WZW Model Primary Fields

1. Primary Fields in Theories with Affine Symmetry

In a general CFT, a primary field is defined by its simple transformation property under the Virasoro algebra. However, WZW models have a much larger symmetry algebra: the affine Kac-Moody algebra $\hat{\mathfrak{g}}_k$. It is natural to define the "most fundamental" fields of the theory as those that transform as simply as possible under this larger algebra. These are often called affine primary fields.

An affine primary field $\Phi_\lambda(w)$ is defined by its Operator Product Expansion (OPE) with the affine currents $J^a(z)$:
$J^a(z){\Phi }_{\lambda }(w)\sim \frac{t_{\lambda }^a{\Phi }_{\lambda }(w)}{z-w}+\text{regular terms}$
Here, $t^a_\lambda$ are the matrices that represent the generators of the finite Lie algebra $\mathfrak{g}$ in the representation labeled by $\lambda$.

This OPE definition has a direct translation into the operator formalism. Using the state-operator correspondence, the field $\Phi_\lambda(w)$ corresponds to a state $|\lambda\rangle = \Phi_\lambda(0)|0\rangle$. The condition above is equivalent to $|\lambda\rangle$ being a highest-weight state of the affine algebra $\hat{\mathfrak{g}}_k$. This means:

1. It is annihilated by all "raising" operators of the affine algebra:
   $J_n^a|\lambda ⟩=0\text{for all }n>0,\text{ and for all }a$
2. It forms a representation of the "horizontal" subalgebra $\mathfrak{g}$ (the zero-modes):
   $J_0^a|\lambda ⟩=(t_{\lambda }^a)|\lambda ⟩$
   Specifically, $|\lambda\rangle$ is a highest-weight state of a representation of $\mathfrak{g}$.

This establishes our first key connection: the primary fields of a WZW model correspond to highest-weight representations of the affine algebra $\hat{\mathfrak{g}}_k$.

2. Conformal Dimension of Primary Fields

A crucial feature of the Sugawara construction is that an affine primary field is automatically a Virasoro primary field. We can compute the conformal dimension $h_\lambda$ of the field $\Phi_\lambda$ by acting on its corresponding state $|\lambda\rangle$ with the $L_0$ operator from the Sugawara construction.

Recall the Sugawara tensor:
$T(z)=\frac{1}{2(k+h^{\vee })}{\sum }_a:J^a(z)J^a(z):$
The corresponding $L_0$ operator involves a sum over modes of the currents. When we act with $L_0$ on a highest-weight state $|\lambda\rangle$, all terms with $J^a_n$ for $n>0$ annihilate the state. The calculation simplifies dramatically.

As you saw, the action of $L_0$ on $|\lambda\rangle$ isolates the action of the zero-modes, $\sum_a J^a_0 J^a_0$ which is precisely the quadratic Casimir operator $C_2$ of the Lie algebra $\mathfrak{g}$. The result is a fundamental formula in WZW theory:
$L_0|\lambda ⟩=h_{\lambda }|\lambda ⟩\text{with}{h}_{\lambda }=\frac{C_2(\lambda )}{2(k+h^{\vee })}$
Here, $C_2(\lambda)$ is the eigenvalue of the quadratic Casimir for the representation $\lambda$. This beautiful formula directly links the representation theory of $\mathfrak{g}$ (through $C_2(\lambda)$ and $h^\vee$) and $\hat{\mathfrak{g}}_k$ (through k) to the physical data of the CFT (the conformal dimensions $h_\lambda$).

3. The Role of Unitarity: Integrable Representations

So far, it seems that any highest-weight representation of $\mathfrak{g}$ gives rise to a primary field in the WZW model. However, WZW models based on compact Lie groups (like SU(N) or SO(N)) are unitary theories. This imposes a powerful physical constraint: the Hilbert space cannot contain any negative-norm states.

When we construct representations of an affine algebra starting from a highest-weight state $|\lambda\rangle$, we generate a Verma module. These modules can contain null states—states with zero norm. For the representation to be unitary, any potential negative-norm states must be absent. This requirement of unitarity restricts the set of allowed highest-weight representations $\lambda$ for a given level k.

The unitary highest-weight representations of an affine algebra are called integrable highest-weight representations. For a WZW model to be unitary, its spectrum of primary fields must correspond precisely to this set.

The condition for a representation $\lambda$ to be integrable at level $k$ is remarkably simple:
$⟨\lambda ,\theta ⟩\le k$
where $\theta$ is the highest root of the Lie algebra $\mathfrak{g}$ and the inner product is the standard one on the weight lattice.

A crucial consequence of this condition is that for any positive integer k, there is only a finite number of representations $\lambda$ that satisfy it. This means the WZW model has a finite number of primary fields. CFTs with this property are known as Rational Conformal Field Theories (RCFTs), and WZW models are the quintessential examples.

4. Example: The SU(2)k WZW Model

Let's make this concrete with the most important example, the $\mathfrak{su}(2)_k$ WZW model.

• Representations of $\mathfrak{su}(2)$: These are labeled by the spin $j \in \{0, 1/2, 1, 3/2, \dots\}$.
• Integrability Condition: For $\mathfrak{su}(2)$, the condition $\langle \lambda, \theta \rangle \le k$ becomes $2j \le k$.
• Allowed Primaries: Therefore, for a given integer level k, the allowed primary fields are those corresponding to spins:
  $j=0,\frac{1}{2},1,\dots ,\frac{k}{2}$
  There are $k+1$ primary fields in total.
• Conformal Dimensions: For $\mathfrak{su}(2)$, the Casimir is $C_2(j) = j(j+1)$ and the dual Coxeter number is $h^\vee = 2$. Plugging this into our formula for the conformal weight gives:
  $h_j=\frac{j(j+1)}{k+2}$

For k=1, the allowed spins are j=0 and j=1/2.

• j=0: $h_0=0$ (the identity operator).
• j=1/2: \(h_{1/2} = \frac{1/2(3/2)}{1+2} = 1/4\.

For k=2, the allowed spins are j=0, 1/2, 1.

• j=0: $h_0=0$
• j=1/2: $h_{1/2} = \frac{1/2(3/2)}{2+2} = 3/16$
• j=1: $h_1 = \frac{1(2)}{2+2} = 1/2$

This provides a complete classification of the primary field content and their scaling dimensions for any $\mathfrak{su}(2)_k$ model.

Conclusion

Today we have uncovered one of the most elegant features of WZW models. By combining the algebraic structure of the affine algebra with the physical requirement of unitarity, we arrived at a complete classification of the theory's primary fields.

Key Takeaways:

• The primary fields of a WZW model are affine primaries, meaning they transform simply under the affine Kac-Moody algebra $\hat{\mathfrak{g}}_k$.
• These fields correspond to states that are highest-weight states of the affine algebra.
• An affine primary is automatically a Virasoro primary, with a conformal dimension given by $h_\lambda = \frac{C_2(\lambda)}{2(k + h^\vee)}$.
• The requirement of unitarity restricts the allowed highest-weight representations to a finite set known as the integrable highest-weight representations, defined by the condition $\langle \lambda, \theta \rangle \le k$.
• This makes WZW models Rational CFTs, theories with a finite number of primary fields, whose properties are computable from the underlying algebra.

We have now identified the "what" of the WZW operator spectrum. In our next and final lesson of this module, we will address the "how"—how do these primary fields interact? We will see that the affine symmetry is so constraining that it leads to a set of linear differential equations, the Knizhnik-Zamolodchikov (KZ) equations, that govern all correlation functions in the theory.