Division algebras and supersymmetry I

Abstract

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.

Division algebras and supersymmetry III

“…We can understand this identity in terms of division algebras, as reviewed in the first paper of this series [5]. In some sense, this solves the puzzle we began with, but leaves us with another: what is the meaning of the 3-ψ's rule itself?…”

“…Mysteriously, for both the superstring and the 2-brane, it is the octonionic cases that are of greatest physical interest-10 dimensions for the superstring, and 11 for the 2-brane, which plays an important role in M-theory. This is the third in a series of papers exploring the relationship between supersymmetry and division algebras [5,6], the first two of which were coauthored with John Baez. In the first paper [5], we reviewed the known story of how the division algebras give rise to the supersymmetry of super-Yang-Mills theory.…”

“…This is the third in a series of papers exploring the relationship between supersymmetry and division algebras [5,6], the first two of which were coauthored with John Baez. In the first paper [5], we reviewed the known story of how the division algebras give rise to the supersymmetry of super-Yang-Mills theory. In the second [6], we showed how the division algebras can be used to construct 'Lie 2-superalgebras' superstring(n + 1, 1), which extend the Poincaré superalgebra in the superstring dimensions n + 2 = 3, 4, 6 and 10.…”

“…Moreover, this fact is exactly what is required to define the classical superstring in those dimensions-technically, it is needed for the Green-Schwarz Lagrangian to have 'Siegel symmetry', which forces the number of bosonic and fermionic degrees of freedom to match [29]. This algebraic story has a beautiful interpretation in terms of division algebras [5,6,23,24,35].…”

Pauli-Lubanski, supertwistors, and the superspinning particle

“…This makes use of the relation of supersymmetric field theories in Minkowski spacetimes of dimension d = 2 + dim K to the normed division algebras K = R, C, H, O [26][27][28][29], although we have not yet seen how to use the K = O case of this relation to extend our (S)PL tensor results to 10D.…”