Hassaan Saleem

This is second of my series of posts on AdS₃/CFT₂ correspondence (for the first post, click here). In this post, I will describe what WZW (Wess-Zumino-Witten) models are. They are relevant here because a WZW model is used to decribe string theory on AdS₃ . Basics of field theory, little familiarity with Lie groups and some familiarity with CFTs is assumed to understand this post.

To start, let me describe what a sigma model is. Consider a field theory that lives on a space that we will call Σ and has a field Φ . Now, for sigma models, the field Φ takes values on a space (manifold) called the target space (or target manifold). For example, the target space for a real scalar field is just the real line.

When we write down the lagrangian for a field theory, we have a kinetic term and a possible potential term. In the kinetic term, two derivetives of the field are contracted with each other (as shown below) via a metric tensor on Σ . For sigma models, this metric tensor itself may depend on the field Φ.

Now, some example of manifolds come from Lie groups. Lie groups are groups that are also manifolds. For example, the group U(1) is a circle and the group SU(2) is a three dimensional sphere (called S³). If a field Φ takes values in a Lie group (let’s call this group G) and Σ is a two dimensional space (e.g. a sphere or a torus), then the sigma model is called a G-WZW model that has the following action.

The WZ (Wess-Zumino) term is required if you want this model to have conformal symmetry. Note that this term doesn’t include an integral over Σ but over a three dimensional space called B whose boundary is Σ.

The parameter k is called the level of the WZW model. It turns out that this level is an integer if G is a compact group (e.g. SU(2)) but it doesn’t need to be an integer if G isn’t compact (e.g. SL(2,R)). We will see the relevance of this fact in a later post.

Now, the first question should be about the equations of motion of this model. The equations of motion just give us two types of conserved currents (see below). Each type of currents are dim(G) in number where dim(G) is the number of dimensions of the Lie group G. If we introduce complex coordinates (z, z*) on the two dimensional Σ, then one type of currents are entirely dependent on z (i.e. it is holomorphic) and the other type of currents are entirely dependent on z* (i.e. it is antiholomorphic).

Now, since WZW models are CFTs, we can talk about their central charge. It turns out that you can write down an expression for the central charge of the WZW model that is built out of G. This expression will be useful in a later post.

It depends on dim(G) and another property of G called the dual Coxeter number (don’t worry about it as it is just a quantity made out of the structure constants of the Lie algebra of G). I will talk more about it later if required. (To be continued)