Unitary Minimal Models: Central Charge, Operator Content, and the Kac Table

Unitarity and the c < 1 Discrete Series

As we discussed previously, a unitary theory requires the Gram matrix of inner products to be positive semi-definite at every level $N$. This means its determinant must be non-negative:

The Kac determinant formula gives us the zeros of this polynomial. Between these zeros, the determinant is either positive or negative. A careful analysis, first performed by Friedan, Qiu, and Shenker in 1984, showed that for c < 1, the only way to satisfy this condition for all $N$ is if (c, h) belongs to a specific discrete set.

To understand this result, let's look at the relevant material.

The result you just read is profound. It tells us that unitary representations of the Virasoro algebra with c < 1 exist only if:

1. The central charge takes one of the discrete values: $$c_m=1-\frac{6}{m(m+1)}\text{for }m=2,3,4,\dots$$
2. For a given $c_m$, the conformal dimension $h$ of a primary field must be one of the values from the Kac formula: $$h_{r,s}=\frac{((m+1)r-ms{)}^2-1}{4m(m+1)}$$ where the integers $r$ and $s$ are restricted to the ranges $1\le r\le m-1$ and $1\le s\le m$.

This discrete series of theories are the unitary minimal models. The integer $m$ labels the model.

The Structure of Minimal Models

Why does having a finite number of primaries relate to these specific values of c? The key lies in the concept of multiply degenerate representations.

To get a finite operator algebra that is closed under the Operator Product Expansion (OPE), we need the fusion of any two primary fields to produce only other fields from our finite set. This truncation happens very effectively when representations have more null vectors than the minimum. This occurs when a conformal dimension h happens to satisfy the condition h = h_{r,s} for multiple pairs (r,s).

This requires a special tuning of the central charge. Let's introduce a useful parametrization you may have seen in your self-study of QFT, which is particularly suited to this analysis.

As you've just read, for a representation to be multiply degenerate, we need coincidences of the form ${\Delta }_{r,s}={\Delta }_{r^{\prime },s^{\prime }}$ (using the notation $\Delta$ for conformal dimension from that text). This happens when the parameter $b^2$ (related to $c$) is a rational number. Let's set $b^2=-q/p$ for coprime integers $p,q$. This leads to:

• A central charge c = 1 - 6(p-q)^2 / (pq)
• An identification between conformal dimensions: ${\Delta }_{r,s}={\Delta }_{p-r,q-s}$.

How does this connect to the unitary series c_m? The unitary models correspond to the special case where |p-q| = 1. If we set q = m and p = m+1, we recover the unitary series:

So, the m-th unitary minimal model is also known as the (m+1, m) minimal model.

The Kac Table and Operator Content

For the (p,q) = (m+1, m) minimal model, the complete set of primary operators is labeled by integer pairs (r,s) within a bounded rectangle, called the Kac table.

The operator content of the m-th unitary minimal model is given by the primary fields with dimensions h_{r,s} where 1 \le r \le p-1 = m and 1 \le s \le q-1 = m-1.

Due to the identification h_{r,s} = h_{p-r, q-s} = h_{m+1-r, m-s}, the number of distinct primary operators is roughly half the size of this rectangle: m(m-1)/2.

Let's make this concrete with the most famous example.

Example: The m=3 Minimal Model (Critical Ising Model)

The simplest non-trivial unitary minimal model corresponds to m=3.

• This is the (p,q) = (4,3) model.
• The central charge is c_3 = 1 - 6/(3 \times 4) = 1/2. This is the CFT that describes a 2D Ising model at its critical temperature.
• The primary operators are labeled by (r,s) with 1 \le r \le 3 and 1 \le s \le 2. This gives a 3x2 Kac table.
• The conformal dimensions are given by h_{r,s} = \frac{(4r - 3s)^2 - 1}{4 \cdot 3 \cdot 4} = \frac{(4r - 3s)^2 - 1}{48}.

Let's calculate the dimensions for the six pairs in the Kac table:

• (r,s)=(1,1): h_{1,1} = ( (4-3)^2 - 1 ) / 48 = 0. This is the identity operator I.
• (r,s)=(1,2): h_{1,2} = ( (4-6)^2 - 1 ) / 48 = 3/48 = 1/16. This is the spin operator σ.
• (r,s)=(2,1): h_{2,1} = ( (8-3)^2 - 1 ) / 48 = 24/48 = 1/2. This is the energy operator ε.
• (r,s)=(2,2): h_{2,2} = ( (8-6)^2 - 1 ) / 48 = 3/48 = 1/16. This is the same as h_{1,2}.
• (r,s)=(3,1): h_{3,1} = ( (12-3)^2 - 1 ) / 48 = 80/48 = 5/3. (Wait, let's use the identity h_{r,s} = h_{4-r, 3-s}).
  h_{3,1} = h_{4-3, 3-1} = h_{1,2} = 1/16.
• (r,s)=(3,2): h_{3,2} = h_{4-3, 3-2} = h_{1,1} = 0.

The Kac table of conformal dimensions looks like this:

The set of distinct primary operator dimensions is {0, 1/16, 1/2}. This CFT contains only three primary fields: the identity I, the spin field σ, and the energy field ε. All other states in the theory are their descendants. The finiteness of this spectrum is what makes the model "minimal" and, ultimately, solvable.

Conclusion

Today we have seen how the physical principle of unitarity acts as a powerful constraint on the mathematical structure of the Virasoro algebra. For c < 1, this leads to a discrete series of physically consistent theories, the unitary minimal models.

Key Takeaways:

• Unitary CFTs with c < 1 only exist for a discrete set of central charges c_m = 1 - 6/(m(m+1)) where m=2,3,....
• These are called the unitary minimal models and can be labeled by m or the coprime integer pair (p,q) = (m+1, m).
• A minimal model has a finite number of primary fields, whose conformal dimensions h_{r,s} are given by the Kac formula.
• The full operator content of the m-th model is classified by the Kac table: a grid of (r,s) values with 1 \le r \le m and 1 \le s \le m-1.
• The symmetry h_{r,s} = h_{m+1-r, m-s} means the number of distinct primary operators is m(m-1)/2.
• The m=3 model, with c=1/2 and operators of dimension {0, 1/16, 1/2}, describes the critical 2D Ising model.

We have now established the classification of these important theories. The next step is to understand how to compute with them. In our next lesson, we will see how the existence of null vectors, which was the key to this classification, gives us powerful differential equations that correlation functions must obey, paving the way to solving these theories completely.