Tag Archives: spinor

(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.

Spinor Helicity Formalism – Twistors to the Rescue

Dang! Didn’t get my teeth into enough supersymmetry today. I’m standing at the gateway though, so I’ll be able to tell you much more tomorrow. For now, let’s backtrack a bit and take a look at spinor helicity formalism.

First things first, I need to remind you that on-shell matter particles (specifically spin \frac{1}{2} fermions) are represented as spinors in quantum field theory. For a massless fermion, the spinor encodes the momentum and helicity of the particle. We introduce the so called spinor helicity notation

\displaystyle [p| = +\textrm{ve helicity particle with momentum }p \\ \langle p| = -\textrm{ve helicity particle with momentum }p

Their hermitian conjugate spinors give the corresponding antiparticles, as you’d expect if you’re familiar with QFT. One can thus naturally define the inner product of a particle and antiparticle state by contracting their corresponding spinors. We see these contractions a lot in the Parke-Taylor formula, for example.

Now it turns out that every null future pointing vector can be represented in terms of it’s corresponding spinor helicity as according to the identification

\displaystyle p \leftrightarrow |p]\langle p|\qquad (A)

This can be made formal easily using the Weyl equation that the spinor states must satisfy. But what exactly is the use of this?

Well we saw that writing down scattering amplitudes in the spinor helicity formalism was particularly easy, since we could keep the amplitudes manifestly “on-shell” throughout the process. However I did sweep under the carpet a little algebraic manipulation. I can be more explicit about that now. The only difficult steps in the simplification I omitted are due to momentum conservation.

Usually momentum conservation for an n particle process takes the form

\displaystyle \sum_{i=1}^n p_i = 0

This is easy to implement in the usual Feynman diagram formalism, because it is linear! But in the spinor helicity world, we see that this formula becomes quadratic on account of the identification (A) above. This quadratic relation is somewhat troublesome to deal with, and requires annoying identity manipulation to impose.

But what if we want to have our cake and eat it? Imagine a world where we could have the spinor helicity simplifications and yet keep the simplicity of linear momentum conservation. Fortunately for us, such a world exists. We can get there by means of an abstract tool called the dual momentum twistor. There’s not enough time to tell you about that now, but watch out for its appearance in a later post.

So tomorrow will be some supersymmetry then, and maybe a short aside on calculating graviton scattering amplitudes easily. Easily, you say? Well it’s a doddle… at tree level… with a small number of particles…

My thanks to Andi Brandhuber for an enlightening discussion on this point.