Fine Structure

The Maldacena Duality

This post is part three in a discussion about Anti de Sitter space / Conformal Field Theory correspondence. You can read more backstory about why we want to know about AdS / CFT here.

Now that we've wrapped our heads around a simplistic view of what Anti de Sitter space is and what a conformal field theory is, we can start to discuss what this whole "correspondence" thing is and why it is so important to the people working in certain areas of theoretical physics.

The Correspondence

Firstly, correspondence is a mathematical term that relates two different things in math. In fact, it relates them so well that whatever you choose in one "thing", you'll be able to choose a match for it in another "thing". I'm calling these vague mathematical objects "things" because they're really proper terms like groups that have actual meanings beyond what we need to know. You can think of it as a translator between two mathematical languages. The messages in both languages mean the same thing but they're represented in significantly different ways. Though correspondence is the usual term you hear, AdS/CFT correspondence is also known as the Maldacena Duality, after Juan Maldacena who first suggested it. The term "duality" is essentially the same as "correspondence" (not to mention a much more clear term in my mind) but the two have subtle differences in mathematics.

This kind of theoretical physics is very much about finding new mathematical tools for dealing with interesting mathematical objects that may turn out to be a good representation of the way the world works. Correspondence is one of those useful mathematical tools and we'll shortly see why.

Anti de Sitter space and Conformal Field Theory are clearly two very different things. One is a description of a curved spacetime and the other is a product of two groups of transformations. What Maldacena found was that for any Anti de Sitter space that represented a certain number of dimensions, there was a conformal field theory of one less dimension that looked very similar. So similar, in fact, that one could create a correspondence between the two and take any representation in AdS and transform it into a representation in CFT, or vice versa. This allowed theorists a new tool in an attempt to solve difficult problems that they had encountered, either while looking at descriptions of the world in AdS or CFT.

Solving the Problems

Many mathematical tools work by approximation. We measure the rate of a falling body by its mass and an approximation of the gravitational force at sea level. If we travel too far above or below sea level, our approximation of gravity is thrown out of whack and we have to find a more complex method to determine the force of gravity at arbitrary distances from the center of the earth. Similarly, a theory of how the universe works might do well on the scale of a human but fall apart when we try to describe the movement of the stars, or perhaps when we try to describe very small particles. Figuring out how to solve these problems with new mathematical tools is at the heart of new discoveries.

When a theorist runs into one of these issues while using a conformal field theory, such as supersymmetry, it can be because calculations get incredibly complex as approximations stop behaving like good approximations and numbers start going to infinity or zero. These two things are usually not good news for a theorist since they represent non-physical characteristics like something disappearing completely or going on forever. This is where the correspondence comes in. By taking a difficult problem in a conformal field theory and, using the correspondence, transforming it into a problem in anti de Sitter space, we can potentially work around occurrences of zero and infinity. Certain problems that can be solved in AdS are more difficult in a CFT, likewise in the opposite direction, and that's where the power of the correspondence lies.

In addition, just the fact that these two very different mathematical concepts relate to one another so well is a fascinating research topic that is studied all on its own.

The Purpose

Although we haven't picked up any of the math related to the correspondence, I hope we've gained a better understanding of what each of the players are and why they're useful individually and in their duality. At least you'll know a thing or two when you see a mention on another blog or in a paper - maybe even enough to get someone started on the way to a new discovery via these tools.

Although I strive to put together facts from many different sources in a way that is understandable and makes sense, this stuff isn't explained easily in any form so please correct me if I've misused a term or have some facts backwards.

Finally, many thanks to the blogger at The First Excited State who helped me out immensely when trying to figure all this stuff out. It was incredibly helpful to get some pointers in the right direction!