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

If you’ve been reading this guide, you know that the (2, 0) theory describes what it’s like to live on a five dimensional membrane in M theory. You know it’s got five scalar fields, and four chiral fermions (and hopefully you have a rough idea of what those things are). And if you’ve been reading for longer, you’ve probably heard me mention that a theory is essentially a list of quantum fields. So if I’m going to define the (2, 0) theory for you, I ought to, at the very least, list its quantum fields.

This is where things get tricky, and where unfortunately I will have to get a big vague. Some of the quantum fields in the (2, 0) theory are things I’ve talked about before: the five scalars and the four fermions. The remaining field, though, is different, and it’s the reason why the (2, 0) theory is so mysterious.

I’ll start by throwing around some terminology. Normally, I’d go back and explain it, but in this case there’s simply too much. My aim here is to give the specialists reading this enough to understand what I’m talking about. Then I’ll take a few paragraphs to talk about what the implications of all this jargon are for a general understanding.

The remaining field in the (2, 0) theory is a two-form, or an antisymmetric, two-index tensor, with a self-dual field strength. It comes from the gauge orientation zero modes of the M5-brane. It is not a Yang-Mills field. However, it is non-abelian, that is, it “interacts with itself” in a similar way to how a Yang-Mills field does.

While I can give examples of familiar Yang-Mills fields, fermions, and now with the Higgs even scalars, I can’t give you a similar example of a fundamental two-form field. That’s because in our four dimensional world, such a field doesn’t make sense. It only makes sense in six or more dimensions.

The problem with understanding this isn’t just a matter of not having examples in the real world, though. We can invent a wide variety of unobserved fields, and in general have no problem calculating their hypothetical properties. The problem is that, in the case of the two-form field of the (2, 0) theory, we don’t know how to properly do calculations about it.

There are a couple different ways to frame the issue. One is that, while we know roughly which fields should interact with which other fields, there isn’t a mathematically consistent way to write down how they do so. Any attempt results in a formula with some critical flaw that keeps it from being useful.

The other way to frame the problem is to point out that every Yang-Mills force has a number that determines how powerful it is, called the coupling constant. As I discuss here, it is the small size of the coupling constant that allows us to calculate only the simplest Feynman diagrams and still get somewhat accurate results.

The (2, 0) theory has no coupling constant. There is no parameter that, if it was small, would allow you to only look at some diagrams and not others. In the (2, 0) theory, every diagram is equally important.

When people say that the (2, 0) theory is “irreducibly quantum”, this is what they’re referring to: we can’t separate out the less-quantum (lower loops) bits from the more quantum (higher loops) bits. The theory simply is quantum, inherently and uniformly so.

This is what makes it so hard to understand, what makes it sexy and mysterious and Mara Jade-like. If we could understand it, the payoffs would be substantial: M theory has a similar problem, so a full understanding of the (2, 0) theory might pave the way to a full understanding of M theory, which, unlike the (2, 0) theory, really is supposed to be a theory of everything.

And there is progress…somewhat, anyway. Twisting one of the six dimensions of the (2, 0) theory around in a circle gives you N=4 super Yang-Mills in five dimensions, while another circle brings it down to four dimensions. Because super Yang-Mills is so well-understood, this gives us a tantalizing in-road to understanding the (2, 0) theory. I’ve worked a bit on this myself.

Perhaps a good way to summarize the situation would be to say that, while N=4 super Yang-Mills is interesting because of how much we know about it, the (2, 0) theory is interesting because, contrary to expectations, we can do something with it at all. Every time someone comes up with a novel method for understanding quantum field theories, you can rest assured that they will end up trying to apply it to the (2, 0) theory. One of them might even work.