(Chiral) Supersymmetry in Different Dimensions

This week I’m at the CERN winter school on supergravity, strings and gauge theory. Jonathan Heckman’s talks about a top-down approach to 6D SCFTs have particularly caught my eye. After all the \mathcal{N}=(2,0) theory is in some sense the mother of my favourite playground \mathcal{N}=4 in four dimensions.

Unless you’re a supersymmetry expert, the notation should already look odd to you! Why do I write down two numbers to classify supersymmetries in 6D, but one suffices for 4D. The answer comes from a subtlety in the definition of the superalgebra, which isn’t often discussed outside of lengthy (and dull) textbooks. Time to set the record straight!

At kindergarten we learn that supersymmetry adds fermonic generators to the Poincare algebra yielding a “unique” extension to the possible spacetime symmetries. Of course, this hides a possible choice – there are many fermionic representations of the Lorentz algebra one could choose for the supersymmetry generators.

Folk theorem: anything you want to prove can be found in Weinberg.
Folk theorem: anything you want to prove can be found in Weinberg.

Fortunately, mathematical consistency restricts you to simple options. For the algebra to close, the generators must live in the lowest dimensional representations of the Lorentz algebra – check Weinberg III for a proof. You’re still free to take many independent copies of the supersymmetry generators (up to the restrictions placed by forbidding higher spin particles, which are usually imposed).

Therefore the classification of supersymmetries allowed in different dimensions reduces to the problem of understanding the possible spinor representations. Thankfully, there are tables of these.

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Reading carefully, you notice that dimensions 2, 6 and 10 are particularly special, in that they admit Majorana-Weyl spinors. Put informally, this means you can have your cake and eat it! Normally, the minimal dimension spinor representation is obtained by imposing a Majorana (reality) or Weyl (chirality) condition. But in this case, you can have both!

This means that in D=2,\ 6 or 10, the supersymmetry generators can be chosen to be chiral. The stipulation \mathcal{N}=(1,0) says that Q should be a left-handed Majorana spinor, for instance. In D = 4 a Majorana spinor must by necessity contain both left-handed and right-handed pieces, so this choice would be impossible! Or, if you like, should I choose Q to be a left-handed Weyl spinor, then it’s conjugate Q\dagger is forced to be right-handed.