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Lesson 36

KZ Equations in WZW Models

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Welcome to the final lesson in our module on Current Algebras and Wess-Zumino-Witten (WZW) models.

In our previous lesson, we identified the primary fields of a WZW model with the finite set of integrable highest-weight representations of the underlying affine Lie algebra $\hat{\mathfrak{g}}_k$. We also derived a precise formula for their conformal dimensions. This provided a complete picture of the operator content of these theories.

Now, we turn to the dynamics. How do these primary fields interact? How can we compute their correlation functions? In a general CFT, this is a difficult problem, but the immensely powerful affine symmetry of WZW models provides an astonishingly complete answer. It constrains the correlation functions to obey a set of linear partial differential equations.

Today's learning objective is to state the Knizhnik-Zamolodchikov (KZ) equations for correlation functions in WZW models. We will derive these equations, understand their deeper geometric meaning, and see what they look like in a concrete example.

1. Deriving the Knizhnik-Zamolodchikov Equations

The logic behind deriving the KZ equations is conceptually similar to how we derived Ward identities for the stress tensor: we find a null operator (an operator that is identically zero) and insert it into a correlation function. Since the operator is zero, the resulting expression must also be zero, yielding a constraint on the correlator.

In our case, the null operator comes from the Sugawara construction, which expresses the Virasoro generators $L_n$ in terms of the affine current modes $J^a_n$. Specifically, we have an exact operator identity relating $L_{-1}$ to a bilinear in currents. Inserting this identity into a correlator of primary fields will give us the KZ equations.

The most direct way to see this is by using the state-operator correspondence and contour integration, a technique you are familiar with from your work with QFT path integrals.

Wess-Zumino-Witten Models

The following resource provides a brilliant and self-contained derivation based on this idea. It presents the task as an exercise and then gives the full solution, which we will work through.

Please study 'Exercise 4' on page 18, including its provided solution. This is the core of today's lesson. Focus on how inserting the identity (3.84) into a correlator (3.87) leads to the final differential equation (3.85). Pay close attention to the steps from (3.90) to (3.94), which use contour integration to evaluate the action of the current bilinear on the fields.

Let's unpack the key steps from that derivation:

  1. The Setup: We insert the operator $L_{-1} - \frac{1}{k + h^\vee}(J^aJ^a)_{-1}$ acting on a primary field $\Phi_{\lambda_i}(z_i)$ inside an n-point correlation function. Since this operator is identically zero, the correlator must vanish.
  2. The $L_{-1}$ term: From the state-operator correspondence, $L_{-1}$ is the translation operator, so its action on $\Phi_{\lambda_i}(z_i)$ simply yields the derivative $\partial_{z_i} \Phi_{\lambda_i}(z_i)$.
  3. The $(J^aJ^a)_{-1}$ term: This is the more complex part. The action of this operator on the field $\Phi_{\lambda_i}(z_i)$ is computed by a contour integral. Using the OPE for the current $J^a(z)$ with the other primary fields $\Phi_{\lambda_j}(z_j)$ in the correlator, the integral picks up poles at each $z_j$.
  4. The Result: Summing the residues from all other fields $j \neq i$ reproduces the sum over $j$ seen in the final equation.

The result is the celebrated Knizhnik-Zamolodchikov (KZ) equation:

Here:

  • The equation holds for each field insertion $i = 1, \dots, n$.
  • $\Phi_{\lambda_i}$ are primary fields in the representation $\lambda_i$ of the Lie algebra $\mathfrak{g}$.
  • $\Omega_{ij} = \sum_a t^a_i \otimes t^a_j$. The term $t^a_i \otimes t^a_j$ means that the generator $t^a$ acts on the $i$-th field in its representation $\lambda_i$ and on the $j$-th field in its representation $\lambda_j$. The operator $\Omega$ is recognized as the quadratic Casimir operator in the tensor product representation.

These equations are a system of first-order linear partial differential equations that completely determine the correlation functions up to an overall normalization constant.

2. Geometric Interpretation: Flat Connections and Conformal Blocks

The KZ equations possess a deep geometric structure, which will resonate with your background in differential geometry. The solutions to these equations, known as conformal blocks, are not single-valued functions on the space of insertion points. Instead, they are sections of a vector bundle over the configuration space of points.

The KZ equations themselves can be elegantly reinterpreted as the condition for a section of this bundle to be flat. In this picture, the equations define a flat connection $\nabla_{KZ}$ on the bundle of conformal blocks.

Wess-Zumino-Witten Models and the Knizhnik-Zamolodchikov ...

The following resource formalizes this powerful geometric viewpoint. It constructs the 'conformal block bundle' and shows how the KZ equations arise as a flatness condition.

Please read Section 6.4, 'The KZ Equations on the Conformal Block Bundle', from page 86 up to the statement of Theorem 6.4.4 on page 91. You need not absorb every detail of the bundle's construction, but focus on the main idea: the KZ equations (6.96 in the text) emerge from demanding that a section $\psi^0$ is flat (\( abla_{\partial_j} \psi^0 = 0\)), where the connection involves the $L_{-1}$ operator (see eq. 6.95).

This geometric interpretation has profound consequences:

  • The space of solutions (conformal blocks) at fixed positions $z_i$ forms a finite-dimensional vector space.
  • When we move the points $z_i$ around each other (i.e., follow a path in the configuration space), the solutions undergo a linear transformation. This transformation, known as monodromy, depends only on the topology of the path.
  • Since the connection is flat, the monodromy around loops provides a representation of the fundamental group of the configuration space, $\pi_1(\text{Conf}_n(\mathbb{C}))$ which is the braid group $B_n$.
  • This establishes a remarkable link between a 2D quantum field theory (the WZW model), representation theory (of $\hat{\mathfrak{g}}_k$), and low-dimensional topology (braid group representations).

