Fine Structure

What is a Conformal Field Theory?

This post is part two 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.

In our second installment focused around an understanding of AdS/CFT correspondence, we're going to try to unravel the term "Conformal Field Theory" to the point where we can put it together with our understanding of Anti de Sitter space. We'll start by breaking up this question into two different pieces, the terms 'conformal' and 'field theory'.

Fields and Quantum Field Theory

Fields are a concept that we should be familiar with; the idea that for a certain force, every point in space has a scalar or vector associated with it which describes the force at that particular point (hopefully you're all still with me, if not, see here for a short introduction to scalar and vector quantities). When you look at all the vectors on a two-dimensional plane, you see a two-dimensional field. Depending on what you care about describing, you may need two or more dimensions. A field theory is simply a description of how a field behaves over time given some initial conditions. As an example, a temperature field theory would describe how temperature at one point transfers to other points and how the overall thermal properties of the system dictate how the field changes. This could be used to theoretically predict how your oven would heat a tasty pie but as far as I know, no manufacturers have begun shipping their wares with mathematical descriptions of how the shape and burner placement effect pie heating.

Although the exact details are much more complex than we need to understand here, a quantum field theory is a similar construct that describes a bunch of stuff at a bunch of different points. The only difference is that in a quantum field theory, such as the Standard Model, you have to replace your familiar vectors for something much more complex (literally, complex vectors), and in the process forgo the concept of one-state/one-outcome for probabilistic behavior. The Standard Model, of course, is currently the most complete description of our physical world. It predicts strong interactions by describing quarks and gluons, and weak and electromagnetic interactions by describing W/Z bosons and photons. If you're counting forces, yes, we did leave out gravity. Gravity simply hasn't been worked into what the Standard Model describes yet, though it is definitely a high priority for many physicists.

Conformal Transformation

We're going to return, for a moment, to the geometrical ideas of our Anti de Sitter space discussion. We spoke of different shaped spaces - some flat, some spherical, etc - and their geometric peculiarities. What if we wanted to display the surface of a spherical space on a euclidean surface? If we were to "unwrap" the surface of a sphere and attempt to paste it onto a flat surface, we would have all sorts of bumps and folds. This is where geometric transformations are useful. I think we've all seen maps of the Earth displayed on flat paper all in various states of disfiguration, this is a result of the different transformations used to expand a spherical surface on to a rectangle. The most notable deformation for me is always just how large or small Greenland appears. In terms of area, Greenland is about 25% of the area of the lower 48 United States. However, when viewed on some maps, Greenland appears as large or larger than the lower 48. This deformation is a result of geometric transformation.

When a mapmaker takes the surface of the Earth and projects it onto a flat surface, the mapmaker performs a geometric transformation. And just as there are lots of ways to show the Earth on a map, there are lots of ways to project a sphere on a flat surface. Some transformations preserve area, others distance or shape. Usually whichever property is useful in a particular task.

We're interested in the subset of these transformations that preserve angles, also known as conformal transformations. To give you an example of a popular conformal projection, check out some various stereographic projections. We notice that even though the landmass may be deformed, the angles at which the lines of latitude and longitude cross are always 90, just like on our original sphere. The Pan Am logo is another good example of a popularized conformal projection.

There are lots of conformal transformations that can occur and lots of differently shaped spaces to apply them to. These transformations can be lumped together into a group based on which spaces are being transformed.

Supersymmetry

Supersymmetry, something you've probably heard of before, is the idea that each elementary particle has a matching particle (a superpartner) that looks very similar in many ways but is beyond our current energy limits to see at colliders. This mathematical concept of a conformal group (a bunch of conformal transformations along with other mathematical structures) is a mathematical representation of supersymmetry when combined with another group, known as an internal symmetry group. Because supersymmetry is built from a conformal group, it is known as one of many conformal field theories, and a very interesting one at that.

The exact mathematical particulars aren't as important to us as the amazing fact that we can take these abstract structures and combine them in a way that predicts something real. When we're looking at supersymmetry, it says that we'll find the particles we already know about, which is great, but it also says that we'll find the superpartners, although the jury is still out at exactly what energy they might be found at.

Supersymmetry is just one of many conformal field theories and although the anti de Sitter space / conformal field theory correspondence applies to many field theories, the important ones are the ones that promise to predict real phenomenon, like supersymmetry. How are CFTs important in the correspondence? We'll wrap up with how anti de Sitter space and conformal field theory correspondence work to help theorists.

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