There’s a cute site out there called Why String Theory. Started by Oxford and the Royal Society, Why String Theory contains lots of concise and well-illustrated explanations of string theory, and it even wades into some of the more complex topics like AdS/CFT and string dualities in general. Their explanation of dualities is a nice introduction to why dualities matter in string theory, but I don’t think it does a very good job of explaining what a duality actually is or how one works. As your fearless host, I’m confident that I can do better.
Why String Theory defines dualities as when “different mathematical theories describe the same physics.” How does that work, though? In what sense are the theories different, if they describe the same thing? And if they describe the same thing, why do we need both of them?
You’ve probably seen the above image before, or one much like it. Look at it one way, and you see a goblet. Another, and you see two faces.
Now imagine that instead of a flat image, these are 3D objects, models you have in your house. You’ve got a goblet, and a pair of clay faces. You’re still pretty sure they fit together like they do in the image, though. Maybe they said they fit together on the packaging, maybe you stuck them together and it didn’t look like there were any gaps. Whatever the reason, you’re confident enough about this that you’re willing to assume it’s true.
Now suppose you want to figure out how long the noses on the faces are. In case you’ve never measured a human nose, I can let you know that it’s tricky. You could put a ruler along the nose, but it would be diagonal rather than straight, so you wouldn’t get an accurate measurement. Even putting the ruler beneath the nose doesn’t work for rounded noses like these.
That said, measuring the goblet is easy. You can run measuring tape around the neck of the goblet to find the circumference, and then calculate the diameter. And if you measure the goblet in this way, you also know how long the faces’ noses are.
You could go further, and build up a list of things you can measure on one object that tell you about the other one. The necks match up to the base of the goblet, the foreheads to the mouth, etc. It would be like a dictionary, translating between two languages: the language of measurements of the faces, and the language of measurements of the goblet.
That sort of “dictionary” is the essence of duality. When two theories have a duality (are dual to each other), you can make a “dictionary” to translate measurements in one theory to measurements in the other. That doesn’t mean, however, that the theories are clearly connected: like 3D models of the faces and the goblet, it may be that without looking at the particular “silhouette” defined by duality the two views are radically different. Rather than physical objects, the theories compare mathematical “objects”, so rather than physical obstructions like the solidity of noses we have to deal with mathematical ones, situations where one quantity or another is easier or harder to calculate depending on how the math is set up. For example, many dualities relate things that require calculations at very high loops to things that can be calculated with fewer loops (for an explanation of loops, check out this post).
As Why String Theory points out, one of the most prominent dualities is called AdS/CFT, and it relates N=4 super Yang-Mills (a Conformal Field Theory, or CFT) to string theory in something called Anti-de Sitter (AdS) space (tricky to describe, but essentially a world in which space is warped like a hyperbola). Another duality relates N=4 super Yang-Mills Feynman diagrams with n particles coming in from outside to diagrams with an n-sided shape and particles randomly coming in from the edges of the shape (these latter diagrams are called Wilson loops). In general N=4 super Yang-Mills is involved in many, many dualities, which is a big part of why it’s so dang cool.