possible. We can view this as the configuration space of a 'rolling spinor': a rolling ball that does not come back to its original orientation after one full rotation, but only after two. To connect this system with the split octonions, it pays to go a step further, and identify antipodal points of the fixed sphere S 2 . This gives RP 2 × SU(2), which is the configuration space of a spinor rolling on a projective plane.This last space explains why the 1:3 ratio of radii is so special. As mentioned, a spinor comes back to its original state only after two full turns. On the other hand, a point moving on the projective plane comes back to its original position after going halfway around the double cover S 2 . Consider a ball rolling without slipping or twisting on a larger fixed ball. What must the ratio of their radii be so that the rolling ball makes two full turns as it rolls halfway around the fixed one? Or put another way: what must the ratio be so that the rolling ball makes four full turns as it rolls once around the fixed one? The answer is 1:3.At first glance this may seem surprising. Isn't the correct answer 1:4? No: a ball of radius 1 turns R + 1 times as it rolls once around a fixed ball of radius R. One can check this when R = 1 using two coins of the same kind. As one rolls all the way about the other without slipping, it makes two full turns. Similarly, in our 365 1 4 day year, the Earth actually turns 366 1 4 times. This is why the sidereal day, the day as judged by the position of the stars, is slightly shorter than the ordinary solar day. The Earth is not rolling without slipping on some imaginary sphere. However, just as with the rolling ball, it makes an 'extra turn' by completing one full revolution ...