Deriving Correlation Function Differential Equations from Null State Conditions

From Null Vectors to Operator Equations

Let's formalize the connection between the algebraic structure we've studied and the fields of our theory.

Recall that a null vector $|\chi ⟩$ is a descendant state that is also primary (i.e., annihilated by all $L_n$ with $n>0$). In a minimal model, the Hilbert space is constructed from irreducible representations of the Virasoro algebra. To get an irreducible representation from a reducible Verma module $V(c,h)$ that contains a null vector $|\chi ⟩$, we "quotient out" the submodule generated by $|\chi ⟩$. Operationally, this is equivalent to setting the null vector to zero within the representation:

This is the null state condition.

The state-operator correspondence provides a direct map from states to fields. If $|h⟩$ is a primary state corresponding to a primary field $V_h(z)$, then the null state $|\chi ⟩$, which is a descendant of $|h⟩$, corresponds to a descendant field. For example, if the null vector appears at level 2, it can be written as:

The null state condition $|\chi ⟩=0$ translates, via the state-operator correspondence, into an operator equation that the primary field $V_h(z)$ must satisfy:

A primary field that satisfies such a condition is called a degenerate field.

The Belavin-Polyakov-Zamolodchikov (BPZ) Equations

This operator equation becomes a powerful computational tool when we insert the degenerate field into a correlation function. The equation must hold inside the correlator:

To make this useful, we need to know how the Virasoro generators act on fields within a correlator. We know from the state-operator map that $L_{-1}$ acts as a derivative, $L_{-1}V(z)={\partial }_{z}V(z)$. The action of $L_{-2}$ can be derived from the Operator Product Expansion of the stress-energy tensor with a primary field. The result is a local Ward identity.

The following reading explains how combining the null state condition with these Ward identities leads directly to a differential equation for the correlation function.

As you have read, for a degenerate field $V_{⟨r,s⟩}(z_1)$ that has a null vector at level $N=rs$, any $M$-point correlation function containing it will obey a partial differential equation of order $rs$.

For the level-2 degenerate field $V_{⟨2,1⟩}(z_1)$, which has dimension ${\Delta }_{⟨2,1⟩}$, the null state condition is $(L_{-2}+b^2{L}_{-1}^2)V_{⟨2,1⟩}(z_1)=0$. Inserting this into an $M$-point function gives the second-order BPZ equation:

Here, $b$ is the parameter related to the central charge by $c=1-6(b-b^{-1}{)}^2=1+6Q^2$ with $Q=b+1/b$, which you've encountered in the readings. This equation is a direct, computable consequence of the underlying Virasoro algebra structure.

Application: Solving the Ising Model Four-Point Function

Let's apply this powerful machinery to the concrete physical example we discussed in our last lesson: the 2D critical Ising model.

• This system is described by the $m=3$ unitary minimal model with $c=1/2$.
• It has three primary fields: the identity $I$ ($h=0$), the energy density $ϵ$ ($h=1/2$), and the spin field $\sigma$ ($h=1/16$).
• The dimension of the spin field, $h=1/16$, corresponds to $h_{1,2}$ (and $h_{2,2}$) in the Kac table. This means $\sigma$ is a degenerate field with a null vector at level $N=1\times 2=2$.

This implies that any correlation function involving the spin field $\sigma$ must satisfy a second-order BPZ equation. Let's see how this allows us to compute the four-point function of spin fields, $⟨\sigma (z_1)\sigma (z_2)\sigma (z_3)\sigma (z_4)⟩$. Global conformal invariance fixes its form up to a function $G(\eta )$ of the cross-ratio $\eta =\frac{(z_1-z_2)(z_3-z_4)}{(z_1-z_3)(z_2-z_4)}$. The BPZ equation will become a solvable ordinary differential equation for $G(\eta )$.

This example is a beautiful illustration of the power of the conformal bootstrap philosophy. Let's summarize the key steps of the calculation you just read:

1. Identify the Null Vector: The spin field $\sigma$ corresponds to $h=h_{1,2}=1/16$, which has a null vector at level 2. In the notation of the text, this is $|{\chi }_{1,2}⟩=\left(L_{-2}-\frac{3}{2(2h_{1,2}+1)}{L}_{-1}^2\right)|h_{1,2}⟩$. With $h_{1,2}=1/16$, the coefficient is $4/3$. The null field condition is $\left(L_{-2}-\frac{4}{3}{L}_{-1}^2\right)\sigma (z)=0$.

2. Write the BPZ Equation: Inserting this into the four-point function $⟨\sigma (z)\sigma (1)\sigma (\eta )\sigma (0)⟩$ yields a second-order PDE. Because of global conformal symmetry, this PDE reduces to a second-order ODE for the function $G(\eta )$ of the cross-ratio.

3. Solve the ODE: The resulting equation is a hypergeometric differential equation. Its solutions are $f_{1/2}(\eta )=(1\pm \sqrt{1-\eta }{)}^{1/2}$.

4. Impose Physical Constraints: The full four-point function must be single-valued. This constraint fixes the linear combination of solutions, leading to the form:

   $$G_{\sigma \sigma \sigma \sigma }(\eta ,\bar{\eta })=\frac{C}{|\eta (1-\eta ){|}^{1/4}}\left(|1+\sqrt{1-\eta }|+|1-\sqrt{1-\eta }|\right)$$

5. Compare with OPE: By taking the limit $\eta \to 0$, we can compare this solution with the operator product expansion $\sigma (\eta )\sigma (0)\sim \frac{1}{|\eta {|}^{1/8}}(I+|\eta |C_{\sigma \sigma ϵ}ϵ(0)+\dots )$. This comparison uniquely fixes the unknown structure constant to be $C_{\sigma \sigma ϵ}=1/2$.

We have, remarkably, determined a fundamental parameter of the theory without ever referring to a Lagrangian. The constraints of symmetry were sufficient.

Conclusion

Today we connected the abstract algebraic structure of the Virasoro algebra to concrete, computable physics. The existence of null states, which define minimal models, is not just a label but a powerful dynamical principle.

Key Takeaways:

• In the irreducible representations used in minimal models, null vectors are set to zero.
• Via the state-operator correspondence, this null state condition becomes an operator equation that a degenerate primary field must satisfy.
• When placed inside a correlator, this operator equation becomes a linear partial differential equation for the correlation function, known as a BPZ equation.
• These equations constrain correlation functions so powerfully that they can often be used to solve for them completely, as demonstrated with the four-point function of the critical Ising model.
• This allows for the direct computation of fundamental CFT data, such as OPE coefficients (structure constants).

We have now seen how to classify minimal models and how to use their special properties to calculate correlation functions. In the next module, we will broaden our focus to develop more general computational tools applicable to a wider range of 2D CFTs. We will begin by deriving one of the most important OPEs in all of conformal field theory: that of the stress-energy tensor with itself.