3. A Concrete Example: The $\mathfrak{sl}_2$ WZW Model

To make the abstract equations more tangible, let's look at their explicit form for the $\mathfrak{sl}_2(\mathbb{C})$ WZW model, which is built on the same Lie algebra as $\mathfrak{su}(2)$. The algebra is spanned by generators $H, E, F$, which correspond to $J^3, J^+, J^-$. The Casimir operator $\Omega_{ij}$ can be written in terms of these generators.

Wess-Zumino-Witten Models and the Knizhnik-Zamolodchikov ...

Let's see what the KZ equations look like for our canonical example. This resource provides an explicit derivation for the $\mathfrak{sl}_2(\mathbb{C})$ case.

Please read Section 4.3, 'The $\hat{sl}_2(\mathbb{C})$ Knizhnik-Zamolodchikov Equations' (pages 54-57). Focus on the final form of the equation (4.105). Notice how the general Casimir term $\Omega_{ij}$ is now written explicitly in terms of the actions of $H_0, E_0, F_0$ on the different fields in the correlator. The dual Coxeter number for $\mathfrak{sl}_2$ is $h^\vee = 2$.

As you saw in the text, the equation becomes:

(Note: The resource uses a slightly different basis and conventions leading to the sign factors $(-1)^{i+j}$; the structure is the same.)

The crucial point is that this provides a concrete system of differential equations. For a given correlator, say a 4-point function, this system can be solved. The solutions typically involve special functions, such as hypergeometric functions. Section 4.4 of the same thesis (LINK) demonstrates this explicitly, showing how the KZ equations can be used as a practical computational tool to determine correlation functions.

Test your understanding!

Consider the KZ equation for a 2-point function $\langle \Phi_\lambda(z_1) \Phi_{\lambda'}(z_2) \rangle$ in a general WZW model.
$$ \left( (k+h^\vee) \frac{\partial}{\partial z_1} + \frac{\Omega_{12}}{z_1 - z_2} \right) \langle \Phi_\lambda(z_1) \Phi_{\lambda'}(z_2) \rangle = 0 $$
We already know from general conformal symmetry that $\langle \Phi_\lambda(z_1) \Phi_{\lambda'}(z_2) \rangle = \frac{C_{12} \delta_{\lambda, \lambda'}}{(z_1-z_2)^{2h_\lambda}}$. How does the KZ equation constrain the conformal dimension $h_\lambda$?

Show answer

First, note that the Casimir operator $\Omega_{12}$ acts on $\Phi_\lambda \otimes \Phi_\lambda$. This state is an eigenstate of $\Omega_{12}$. The eigenvalue is related to the Casimirs of the individual representations and the tensor product. Specifically, $2\Omega_{12} = C_2(\lambda \otimes \lambda) - C_2(\lambda) - C_2(\lambda)$. For this problem, we just need its action on the state. Acting on the state $|\lambda\rangle \otimes |\lambda\rangle$, we have $\Omega_{12} |\lambda\rangle \otimes |\lambda\rangle = C_2(\lambda) |\lambda\rangle \otimes |\lambda\rangle$.

Let's plug the form of the 2-point function into the KZ equation (and ignore the delta function for now):
$$ \left( (k+h^\vee) \frac{\partial}{\partial z_1} + \frac{C_2(\lambda)}{z_1 - z_2} \right) \frac{1}{(z_1-z_2)^{2h_\lambda}} = 0 $$
$$ (k+h^\vee) \frac{-2h_\lambda}{(z_1-z_2)^{2h_\lambda+1}} + \frac{C_2(\lambda)}{z_1-z_2} \frac{1}{(z_1-z_2)^{2h_\lambda}} = 0 $$
Multiplying by $(z_1-z_2)^{2h_\lambda+1}$, we get:
$$ -2h_\lambda (k+h^\vee) + C_2(\lambda) = 0 $$
$$ h_\lambda = \frac{C_2(\lambda)}{2(k+h^\vee)} $$
This is exactly the formula for the conformal dimension of a primary field that we derived in the last lesson using the Sugawara $L_0$ operator! The KZ equations are beautifully consistent with the rest of the theory.

Conclusion

This lesson concludes our exploration of WZW models, and it ends on a powerful note. The affine symmetry that defines these theories is so restrictive that it dictates their entire correlator structure through a set of solvable differential equations.

Key Takeaways:

  • The correlation functions of primary fields in a WZW model obey the Knizhnik-Zamolodchikov (KZ) equations.
  • These are first-order linear partial differential equations derived by enforcing the operator relations of the Sugawara construction inside a correlator.
  • The KZ equations have a deep geometric meaning, defining a flat connection on the bundle of conformal blocks, whose monodromy gives representations of the braid group.
  • These equations are not just of theoretical interest; they provide a concrete computational tool for calculating correlation functions in WZW models.

We have now completed our foundational study of CFT, from the basic principles of conformal symmetry to the intricate algebraic structures of minimal models and WZW models.

Next Lesson Preview:
In the next module, we will venture into the realm of Superconformal Field Theory (SCFT). We will introduce a new type of symmetry—supersymmetry—which relates bosons and fermions. This will lead us to construct superconformal algebras and explore their representations. Given your prior study of classical supersymmetry, you will find it fascinating to see how these concepts are realized at the quantum level in CFT.