Part Two of a Series on the (2, 0) Theory

So now that you know that the (2, 0) theory is the **world-volume theory of an M5-brane**, you might be asking what the hell (2, 0) means. Why is this theory labeled with an arcane bunch of numbers rather than words like any sensible theory?

To explain that, we have to talk a bit about how we count supersymmetries. As I talked about with N=4 super Yang-Mills, supersymmetry is a relationship between particles of different spins, and since one particle can be related in this way to more than one other particle, we indicated the number of different related particles by the number N. (I’d recommend reading those posts to understand this one. If you need a quick summary, spin is a way of categorizing particles, with spin 1 corresponding to forces of nature like electromagnetism and the Yang-Mills forces in general, while spin ½ corresponds to the types of particles that make up much of everyday matter, like electrons and quarks.)

As it turns out, we count the number of supersymmetries N differently in different dimensions. The reasons are fairly technical, and are related to the fact that spin ½ particles are more complicated in higher dimensions. The end result is that while super Yang-Mills has N=4 in four dimensions (three space one time), in six dimensions it only has N=2 (in case you’re curious, it goes all the way down to N=1 in ten dimensions).

The “2” in the (2, 0) theory means the same thing as that N=2. However, the (2, 0) theory is very different from super Yang-Mills, and that’s where the other number in the pair comes in. To explain this, I have to talk a bit about something called **chirality**.

Chirality is a word for handedness. If you’re given a right-handed glove, no matter what you do you can’t rotate it to turn it into a left-handed glove. The only way you could change a right-handed glove into a left-handed glove would be to flip it through a mirror, like Alice through the looking glass.

Particles often behave similarly. While they don’t have fingers to flip, they do have spin.

I told you earlier to think of spin as just a way to classify particles. That’s still the best way for you to think about it, but in order to explain chirality I have to mention that spin isn’t just an arbitrary classification scheme, it’s a number that corresponds to how fast a particle is “spinning”.

Here I have to caution that the particle isn’t necessarily literally spinning. Rather, it acts as if it were spinning, interacting with other objects as if it were spinning with a particular speed. If you’ve ever played with a gyroscope, you know that a spinning object behaves differently from a non-spinning one: the faster it spins, the harder it is to change the direction in which it is spinning.

Suppose that a particle is flying at you head-on. If you measured the particle’s spin, it would appear to be spinning either clockwise or counterclockwise, to the left or to the right. This choice, left or right, is the particle’s chirality.

The weird thing is that there are some particles that only spin one way. For example, every neutrino that has been discovered has left-handed chirality. In general when a fermion only spins one way we call it a **chiral fermion**.

What does this have to do with the (2, 0) theory?

Supersymmetry relates particles of spin X to particles of spin X-½. As such, you can look at supersymmetry as taking the original particle, and “subtracting” a particle of spin ½. These aren’t really particles, but they share some properties, and those properties can include chirality. You can have left-supersymmetry, and right-supersymmetry.

So what does (2, 0) mean? It means that not only is the (2, 0) theory an N=2 theory in six dimensions, but **those two supersymmetries are chiral**. They are only left-handed (or, if you prefer, only right-handed). By contrast, super Yang-Mills in six dimensions is a (1, 1) theory. It has one left-handed supersymmetry, and one right-handed supersymmetry.

We can now learn a bit more about the sorts of particles in the (2, 0) theory.

As I said when discussing N=4 super Yang-Mills, N=4 is the most supersymmetry you can have in Yang-Mills in four dimensions. Any more, and you need to include gravity. Recall that the (2, 0) theory comes from the behavior of M5-branes in M theory. M theory includes gravity, **which means that it can go higher than N=4**.

How high? As it turns out, the maximum including gravity (which I will explain a bit more when I do a series on supergravity) is N=8. That’s in four dimensions, however. In M theory’s native eleven dimensions, this is just N=1. In six dimensions, where the (2, 0) theory lives, this becomes N=4. More specifically, including information about chirality, its supersymmetry is **(2, 2)**.

So if M theory in six dimensions has (2, 2) symmetry, how to we get to (2, 0)? What happens to the other ( ,2)?

As I talked about in the last post, the varying position of the M5-brane in the other five dimensions gives rise to five scalar fields. In a way, we have broken the symmetry between the eleven dimensions of M theory, treating five of them differently from the other six.

It turns out that supersymmetry is closely connected to the symmetry of space and time. What this means in practice is that when you break the symmetry of space-time, you can also break supersymmetry, reducing the number N of symmetries. Here, the M5-brane breaks supersymmetry from (2, 2) to (2, 0), so two of the supersymmetries are broken.

Just like the position of the M5-brane can vary, so too can the specific supersymmetries broken. What this means is that just like the numbers for the positions become scalar fields, **the choices of supersymmetry to be broken become new fermion fields**. Because supersymmetry is broken in a chiral way, these new fermion fields are chiral, which for technical reasons ends up meanings that **because of the two broken supersymmetries, there are four new chiral fermions**.

So far, we know that the (2, 0) theory has five scalar fields, and four chiral fermions. But scalar fields and chiral fermions are pretty ordinary, surely not as mysterious as the Emperor, or even Mara Jade. What makes the (2, 0) theory so mysterious, so difficult to deal with? What makes it, in a word, sexy? Tune in to Part Three to find out!

ravimohanGreat article! The concepts are introduced in very lucid and creative manner. I have one question regarding the last paragraph. The (2,0) theory (in 6 dimensions) had five scalar fields (thus 5 bosonic degrees of freedom (doc)) and four fermions (which on-shell give 12 fermionic dof). So how does the susy relate these degrees of freedom? Or am I counting wrong?

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4gravitonsandagradstudentPost authorFirst of all, you’re counting the fermionic degrees of freedom wrong, there are 8, not 12. But the main thing is that you need to read part 3, as the scalars aren’t the only bosonic degrees of freedom. 😉

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ravimohanI think I am getting little confused while calculating the dof. I think I will understand it better after studying the Spinors in general n dimensions.

Also do (2,0) and (0,2) represent the same theory?

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4gravitonsandagradstudentPost authorYes, for all practical purposes they’re the same theory, just picking out either the chiral or anti-chiral supersymmetries.

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