Minimal Models, Ward Identities, and Correlators

The Kac determinant tells us when a Virasoro module develops null states. Minimal models are the theories where this mechanism becomes so strong that only finitely many primary fields remain.

This is one of the most striking lessons of two-dimensional CFT: in special theories, symmetry plus null-vector decoupling can determine correlation functions almost completely. The same contour methods are also the everyday technology of string perturbation theory, where scattering amplitudes are worldsheet correlators of vertex operators.

The unitary minimal series

The unitary minimal models have central charges

$$c_m=1-\frac{6}{m(m+1)}, \qquad m=3,4,5,\ldots.$$

Their primary fields are labeled by pairs of integers

$$(r,s), \qquad 1\leq r\leq m-1, \qquad 1\leq s\leq m,$$

with weights

$$h_{r,s} = \frac{\left((m+1)r-ms\right)^2-1}{4m(m+1)}.$$

The labels satisfy the reflection identification

$$(r,s)\sim(m-r,m+1-s).$$

Therefore the number of distinct primaries is

$$\frac{m(m-1)}{2}.$$

The unitary minimal models form a discrete series with $c_m<1$ and $c_m\to1$ as $m\to\infty$.

The first few examples are famous:

$$\begin{array}{c|c|c} m & c_m & \text{model}\\ \hline 3 & \frac12 & \text{critical Ising}\\[2pt] 4 & \frac{7}{10} & \text{tricritical Ising}\\[2pt] 5 & \frac45 & \text{three-state Potts} \end{array}$$

For $m=3$, the Ising model has

$$c=\frac12, \qquad h_{r,s}=\frac{(4r-3s)^2-1}{48}.$$

The allowed labels are $r=1,2$ and $s=1,2,3$. After the reflection identification, there are three primaries:

$$\mathbf 1: h=0, \qquad \sigma: h=\frac{1}{16}, \qquad \varepsilon: h=\frac12.$$

The Ising Kac table contains six labels but only three independent primary weights after the reflection identification.

The field $\sigma$ is the spin field of the Ising model; $\varepsilon$ is the energy operator. The fact that a lattice statistical model is controlled by a tiny Virasoro representation table is one reason two-dimensional CFT is so powerful.

From null states to null fields

A null state is a statement in the Hilbert space. The state-operator correspondence converts it into a local operator identity inside correlation functions.

For a primary field $\phi(z)$ of weight $h$,

$$|h\rangle\longleftrightarrow \phi(0)|0\rangle.$$

The descendant $L_{-1}|h\rangle$ corresponds to

$$\partial\phi(0).$$

More generally,

$$(L_{-n}\phi)(z) = \frac{1}{2\pi i} \oint_z dw\,(w-z)^{1-n}T(w)\phi(z), \qquad n\geq1.$$

For $n=2$ this means

$$(L_{-2}\phi)(z) = \lim_{w\to z} \left[ T(w)\phi(z) - \frac{h}{(w-z)^2}\phi(z) - \frac{1}{w-z}\partial\phi(z) \right].$$

If the state has a level-two null vector,

$$\left( L_{-2}-\frac{3}{2(2h+1)}L_{-1}^2 \right)|h\rangle=0,$$

then the corresponding field satisfies

$$\left( L_{-2}\phi - \frac{3}{2(2h+1)}\partial^2\phi \right)(z)=0$$

inside every correlator.

A null state becomes a null field under the state-operator correspondence. The Ward identity then turns the null-field relation into a differential equation for correlators.

The conformal Ward identity

Let $\phi_i(z_i,\bar z_i)$ be primary fields of holomorphic weights $h_i$. Their OPE with the stress tensor is

$$T(w)\phi_i(z_i,\bar z_i) \sim \frac{h_i}{(w-z_i)^2}\phi_i(z_i,\bar z_i) + \frac{1}{w-z_i}\partial_{z_i}\phi_i(z_i,\bar z_i).$$

Insert $T(w)$ in a correlation function

$$G(z_i,\bar z_i) = \left\langle\prod_{i=1}^n\phi_i(z_i,\bar z_i)\right\rangle.$$

The OPE gives the Ward identity

$$\left\langle T(w)\prod_i\phi_i(z_i,\bar z_i)\right\rangle = \sum_i \left( \frac{h_i}{(w-z_i)^2} + \frac{1}{w-z_i}\partial_{z_i} \right)G.$$

The Ward identity follows by deforming a stress-tensor contour. Double poles measure conformal weights; simple poles generate derivatives with respect to insertion points.

This identity is the workhorse of two-dimensional CFT. It says that an insertion of $T$ is equivalent to a differential operator acting on the correlator.

