Category Archives: (2, 0) Theory

Hype versus Miscommunication, or the Language of Importance

A fellow amplitudes-person was complaining to me recently about the hype surrounding the debate regarding whether black holes have “firewalls”. New York Times coverage seems somewhat excessive for what is, in the end, a fairly technical debate, and its enthusiasm was (rightly?) mocked in several places.

There’s an attitude I often run into among other physicists. The idea is that when hype like this happens, it’s because senior physicists are, at worst, cynically manipulating the press to further their positions or, at best, so naïve that they really see what they’re working on as so important that it deserves hype-y coverage. Occasionally, the blame will instead be put on the journalists, with largely the same ascribed motivations: cynical need for more page views, or naïve acceptance of whatever story they’re handed.

In my opinion, what’s going on there is a bit deeper, and not so easily traceable to any particular person.

In the articles on the (2, 0) theory I put up in the last few weeks, I made some disparaging comments about the tone of this Scientific American blog post. After exchanging a few tweets with the author, I think I have a better idea of what went down.

The problem here is that when you ask a scientist about something they’re excited about, they’re going to tell you why they’re excited about it. That’s what happened here when Nima Arkani-Hamed was interviewed for the above article: he was asked about the (2, 0) theory, and he seems to have tried to convey his enthusiasm with a metaphor that explained how the situation felt to him.

The reason this went wrong and led to a title as off-base and hype-sounding as “the Ultimate Ultimate Theory of Physics” was that we (scientists and science journalists) are taught to express enthusiasm in the language of importance.

There has been an enormous resurgence in science communication in recent years, but it has come with a very us-vs.-them mentality. The prevailing attitude is that the public will only pay attention to a scientific development if they are told that it is important. As such, both scientists and journalists try to make whatever they’re trying to communicate sound central, either to daily life or to our understanding of the universe. When both sides of the conversation are operating under this attitude, it creates an echo chamber where a concept’s importance is blown up many times greater than it really deserves, without either side doing anything other than communicating science in the only way they know.

We all have to step back and realize that most of the time, science isn’t interesting because of its absolute “importance”. Rather, a puzzle is often interesting simply because it is a puzzle. That’s what’s going on with the (2, 0) theory, or with firewalls: they’re hard to figure out, and that’s why we care.

Being honest about this is not going to lose us public backing, or funding. It’s not just scientists who value interesting things because they are challenging. People choose the path of their lives not based on some absolute relevance to the universe at large, but because things make sense in context. You don’t fall in love because the target of your affections is the most perfect person in the universe, you fall in love because they’re someone who can constantly surprise you.

Scientists are in love with what they do. We need to make sure that that, and not some abstract sense of importance, is what we’re communicating. If we do that, if we calm down and make a bit more effort to be understood, maybe we can win back some of the trust that we’ve lost by appearing to promote Ultimate Ultimate Theories of Everything.

The (2, 0) Theory: What is it, though?

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

If you’ve been following this series, 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.

The (2, 0) Theory: What does it mean?

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

Apologies in advance. This is going to be a long one.

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.

L for left, R for right, V and p show which way the particle is going.

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 next week to find out!

The (2, 0) Theory: Where does it come from?

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

By semi-popular demand, I’m doing a guide on the (2, 0) theory. Over the course of this guide I’ll try to explain where the (2, 0) theory comes from, what its name means, and, more elusively, what it’s actually about.

The (2, 0) theory doesn’t get much press coverage, and when it does, it’s a bit silly. The article I just linked compares it to Star Wars’ Emperor Palpatine, in analogy with what George Musser and perhaps others call the Darth Vader theory, or N=4 super Yang-Mills.

The metaphor, as far as I can parse it, is the following: while N=4 super Yang-Mills is solid, powerful, and important (like Darth Vader), the (2, 0) theory is mysterious and yet somehow even more central (like the Emperor).

The thing is, while the (2, 0) theory is indeed sexy and mysterious, it isn’t especially central. Laymen haven’t heard of it for good reason: it’s really only specialists in the field who have a reason to be interested in it. So really, it’s more like the Mara Jade theory.

If you don’t know who this is, that’s my point

The (2, 0) theory is very much a theory, in the same sense as N=4 super Yang-Mills. It isn’t a “theory of everything”, and it isn’t supposed to describe the real world. With that in mind, let’s talk about the sort of world it does describe.

There are two ways to “define” the (2, 0) theory. One of them is to take a particular type of string theory (type IIB) with ten dimensions (nine space dimensions and one dimension for time), and twist four of those dimensions into a particular shape (called a K3 surface). There are six dimensions left (five space, one time), and in those six dimensions the world obeys the (2, 0) theory.

That definition may not seem particularly illuminating, and it really isn’t. You can get almost any theory in physics by taking some type of string theory and twisting up some of the dimensions in a particular way, so unless you’re familiar with that particular type of string theory or the particular shape of the dimensions, you don’t learn anything from that definition.

The second definition, though, is more appealing. The (2, 0) theory can be defined as the world-volume theory of a five-dimensional object called an M5-brane.

A world-volume theory is a theory that describes what it is like to live inside of the volume of some object, so that the object is your whole world. To understand what that means, think about Flatland.

Think upward, not northward

In Edwin A. Abbott’s Flatland, the characters are two-dimensional shapes living in a two-dimensional world. Because their whole world is two-dimensional, they cannot imagine a third dimension. Despite that, there is a third dimension, as demonstrated by a sphere who floats through the world one day and upsets the main character’s life. The theory of physics in Flatland, then, is the world-volume theory of a two-dimensional plane in three-dimensional space.

Imagine that the two-dimensional plane of Flatland was flexible, that is, more like a two-dimensional membrane. Such a membrane could move back and forth in the third dimension, rippling up and down.

Now remember that, in Flatland, nobody can imagine a third dimension. So if you are within Flatland, and the world around you is bouncing up and down, can you notice?

The answer is a counter-intuitive yes. It’s easy if there is gravity in the third dimension: when the world curves up, it would get harder to climb up, while if the world curves down, it would be easier. Even if there isn’t gravity, though, you can still notice the changes in energy. It takes energy to set the world vibrating, and that energy has to come from somewhere. That energy can come from movement within your dimension. What a Flatlander would observe, then, would be processes that seem to violate conservation of energy, by losing more energy than they put in: instead, that energy would go to making the world wiggle.

What a Flatland scientist would observe, then, would be a world in which there is some number that can change from place to place, and that can oscillate, carrying energy as it does so. Those of you who remember my older posts might recognize what’s going on here: this is precisely the way in which you discover the existence of a scalar field!

An M5-brane is a five-dimensional membrane that lives in M theory, a theory with eleven dimensions (ten space and one time). The world-volume theory of an M5-brane, then, is the theory of what it is like to have your whole world inside the five dimensions of the M5-brane, just like a person in Flatland has their whole world within the two dimensions of Flatland. And just like the two-dimensional Flatland would have one scalar field corresponding to its ability to vibrate in the third dimension, the five space dimensions of the (2, 0) theory have five scalar fields, corresponding to the five other directions (ten minus five) in which the M5-brane can move.

So the (2, 0) theory is the theory of what it’s like to live on a five-dimensional membrane in a ten-dimensional space, and because of that, the theory contains five scalar fields. But if it was just five scalar fields, it would hardly be mysterious. What else does the theory contain? And what does “(2, 0)” mean anyway? Tune in next week to find out!

Edit: I am informed by George Musser (@gmusser on twitter) that the Darth Vader thing was apparently all Nima Arkani-Hamed’s idea. So don’t blame him for the somewhat misleading metaphor!