WZW Model: Action, Equations of Motion, and Conserved Currents

1. The Challenge: A Naive Sigma Model

Let's start by constructing the most straightforward field theory with a Lie group symmetry. Consider a 2D field theory where the field g(z, \bar{z}) is a map from our 2D spacetime (which we take to be a Riemann sphere, $S^2$) to a compact, simple Lie group G.

This is a type of non-linear sigma model, often called the Principal Chiral Model. Its action is given by:
$S_0=\frac{1}{4{\lambda }^2}{\int }_{S^2}{d}^{2}z\text{tr}\left((g^{-1}{\partial }_{\mu }g)(g^{-1}{\partial }^{\mu }g)\right)$
Here, $\lambda^2$ is a coupling constant. The quantity J_\mu = g^{-1}\partial_\mu g is a Lie algebra-valued field, which you may recognize as the pullback of the Maurer-Cartan form on G. This action has a global $G_L\times G_R$ symmetry, under which g \to g_L g g_R^{-1}.

Varying this action yields the classical equation of motion:
${\partial }^{\mu }(g^{-1}{\partial }_{\mu }g)=0\text{or}{\partial }^{\mu }{J}_{\mu }=0$
This is a conservation equation for the current J_\mu. In complex coordinates (z, \bar{z}), this becomes \partial J_{\bar{z}} + \bar{\partial} J_z = 0.

However, for a 2D CFT, we expect something stronger: the existence of separately conserved holomorphic (\bar{\partial} J_z = 0) and anti-holomorphic (\partial J_{\bar{z}} = 0) currents. The equation of motion for the Principal Chiral Model does not guarantee this. In fact, one can show that this factorization only occurs if the Lie algebra is abelian. For a non-abelian group, the theory is not conformally invariant at the quantum level; it has a non-zero beta function and is only asymptotically free.

To build a true CFT, we must modify the action.

2. The Solution: The Wess-Zumino Term

The key insight, due to Wess, Zumino, and Witten, is to add a topological term to the action. This Wess-Zumino (WZ) term is defined in a peculiar way. We view our 2D spacetime S^2 as the boundary of a 3-dimensional ball, B. We then extend the field g from S^2 into B. The WZ term is an integral over this 3D bulk:

$\Gamma [g]=-\frac{i}{12\pi }{\int }_B{d}^{3}y{ϵ}^{\alpha \beta \gamma }\text{tr}\left((g^{-1}{\partial }_{\alpha }g)(g^{-1}{\partial }_{\beta }g)(g^{-1}{\partial }_{\gamma }g)\right)$

This definition immediately raises a question: the extension of g from the boundary S^2 to the bulk B is not unique. How can the action be well-defined?

This is where topology provides a beautiful answer.

1. Existence of Extension: For any simple Lie group G, the second homotopy group is trivial: $\pi_2(G) = 0$. This guarantees that any map from S^2 to G can be smoothly contracted to a point, and thus an extension to the interior of the ball B always exists.
2. Ambiguity of Extension: The ambiguity in choosing an extension is classified by the third homotopy group, $\pi_3(G)$. For any compact, simple Lie group, $\pi_3(G) = \mathbb{Z}$. This means that any two different extensions, say g_1 and g_2, are related by an integer "winding number".
3. Path Integral Quantization: One can calculate that if two extensions differ by one unit of this winding number, the value of the WZ term \Gamma[g] changes by exactly $2\pi i$: $\Delta\Gamma = 2\pi i$.

The full WZW action is then defined as $S_{WZW} = S_0 + k\Gamma[g]$, where k is a constant. In a quantum theory, we compute observables using the path integral, which involves the weighting factor $e^{-S_{WZW}}$. For this factor to be uniquely defined, the action S_{WZW} only needs to be well-defined up to integer multiples of $2\pi i$.

Since an ambiguity in $\Gamma$ introduces a shift of $k \cdot 2\pi i$, the path integral is well-defined if and only if k is an integer. This is a profound result: the level k, which we saw classifying representations of the affine algebra, is quantized because of the topology of the Lie group G.

3. Equations of Motion and Conserved Currents

Now, let's derive the equations of motion for the full WZW action and see how the WZ term achieves our goal. The following reading guides you through this calculation.

As the resource demonstrates, the variation of the full action $S = S_0 + k\Gamma$ leads to the equation of motion:
$\left(\frac{1}{2{\lambda }^2}-\frac{k}{4\pi }\right)\bar{\partial }{J}_z+\left(\frac{1}{2{\lambda }^2}+\frac{k}{4\pi }\right)\partial J_{\bar{z}}=0$
where we have used the complex coordinate definitions from the text, $J_z = g^{-1}\partial g$ and $J_{\bar{z}} = g^{-1}\bar{\partial} g$.

This is the crucial result. Unlike the principal chiral model, we now have two distinct terms. By making a specific choice for the coupling constant $\lambda^2$ relative to the level k, we can eliminate one of them. The standard convention for the WZW model is to set $\lambda^2 = \pi/k$, which corresponds to setting the coefficient of $\bar{\partial}J_z$ to zero (after an overall rescaling). This yields:
$\partial (g^{-1}\bar{\partial }g)=0$

This simple equation has powerful consequences:

1. The Lie-algebra valued current $\bar{J} \propto g^{-1}\bar{\partial}g$ is anti-holomorphic: $\partial \bar{J} = 0$.
2. One can also show this implies that the "left" current $J \propto \partial g g^{-1}$ is holomorphic: $\bar{\partial} J = 0$.

By adding the WZ term and fine-tuning the coupling, we have engineered a theory that possesses exactly the conserved holomorphic and anti-holomorphic currents required to build a Kac-Moody algebra. These are the currents $J^a(z)$ and $\bar{J}^a(\bar{z})$ whose modes $J^a_n$ and $\bar{J}^a_n$ generate the affine algebra we studied.

Conclusion

In this lesson, we have constructed the Wess-Zumino-Witten model, a concrete physical system that realizes an affine Lie algebra symmetry.

Key Takeaways:

• The Principal Chiral Model, a simple non-linear sigma model on a Lie group G, is not conformally invariant for non-abelian G because its currents are not separately holomorphic and anti-holomorphic.
• The Wess-Zumino (WZ) term is a topological term added to the action. It is defined as an integral over a 3D manifold whose boundary is the 2D spacetime.
• The topological ambiguity of the WZ term under different extensions of the field leads to the quantization of the level k to an integer in the quantum theory.
• The full WZW action, at a specific value of the coupling, yields equations of motion that guarantee the existence of separate holomorphic and anti-holomorphic conserved currents, $J(z)$ and $\bar{J}(\bar{z})$.

We now have a classical action and its associated conserved currents. The next vital step is to understand how this model fits into the general framework of 2D CFT, which is governed by the Virasoro algebra. In our next lesson, we will see how the energy-momentum tensor $T(z)$ can be constructed directly from the currents $J^a(z)$ using the Sugawara construction. This will allow us to compute the central charge of the WZW model and show that it is determined by the level k and properties of the Lie group G.