Correlation Functions in Minimal Models

Hello! Let's dive into our next lesson.

In our previous explorations of 2D CFT, we've established the foundational role of the Virasoro algebra and its representations, the Verma modules. Most recently, we performed a crucial calculation—the Operator Product Expansion (OPE) of the stress-energy tensor with itself—which revealed the central charge $c$ and confirmed that $T (z)$ is not a primary field.

Today, we will uncover a profound connection between the abstract representation theory of the Virasoro algebra and the practical computation of physical observables. We will investigate what happens when Verma modules are reducible, meaning they contain special states known as null states. The physical requirement that these unphysical states decouple from the theory imposes powerful constraints, leading to differential equations for correlation functions. This technique, known as null state decoupling, allows us to solve for correlation functions and determine the fundamental structure constants in a special class of theories called minimal models.

Your objective for this lesson is to learn how to calculate correlation functions in minimal models using null state decoupling. This is a cornerstone computational skill in 2D CFT, bridging the gap between abstract symmetry and concrete prediction.

1. Null Vectors and Degenerate Fields

Recall that a Verma module $V (c, h)$ is generated by acting with the raising operators $L_{- n}$ ( $n > 0$ ) on a highest-weight state $∣ h ⟩$ . While we often assume these modules are irreducible, for certain values of $c$ and $h$ , they are not.

A Verma module is reducible if it contains a null vector (or singular vector), which is a state $∣ χ ⟩$ that is simultaneously:

A descendant of $∣ h ⟩$ , meaning $∣ χ ⟩ = L_{- n_{k}} \dots L_{- n_{1}} ∣ h ⟩$ for some sequence of operators.
A primary state itself, meaning it is annihilated by all lowering operators: $L_{n} ∣ χ ⟩ = 0$ for all $n > 0$ .

A physical CFT is built upon a spectrum of irreducible representations. To achieve this when a Verma module is reducible, we quotient out the submodule generated by the null vector. In practice, this means we impose the condition that the null vector and all of its descendants are zero:

∣ χ ⟩ = 0

This is the principle of "decoupling".

The state-operator correspondence promotes this algebraic condition into a powerful statement about fields. A primary field $ϕ_{h} (z)$ whose corresponding state $∣ h ⟩$ generates a reducible Verma module is called a degenerate primary field. The null state condition $∣ χ ⟩ = 0$ implies that the corresponding null field $χ (z)$ must vanish inside any correlation function:

⟨ \dots χ (z) \dots ⟩ = 0

To make this concrete, let's start with a foundational text that introduces null vectors and their corresponding fields.

Introduction to two-dimensional conformal field theory

The following sections from the lecture notes 'Introduction to two-dimensional conformal field theory' by Sylvain Ribault define null vectors and degenerate representations, and then connect them to degenerate fields via the state-field correspondence.

Please read Section 1.3, 'Null vectors and degenerate representations', and the beginning of Section 2.1 up to equation (2.3). Focus on understanding: The definition of a null vector. How the existence of a null vector makes a Verma module reducible. The definition of a degenerate representation as a quotient, where the null vector is set to zero (Eq. 1.18). How this condition translates to an operator equation for the corresponding degenerate field (Eq. 2.3).

The simplest non-trivial example of a null vector occurs at level 2 in a Verma module. A generic level-2 descendant has the form $∣ χ ⟩ = (a L_{- 2} + b L_{- 1}^{2}) ∣ h ⟩$ . Requiring $L_{1} ∣ χ ⟩ = 0$ and $L_{2} ∣ χ ⟩ = 0$ imposes constraints on $a, b, c,$ and $h$ . This leads to a specific relationship between $c$ and $h$ , which determines when the module is reducible at level 2.

2. From Null Fields to Differential Equations

The operator condition $χ (z) = 0$ is the key to our new computational method. Since $χ (z)$ is built from operators like $L_{- n} ϕ (z)$ , we need to know how these act within a correlation function.

