Sugawara Construction and Central Charge Calculation

1. The Sugawara Ansatz

In a theory with an affine Lie algebra symmetry, we have an infinite set of conserved quantities associated with the currents $J^a(z)$. How does the stress-energy tensor $T(z)$, which generates conformal transformations, relate to these currents?

The Sugawara construction proposes that $T(z)$ is built quadratically from the currents themselves. The most natural guess is an expression of the form:

$T(z)=\gamma {\sum }_a:J^a(z)J^a(z):$

Let's break this down:

• $\sum_a$ is a sum over an orthonormal basis of the Lie algebra $\mathfrak{g}$.
• The $::$ notation denotes normal ordering. As you know from QFT, when we bring two operator-valued fields to the same point, we get singularities. Normal ordering is a procedure to subtract these singularities, leaving a well-defined local operator. In the context of the Operator Product Expansion (OPE), it means we subtract the singular terms.
• $\gamma$ is a normalization constant. Our first task will be to determine its value by imposing consistency conditions.

For this $T(z)$ to be a valid stress-energy tensor for our CFT, two crucial properties must hold:

1. The currents $J^a(z)$ must be primary fields of conformal dimension $h=1$ with respect to this $T(z)$.
2. $T(z)$ must satisfy the Virasoro algebra OPE with itself.

We will now verify these two properties in turn.

2. Making Currents Primary: Fixing $\gamma$

Recall that the OPE between the stress tensor and a primary field $\phi(w)$ of dimension $h$ takes the universal form:
$T(z)\varphi (w)\sim \frac{h\varphi (w)}{(z-w{)}^2}+\frac{\partial \varphi (w)}{z-w}+\dots$
We need our currents $J^a(w)$ to satisfy this with $h=1$. To check this, we must compute the OPE of our $T(z)$ ansatz with $J^b(w)$ and demand it has the correct form. This calculation relies on the fundamental OPE of the affine Lie algebra, which we saw in a previous lesson:
$J^a(z)J^b(w)\sim \frac{k{\delta }^{ab}}{(z-w{)}^2}+\frac{i{f}^{abc}{J}^c(w)}{z-w}+\dots$

The following reading walks through this calculation. It shows that the OPE $T(z)J^b(w)$ does indeed produce the correct structure for a dimension-1 primary field, and this requirement uniquely fixes the value of $\gamma$.

As you saw in the reading, the calculation involves applying Wick's theorem to the expression $:J^a(z)J^a(z): J^b(w)$. A key algebraic identity that appears is:
$f^{acd}{f}^{bcd}=2h^{\vee }{\delta }^{ab}$
where $h^\vee$ is the dual Coxeter number of the Lie algebra $\mathfrak{g}$, a characteristic integer associated with the algebra's structure.

By performing the OPE calculation, one arrives at:
$T(z)J^a(w)\sim 2\gamma (k+h^{\vee })\left(\frac{J^a(w)}{(z-w{)}^2}+\frac{\partial J^a(w)}{z-w}\right)+\dots$
Comparing this to the standard form for a dimension-1 primary, we must have $2\gamma(k + h^\vee) = 1$. This fixes our normalization constant:
$\gamma =\frac{1}{2(k+h^{\vee })}$

With this, our stress tensor is fully defined as:
$T(z)=\frac{1}{2(k+h^{\vee })}{\sum }_a:J^a(z)J^a(z):$

3. Computing the Central Charge

Now that $T(z)$ is properly normalized, we can perform the final and most important check: computing its OPE with itself. This will verify that it generates a Virasoro algebra and will allow us to read off the central charge $c$.

The calculation is conceptually similar to the last one but algebraically more intensive. We need to compute the OPE of $T(z)T(w)$ by applying Wick's theorem to the product of two Sugawara tensors.

The calculation yields the standard Virasoro OPE:
$T(z)T(w)\sim \frac{c/2}{(z-w{)}^4}+\frac{2T(w)}{(z-w{)}^2}+\frac{\partial T(w)}{z-w}+\dots$
where the central charge $c$ is found to be:
$c=\frac{k\text{dim}(\mathfrak{g})}{k+h^{\vee }}$

This is a beautiful and powerful result. It explicitly shows that a WZW model is a CFT whose central charge is completely determined by the properties of its underlying affine Lie algebra: the level k, the dimension of the group $\text{dim}(\mathfrak{g})$, and the dual Coxeter number $h^\vee$.

4. Properties of the WZW Central Charge

Let's briefly examine the formula for the central charge.

A few key points are worth highlighting:

• For non-trivial $\mathfrak{g}$ and finite $k$ and $h^\vee$, the central charge is non-zero, confirming these are interacting CFTs (unless $k=0$).
• For a fixed algebra $\mathfrak{g}$, as the level $k \to \infty$, the central charge approaches $\text{dim}(\mathfrak{g})$. This corresponds to a semi-classical limit.
• The bound $\text{rank}(\mathfrak{g}) \le c < \text{dim}(\mathfrak{g})$ holds for positive integer k. The lower bound $c = \text{rank}(\mathfrak{g})$ is saturated for simply-laced algebras (A, D, E series) at level $k=1$. These k=1 models are special because they can be described by $\text{rank}(\mathfrak{g})$ free bosons, a result known as the Frenkel-Kac-Segal construction.

For example, for the $\mathfrak{su}(2)_k$ WZW model, we have $\text{dim}(\mathfrak{su}(2))=3$ and $h^\vee=2$. The central charge is:
$c=\frac{3k}{k+2}$
For $k=1$, we get $c=1$, corresponding to a single free boson. For $k=2$, $c=3/2$, which you may recognize as the central charge of the critical 3-state Potts model or the $\mathcal{N}=1$ superconformal minimal model.

Conclusion

Today, we have demonstrated that any WZW model based on an affine Lie algebra $\hat{\mathfrak{g}}_k$ is a conformal field theory.

Key Takeaways:

• The Sugawara construction provides an explicit formula for the stress-energy tensor $T(z)$ of a WZW model as a normal-ordered quadratic expression in the currents $J^a(z)$.
• The normalization of the Sugawara tensor is fixed by requiring the currents to be primary fields of dimension 1. This introduces the dual Coxeter number $h^\vee$ into the expression.
• The $T(z)T(w)$ OPE confirms that the Sugawara tensor generates a Virasoro algebra, with a central charge given by $c = \frac{k \, \text{dim}(\mathfrak{g})}{k + h^\vee}$.
• This result solidifies the connection between affine Lie algebras and 2D CFT, showing that the algebraic data ($k, \mathfrak{g}, h^\vee$) directly determines a key physical parameter (c) of the theory.

We have now established the full conformal and affine symmetry structure of WZW models. In our next lesson, we will begin to explore the consequences of this structure by identifying the primary fields of the theory, which correspond to the integrable representations of the affine Lie algebra that we studied previously.