# (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.

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.

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.

# Calabi-Yau Manifolds and Moduli Stabilization

For better or worse, string theory dominates modern research in theoretical physics. Naively, you might expect a theory consisting of tiny strings to be pretty simple. But the subject has grown into a vast and exciting playground for new ideas.

String theory is popular not just because it might unify physics or quantize gravity. In fact, many unexpected offshoots have proved more successful than the original idea! From particle physics to superconductors, string theory is having a surprising indirect impact. It’s certainly useful, even if it doesn’t prove to be the ultimate description of reality.

But what of the original plan – to describe nature using strings? A key sticking point is the existence of extra dimensions. String theory needs 6 of these to work consistently. Another problem is supersymmetry. 10 dimensional string theory must have lots of this to work correctly. But 4 dimensional physics only has a little bit of supersymmetry at most!

It turns out that these problems can be solved in one step. By coiling up the extra dimensions into a Calabi-Yau manifold we can make the extra dimensions effectively invisible, while reducing the supersymmetry we end up with in our 4D world. So what is this Calabi-Yau manifold, I hear you ask!

Well, Calabi-Yau is just a technical term for the shape of the compact extra dimensions. Different shapes break different amounts of symmetries, leaving us with different theories. Calabi-Yau’s are just symmetrical enough to break the right amount of supersymmetry, giving us a sensible theory in the end!

Technically Calabi-Yau manifolds must have a metric which is Kaehler and Ricci flat. These properties provide the correct information about the shape of the curled up dimensions. So we must look for 6-real dimensional manifolds with these properties.

Generically, you don’t have to put a notion of distance on a space. When I go for a walk, I don’t always carry around a yardstick so I can measure how far I’ve gone! You can have a perfectly good manifold without giving it a metric, but you get extra information once you have defined what distance means.

As it happens, finding a metric which is Calabi-Yau is quite difficult. But due to the genius of Shing-Tung Yau, we know that you don’t need to do this! There’s an equivalent definition of a Calabi-Yau manifold which doesn’t depend on metrics at all. All you need to know is the topological information about the manifold – roughly speaking, how “holey” it is.

If you know something about differential geometry, this kind of equivalence might sound familiar. Yau’s theorem relating geometry and topology is like a (much) more complicated version of the classic Gauss-Bonnet theorem!

It’s a darn sight easier to discover Calabi-Yau’s when you know it’s only the topological data that matters. At first people thought there might only be a few, but now we know there’s a huge number of potential candidates! The problem then becomes choosing one which produces the physics of our universe.

While people have made progress on this, the going is tough. One reason is that nobody knows the metric on a compact Calabi-Yau. This isn’t so important for string calculations, but it makes a big difference when you need to consider branes. So people have come up with various workarounds, which give promising physical results. One such success story is provided by my colleague Zac Kenton, who recently wrote a paper on brane inflation with his PhD supervisor, Steve Thomas.

There’s one final complication that I should mention. If string theory is to be a fundamental theory, then the Calabi-Yau shape should be dynamic. More specifically it will squeeze and stretch over time, unless there’s some mechanism to keep it stable. From the perspective of the 4 large dimensions, this freedom is seen as free scalar fields. These so-called “moduli” fields are bad, because we don’t observe anything like them in nature!

To solve this problem, we must find a way of constraining the fluctuations of the Calabi-Yau. Put another way we have to stabilize the moduli fields, by giving them potential terms, so that their fluctuations are small and essentially negligible at low energies. Hence this is known as the problem of moduli stabilization.

One popular way to solve the conundrum is to turn on some supergravity fields at high energy. These so-called fluxes generate potential terms for the moduli, solving the stabilization problem. Initially this idea was unpopular because of a famous no-go theorem by Witten. But since the advent of the D-brane revolution, the concept is back in vogue!

So there you have it – a 5 minute snapshot of “real” string theory. Now it’s time to get back to my calculations, where string theory is more the background Muse, and certainly not the main protagonist!

# Why I Like Supersymmetry

Supersymmetry can be variously described as beautiful, convenient, unphysical and contrived. The truth is that nobody really knows whether we’re likely to find it in our universe. Like most theoretical physicists I hope we do, but even if we don’t it can still be a useful mathematical tool.

There are tons of reasons to like supersymmetry, as well as a good many arguments against it. I can’t cover all of these in a brief post, so I’m just going to talk about one tiny yet pretty application I glanced at today.

Let’s talk about scattering processes again, my favourite topic of (physics) conversation. These are described by quantum field theory, which is itself based on very general principles of symmetry. In the standard formulation (imaginatively called the Standard Model) these symmetries involve physical motions in spacetime, as well as more abstract transformations internal to the theory. The spacetime symmetries are responsible for giving particles mass, spin and momentum, while the internal ones endow particles with various charges.

At the quantum level these symmetries actually provide some bonus information, in the form of certain identities that scattering processes have to satisfy. These go by the name of Ward identities. For example QED has a both a gauge and a global $U(1)$ symmetry. The Ward identity for the global symmetry tells you that charge must be conserved. The Ward identity for the gauge symmetry tells you that longitudinally polarized photons are unphysical.