An operator $L_{- n}$ acting on a field $ϕ (w)$ can be expressed as a differential operator acting on the entire correlation function.

The Structure Constants of the Minimal Models

This section from Mark Gerraty's thesis, 'The Structure Constants of the Minimal Models', provides the explicit formula for the action of L-n as a differential operator inside a correlator. This is the crucial technical step to turn an operator equation into a differential equation.

Please read Section 2.5.3, 'Correlators of Secondary Fields'. Pay close attention to the derivation leading to equation (2.56). This equation shows that calculating a correlator with a descendant field is equivalent to applying a differential operator, defined in (2.55), to the correlator of its primary.

As you've just read, the action of $L_{- n}$ on a field $ϕ_{1} (z_{1})$ inside an $N$ -point correlator is given by:

⟨(L_{- n} ϕ_{1}) (z_{1}) ϕ_{2} (z_{2}) \dots ϕ_{N} (z_{N})⟩ = L_{- n}^{(1)} ⟨ ϕ_{1} (z_{1}) ϕ_{2} (z_{2}) \dots ϕ_{N} (z_{N})⟩

where $L_{- n}^{(1)}$ is the differential operator:

L_{- n}^{(1)} = j = 2 \sum N (\frac{( 1 - n ) h _{j}}{( z _{j} - z _{1} ) ^{n}} + \frac{1}{( z _{j} - z _{1} ) ^{n - 1}} \frac{\partial}{\partial z _{j}})

Now we combine these ideas. Consider a 4-point function containing a degenerate field $ϕ_{h_{2, 1}} (z_{1})$ that has a null vector at level 2, of the form $∣ χ_{2, 1} ⟩ = (L_{- 2} + a L_{- 1}^{2}) ∣ h_{2, 1} ⟩$ . The null state decoupling condition is:

⟨(L_{- 2} + a L_{- 1}^{2}) ϕ_{h_{2, 1}} (z_{1}) ϕ_{2} (z_{2}) ϕ_{3} (z_{3}) ϕ_{4} (z_{4})⟩ = 0

Applying the differential operator rule, this becomes a second-order partial differential equation for the 4-point function. These are the celebrated Belavin-Polyakov-Zamolodchikov (BPZ) equations.

For a 4-point function, which depends only on the cross-ratio $z$ , the BPZ equation reduces to an ordinary differential equation.

Introduction to two-dimensional conformal field theory

The following resources detail the derivation of the BPZ equation for a 4-point function and its application to calculating structure constants in minimal models.

First, please read Section 2.3, 'Belavin–Polyakov–Zamolodchikov equations', in the notes by Ribault. This section shows how the null vector condition leads directly to a differential equation for the N-point function (Eq. 2.28).

The Structure Constants of the Minimal Models

Now, for the main event. Gerraty's thesis provides a detailed, step-by-step calculation for a 4-point function, culminating in the explicit determination of structure constants for two key minimal models.

Please read Section 3.3, 'Belavin-Polyakov-Zamolodchikov Equations', and Section 4.2, 'Structure Constants', up to the end of 4.2.2. Your reading should focus on the following logic: Derivation (Sec 3.3): Follow how the operator equation (3.27) is transformed into a second-order ODE for a function H(z) related to the 4-point correlator (Eq. 3.32). You don't need to check every algebraic step, but understand the flow. Application (Sec 4.2.1, Yang-Lee): This is the core calculation. Track how the 4-point function is expanded in two different ways: a) Using the OPE and fusion rules (Eq. 4.8). b) Using the solutions to the BPZ equation, which are conformal blocks (Eq. 4.10). Solution: See how comparing these two expansions and enforcing single-valuedness (crossing symmetry) leads to a constraint that determines the structure constant C^2_{\Phi\Phi\Phi} (Eq. 4.14). Follow the same logic for the Ising Model in 4.2.2.