For a descendant field $(L_{-n}\phi)(z)$ inserted at $z$, with $n\geq2$, contour deformation gives

$$\left\langle (L_{-n}\phi)(z)\prod_i\phi_i(z_i)\right\rangle = \sum_i \left[ \frac{(n-1)h_i}{(z_i-z)^n} - \frac{1}{(z_i-z)^{n-1}}\partial_{z_i} \right] \left\langle\phi(z)\prod_i\phi_i(z_i)\right\rangle.$$

For $n=2$ this becomes

$$\left\langle (L_{-2}\phi)(z)\prod_i\phi_i(z_i)\right\rangle = \sum_i \left[ \frac{h_i}{(z_i-z)^2} - \frac{1}{z_i-z}\partial_{z_i} \right] \left\langle\phi(z)\prod_i\phi_i(z_i)\right\rangle.$$

Combining this with the level-two null-field relation gives the BPZ equation

$$\left[ \sum_i \left( \frac{h_i}{(z_i-z)^2} - \frac{1}{z_i-z}\partial_{z_i} \right) - \frac{3}{2(2h+1)}\partial_z^2 \right] \left\langle\phi(z)\prod_i\phi_i(z_i)\right\rangle =0.$$

Here $\phi(z)$ is the degenerate primary. Higher-level null vectors give higher-order differential equations.

This is the Belavin-Polyakov-Zamolodchikov mechanism: representation-theoretic null states become differential equations for correlation functions.

Global conformal constraints

Before using null states, global conformal invariance already fixes much of the coordinate dependence of low-point functions.

For primary fields, the two-point function is fixed up to normalization:

$$\left\langle \phi_i(z_1,\bar z_1)\phi_j(z_2,\bar z_2) \right\rangle = \frac{C_{ij}\,\delta_{h_i,h_j}\delta_{\bar h_i,\bar h_j}} {z_{12}^{2h_i}\bar z_{12}^{2\bar h_i}},$$

where

$$z_{ij}=z_i-z_j.$$

The three-point function is also fixed up to constants:

$$\left\langle \phi_1(z_1)\phi_2(z_2)\phi_3(z_3) \right\rangle = \frac{C_{123}} {z_{12}^{h_1+h_2-h_3} z_{13}^{h_1+h_3-h_2} z_{23}^{h_2+h_3-h_1}},$$

times the analogous antiholomorphic factor. The constants $C_{123}$ are the OPE coefficients.

Four-point functions are the first place where nontrivial functions appear. The group $SL(2,\mathbb C)$ can move three points to convenient locations, for instance

$$(z_1,z_2,z_3,z_4) \longmapsto (0,x,1,\infty),$$

leaving the cross-ratio

$$x=\frac{z_{12}z_{34}}{z_{13}z_{24}}, \qquad \bar x=\frac{\bar z_{12}\bar z_{34}}{\bar z_{13}\bar z_{24}}.$$

Four marked points on the sphere leave one complex invariant, the cross-ratio $x$. The remaining dependence of a four-point function is dynamical.

For identical scalar primaries of scaling dimension $\Delta=h+\bar h$, one convenient form is

$$\left\langle \phi(z_1)\phi(z_2)\phi(z_3)\phi(z_4) \right\rangle = \frac{1}{|z_{12}|^{2\Delta}|z_{34}|^{2\Delta}} \mathcal G(x,\bar x).$$

The function $\mathcal G(x,\bar x)$ contains the dynamical information. In a generic CFT it is expanded in conformal blocks. In a minimal model, null-vector equations restrict the possible conformal blocks to solutions of finite-order differential equations.

OPE, conformal blocks, and crossing

The operator product expansion has the schematic form

$$\phi_i(z,\bar z)\phi_j(0,0) = \sum_k C_{ij}^{\phantom{ij}k} z^{h_k-h_i-h_j}\bar z^{\bar h_k-\bar h_i-\bar h_j} \left[\phi_k(0,0)+\text{descendants}\right].$$

In a four-point function, expanding in the $12\to34$ channel gives one decomposition into conformal blocks. Expanding in the $13\to24$ channel gives another. Equality of these decompositions is crossing symmetry.

String amplitudes will use the same logic in a different guise. When vertex operators approach one another on the worldsheet, the OPE reveals which intermediate string states propagate. Factorization of amplitudes is the spacetime interpretation of this local worldsheet statement.

Exercises

Exercise 1. The Ising Kac table

For $m=3$, show that the Kac formula gives the distinct weights $0$, $1/16$, and $1/2$.

