Conformal field theory

A conformal field theory (CFT) is a special kind of quantum field theory that stays the same even when shapes are stretched or squished. This happens because it is invariant under conformal transformations.

In just two dimensions, there are many ways to change shapes, and this creates a very rich structure that helps scientists solve and organize these theories exactly in some cases.

Conformal field theory is very useful in many areas of physics, such as condensed matter physics, where it helps us understand how materials behave, and in statistical mechanics and quantum statistical mechanics. It also plays an important role in string theory. Often, systems in these fields show conformal symmetry at special points, like their thermodynamic or quantum critical points.

Scale invariance vs conformal invariance

In quantum field theory, scale invariance is a common symmetry. This means that some theories look the same no matter how much you zoom in or out. However, conformal symmetry is even stronger than scale invariance. It needs extra ideas to explain why it might appear in nature.

In some special cases, scale invariance actually means the theory is also conformally invariant. This is true for certain types of theories in two dimensions. While it is possible for a theory to be scale invariant but not conformally invariant, such examples are rare. Because of this, the two terms are often used interchangeably in quantum field theory.

Two dimensions vs higher dimensions

In two dimensions, there are infinitely many ways to change the shape of things while keeping angles the same. This makes studying these special theories much more detailed and powerful. In higher dimensions, there are fewer ways to do this, so scientists often need computers to study these theories.

Scientists have been studying these special theories in two dimensions for a long time, especially since a big paper in 1983. These theories became more popular again later when scientists found new ways to connect theories in different dimensions.

Global vs local conformal symmetry in two dimensions

In two dimensions, there are two kinds of shape changes. One kind has only a few options, like moving, rotating, or zooming in and out. The other kind has infinitely many options, like twisting and stretching in very smooth ways. These smooth changes create many special patterns that help scientists understand these theories better.

Conformal field theories with a Virasoro symmetry algebra

Main article: Two-dimensional conformal field theory

When studying these special theories in two dimensions, scientists found that the rules for shape changes include something extra called the central charge. This helps them understand how these theories change over time. These theories have two sets of rules that work together, and they can describe very simple or very complex situations.

The space of all possible states in these theories follows special patterns, and scientists use these patterns to learn more about how these theories behave.

Conformal symmetry

Main article: Conformal symmetry

Conformal symmetry is a special kind of transformation in physics. In simple terms, a conformal transformation is one that keeps angles the same, even if it changes the size of shapes. This idea is important in studying spaces like flat Euclidean space or Minkowski space, which is used to describe space and time in physics.

The conformal group includes many types of transformations, such as moving things around (translations), turning them (rotations or Lorentz transformations), and changing their size (scale transformations). These transformations help scientists understand how things behave in different spaces and are used in many areas of physics, from studying materials to theories about the universe.

Correlation functions and conformal bootstrap

A conformal field theory is a special kind of quantum theory where the way things change with distance follows specific rules. In these theories, we look at how different points in space relate to each other through "correlation functions." These functions tell us how fields — which are like spreads of properties through space — interact at different points.

The "conformal bootstrap" approach helps us understand these theories by focusing on the rules that these correlation functions must follow. These rules are like puzzle pieces that fit together to describe the whole picture, without needing to know every tiny detail about the fields themselves. This method is powerful because it can sometimes solve or classify these theories exactly, especially in two dimensions where the rules become even clearer.

Examples

Mean field theory

A generalized free field is a special kind of field where its relationships between different points can be figured out using a simple rule called Wick's theorem. For example, if ϕ is a type of field with a certain size Δ, we can describe how it behaves at four different points.

Mean field theory is a name for conformal field theories made from generalized free fields. For instance, we can build a mean field theory from one field ϕ. This theory includes ϕ, fields related to ϕ, and other fields that show up when we combine ϕ with itself.

It is also possible to make mean field theories from fields that twist in certain ways. For example, a theory in four dimensions called Maxwell theory is a mean field theory made from a special kind of field Fμν with a size of 2.

Mean field theories can be described using math that involves a special operator raised to any power, which decides the size of the field. For some sizes, this power isn't a whole number, making the theory a bit more complex.

Critical Ising model

The critical Ising model is a special state of the Ising model, which is usually studied on a grid in two or three dimensions. It has a symmetry that lets you flip all the "spins" (or points) at once. The two-dimensional version of this model can be solved exactly and is linked to special math structures.

Critical Potts model

The critical Potts model is a general version of the Ising model for q = 2, 3, 4, and so on colors. It is unchanged when we rearrange the colors using a special math group Sq. This model is a broader version of the Ising model, which is like the Potts model when q = 2.

The critical Potts model can be made by looking at the Potts model on a grid in d dimensions as we move closer and closer to a special point. In some math ways, the Potts model can use colors that aren't whole numbers, but it only works well when the number of colors is a whole number.

Critical O(N) model

The critical O(N) model is a conformal field theory that keeps a certain symmetry. For any whole number N, it exists as a special theory in three dimensions (and for N = 1 also in two dimensions). It is a broader version of the Ising model, which is the O(N) model when N = 1.

The O(N) model can be made by looking at a lattice model with points that are N-dimensional vectors as we move closer to a special point.

Phase transition

Special changes in materials, called continuous phase transitions, are often described by certain kinds of field theories. For this to work, the material must look the same in all directions and positions. However, this isn't always enough: some special points are described by theories that save distances but not angles. If the material has a special property called reflection positivity, the field theory describing its special point will follow special rules.

Continuous changes in quantum materials with D spatial directions can sometimes be described by field theories in D+1 dimensions that save angles and follow special time rules. Besides saving positions and directions, another rule must be followed: a special exponent z must be 1. Field theories describing these quantum changes (when there are no special disorders) always follow special rules.

String theory

The study of strings includes a two-dimensional conformal field theory linked to how the world bends. For string theory to work, this field theory must have a certain value, called the central charge, which is 26 for basic string theory and 10 for superstring theory. The places where strings exist match up with certain fields in this theory.

AdS/CFT correspondence

Conformal field theories are very important in the AdS/CFT correspondence. This is a way to match a theory of gravity in a special space called anti-de Sitter space (AdS) to a conformal field theory on the edge of that space. For example, a special theory in 4 dimensions called N=4 supersymmetric Yang–Mills theory matches up with a type of string theory in a five-dimensional AdS space and a five-dimensional sphere. Another example is a special theory in 3 dimensions called N=6 super-Chern–Simons theory, which matches up with a theory called M-theory in a four-dimensional AdS space and a seven-dimensional sphere.

Conformal perturbation theory

By changing a conformal field theory a little, we can build other theories, some of which save angles and some of which don't. We can figure out how these new theories behave using a method called conformal perturbation theory, which uses the behavior of the original theory.

For example, we can study a conformal field theory on a grid instead of smooth space. The small changes that come from this can be calculated using conformal perturbation theory.