If you’re a layman and got lost above then don’t worry. All you need to know is that Ward identities are cool because they tell you extra things about a theory. The more information you have, the more constrained the answer must be, so the less work you have to do yourself! And this is where supersymmetry comes into the picture.

Supersymmetry is another (very special) type of symmetry that pairs up fermions (matter) and bosons (forces). Because it’s a symmetry it has associated Ward identities. These relate different scattering amplitudes. The upshot is that once you compute one amplitude you get more for free. The more supersymmetry you have, the more relations there are, so the easier your job becomes.

So what’s the use if supersymmetry isn’t true then? Well, in general terms it’s still useful to look at these simplified situations because it might help us discover tools that would be hard to uncover otherwise. Take the analogy of learning a language, for example. One way to do it is just to plunge headlong in and try to pick things up as you go along. This way you tend to get lots of everyday phrases quickly, but won’t necessary understand the structure of the language.

Alternatively you can go to classes that break things down into simpler building blocks. Okay spending one hour studying the subjunctive alone might not seem very useful at first, but when you go back to having a real conversation you’ll pick up new subtleties you never noticed before.

If you’re still unconvinced here’s a (somewhat trivial) concrete example. Recall that you can show that purely positive helicity gluon amplitudes must vanish at tree level in QCD. The proof is easy, but requires some algebraic fiddling. The SUSY Ward identity tells us immediately than in a Super-Yang-Mills (SYM) theory this amplitude must vanish to all orders in the loop expansion. So how do we connect back to QCD?

Well the gluon superpartners (gluinos) have quadratic coupling to the gluon, so an all gluon scattering amplitude in SYM can’t include any gluinos at tree level. (Just think about trying to draw the diagram if you’re confused)! In other words, at tree level the SYM amplitude is exactly the QCD amplitude, which proves our result.

Not sure what will be on the menu tomorrow – I’m guessing that either color-ordering or unitarity methods will feature. Drop me a comment if you have a preference.

# Recursion Relations for Amplitudes

So yesterday’s discussion was all about how simple some gluon scattering processes (or amplitudes) look. In particular the maximal helicity violating (MHV) ones are special, because hundreds of terms cancel down to give a single neat result. Today I’ve been looking at how to prove this result, so I can now sketch the main ingredients for you.

If you’re from a mathsy background you won’t be surprised to learn that the $n$-gluon Parke-Taylor identity is proved using induction. For the uninitiated there’s a simple analogy with climbing stairs. If you can get up the first one, and you can get from every one to the next one then you can get to the top!

With this in mind, our first task is to prove the simplest case, which turns out to be $n=3$. Why exactly? Well the Feynman rules for QCD have 3 and 4 point vertices at tree level, so there’s no tree level 2 point amplitudes! Turns out that the $n = 3$ case is neatly dealt with using spinor helicity formalism. Roughly speaking this takes into account the special helicity structure of the MHV amplitudes to lock in simplifications right from the start of the calculation! Add in momentum conservation and hey presto the $n=3$ Parke-Taylor identity drops right out.

So now we need to climb from one stair to the next. This is where recursion relations come in handy. Nine years ago, a group of theoreticians spotted a cunning way to break apart tree level gluon amplitudes into smaller, more manageable pieces. Mathematically they spotted that the $n$-gluon scattering amplitude factorized into the product of two distinct on-shell amplitudes, each with a complementary subset of the original external legs plus an extra leg with momentum $\hat{P}$. The only added ingredient needed was a factor of $1/P^2$ corresponding to a propagator between the two diagrams.

Woah – hold up there! What’s all this terminology all of a sudden. For the uninitiated I’m guessing that on-shell sounds a bit confusing. But it’s no cause for alarm. In general an on-shell quantity is one which obeys the equations of motion of the system involved. Here the relevant equation is the Weyl equation, which implies that $\hat{P}^2 = 0$.

Why are these recursion relations so useful? Well, they give us exactly the ingredient we needed for the induction step. And we’re done – the Parke-Taylor identity is proved, with a little bit of algebra I’ve shoved under the rug.

There’s one more point I’ve neglected to mention. How do you go about finding these mythical on-shell recursion relations? The answer comes from doing some subtle complex analysis, transforming momenta into the complex plane. It might not sound very physical to do that, but in fact the method opens up oodles of new possibilities. One reason is that complex integration is both more powerful and easier than its real counterpart, so it can be used to extract valuable identities from the world of scattering processes.

I’ll leave you with this great basic article clarifying the subtleties of the on-shell/off-shell distinction. It goes a bit deeper than that too, so is worth a read even if you’re more of an expert!

Tomorrow I hope I’ll bring you something more supersymmetrical in nature. $\mathcal{N}=4$ SUSY is a favourite playground for scattering enthusiasts because it is finite (no renormalization needed) and very simple (no free parameters). We’ll encounter its stark beauty in due course.