Solution

For $m=3$,

$$h_{r,s}=\frac{(4r-3s)^2-1}{48}.$$

The six labels give

$$\begin{array}{c|ccc} & s=1 & s=2 & s=3\\ \hline r=1 & 0 & \frac{1}{16} & \frac12\\[2pt] r=2 & \frac12 & \frac{1}{16} & 0 \end{array}$$

The identification $(r,s)\sim(3-r,4-s)$ leaves three independent primaries:

$$0, \qquad \frac{1}{16}, \qquad \frac12.$$

Exercise 2. Derive the Ward identity

Use the OPE $T(w)\phi_i(z_i)\sim h_i\phi_i/(w-z_i)^2+\partial_i\phi_i/(w-z_i)$ to derive the conformal Ward identity for $\langle T(w)\prod_i\phi_i(z_i)\rangle$.

Solution

The correlator as a function of $w$ is meromorphic away from operator insertions. Near $w=z_i$, the singular terms are fixed by the OPE:

$$T(w)\phi_i(z_i) \sim \frac{h_i}{(w-z_i)^2}\phi_i(z_i) + \frac{1}{w-z_i}\partial_{z_i}\phi_i(z_i).$$

Summing the singular behavior around all insertions gives

$$\left\langle T(w)\prod_i\phi_i(z_i)\right\rangle = \sum_i \left( \frac{h_i}{(w-z_i)^2} + \frac{1}{w-z_i}\partial_{z_i} \right) \left\langle\prod_i\phi_i(z_i)\right\rangle,$$

assuming no additional singularity at infinity.

Exercise 3. The BPZ equation

Starting from the level-two null-field relation

$$\left(L_{-2}\phi-a\partial^2\phi\right)(z)=0, \qquad a=\frac{3}{2(2h+1)},$$

derive the second-order BPZ equation for $\langle\phi(z)\prod_i\phi_i(z_i)\rangle$.

Solution

Insert the null field into the correlator:

$$\left\langle(L_{-2}\phi)(z)\prod_i\phi_i(z_i)\right\rangle - a\partial_z^2 \left\langle\phi(z)\prod_i\phi_i(z_i)\right\rangle =0.$$

Contour deformation gives

$$\left\langle(L_{-2}\phi)(z)\prod_i\phi_i(z_i)\right\rangle = \sum_i \left[ \frac{h_i}{(z_i-z)^2} - \frac{1}{z_i-z}\partial_{z_i} \right] \left\langle\phi(z)\prod_i\phi_i(z_i)\right\rangle.$$

Substituting this expression and $a=3/[2(2h+1)]$ gives the BPZ equation displayed above.

Exercise 4. Invariance of the cross-ratio

Show that

$$x=\frac{z_{12}z_{34}}{z_{13}z_{24}}$$

is invariant under the Möbius transformation $z\mapsto f(z)=(az+b)/(cz+d)$.

Solution

For $f(z)=(az+b)/(cz+d)$,

$$f(z_i)-f(z_j) = \frac{(ad-bc)(z_i-z_j)}{(cz_i+d)(cz_j+d)}.$$

Substituting this into

$$\frac{f(z_1)-f(z_2)}{f(z_1)-f(z_3)} \frac{f(z_3)-f(z_4)}{f(z_2)-f(z_4)}$$

shows that every factor of $ad-bc$ and every factor of $cz_i+d$ cancels. Hence $x$ is invariant.

Exercise 5. Three-point exponents

Use scaling and translation invariance to show that the holomorphic three-point function has the form

$$\langle\phi_1(z_1)\phi_2(z_2)\phi_3(z_3)\rangle = \frac{C_{123}} {z_{12}^{a}z_{13}^{b}z_{23}^{c}},$$

with

$$a=h_1+h_2-h_3, \qquad b=h_1+h_3-h_2, \qquad c=h_2+h_3-h_1.$$

Solution

Translation invariance implies dependence only on differences $z_{ij}$. Under a common scale transformation $z_i\mapsto\lambda z_i$, a primary correlator scales as

$$\lambda^{-(h_1+h_2+h_3)}.$$

The ansatz scales as $\lambda^{-(a+b+c)}$, so $a+b+c=h_1+h_2+h_3$.

Now demand the correct scaling when one point is moved while the others are fixed, equivalently impose covariance under the global conformal generators $L_0$ and $L_1$. The resulting linear equations are

$$a+b=2h_1, \qquad a+c=2h_2, \qquad b+c=2h_3.$$

Solving them gives

$$a=h_1+h_2-h_3, \quad b=h_1+h_3-h_2, \quad c=h_2+h_3-h_1.$$