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. Since the (2, 0) theory involves some more complicated topics than N=4 super Yang-Mills, you should read my guide on that theory before reading this one.

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 Nima Arkani-Hamed referred to as the Darth Vader theory, or N=4 super Yang-Mills to most people.

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? Read Part Two to find out!

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