3. The Calculation: A Summary

Let's synthesize the procedure you just studied. The calculation of the structure constant $C_{ΦΦΦ}$ in the $M (2, 5)$ minimal model (Yang-Lee singularity) is a perfect template for this method.

Step 1: OPE Expansion
We consider the 4-point function $⟨ Φ (\infty) Φ (1) Φ (z) Φ (0)⟩$ . We perform an OPE on the two inner fields, $Φ (z) Φ (0)$ . The fusion rules for this model, $Φ \times Φ = I + Φ$ , tell us which primary fields can appear in the expansion.

⟨ ΦΦΦΦ ⟩ \approx ⟨ ΦΦ (I + C_{ΦΦΦ} Φ)⟩ = ⟨ ΦΦ ⟩ + C_{ΦΦΦ} ⟨ ΦΦΦ ⟩

Working out the coordinate dependence yields an expansion in powers of $z$ and $z ˉ$ , with coefficients related to the unknown structure constant $C_{ΦΦΦ}$ . For the $M (2, 5)$ model, this gives (schematically):

F^{(4)} (z, z ˉ) = ∣ z ∣^{4/5} + C_{ΦΦΦ}^{2} ∣ z ∣^{2/5} + \dots

Step 2: BPZ Equation and Conformal Blocks
The field $Φ$ is the degenerate field $ϕ_{2, 1}$ , so the 4-point function must satisfy a 2nd-order BPZ equation. The solutions to this ODE are hypergeometric functions, which form the basis of conformal blocks. A general solution is a linear combination of these blocks. For the $M (2, 5)$ case, the 4-point function is:

F^{(4)} (z, z ˉ) = C_{1, 1} F_{1} (z) F_{1} (z ˉ) + C_{2, 2} F_{2} (z) F_{2} (z ˉ)

where we've enforced that the physical correlator must be single-valued. The $F_{i}$ are the conformal blocks, which are known functions. Expanding for small $z$ gives:

F^{(4)} (z, z ˉ) = C_{1, 1} ∣ z ∣^{4/5} + C_{2, 2} ∣ z ∣^{2/5} + \dots

Step 3: Matching and Solving
By comparing the coefficients of the two expansions, we immediately identify:

C_{1, 1} = 1 and C_{2, 2} = C_{ΦΦΦ}^{2}

This fixes the correlator in terms of the unknown structure constant.

Step 4: Imposing Crossing Symmetry
The OPE is associative. This means we should get the same 4-point function if we expand $Φ (\infty) Φ (1)$ and $Φ (z) Φ (0)$ (s-channel) as we do if we expand $Φ (\infty) Φ (z)$ and $Φ (1) Φ (0)$ (t-channel). In terms of the cross-ratio, this means the function must have specific transformation properties as $z \to 1 - z$ . By demanding that the solution expressed in terms of conformal blocks respects this symmetry, we derive an algebraic equation for the coefficients, which solves for $C_{ΦΦΦ}^{2}$ .

Test your understanding!

Why does the BPZ equation for a 4-point function containing a level-2 degenerate field $ϕ_{2, 1}$ reduce the infinite sum over conformal blocks to a finite, two-term sum?

Show answer

The BPZ equation is a second-order ordinary differential equation for the function of the cross-ratio that determines the 4-point function. A second-order ODE has a two-dimensional space of solutions. The conformal blocks are, by definition, a basis of solutions to the Ward identities. The BPZ equation provides an additional constraint. Therefore, the physical correlation function must be built from a linear combination of just two conformal blocks that span the solution space of this specific ODE.

Conclusion

In this lesson, we have unlocked one of the most powerful computational tools in 2D CFT. By starting with the abstract concept of reducible Virasoro representations, we derived a concrete method for calculating correlation functions.

Key Takeaways:

For specific values of $c$ and $h$ , Verma modules contain null states, making them reducible.
Physical theories are built from irreducible representations, so we enforce null state decoupling: $∣ χ ⟩ = 0$ .
Via the state-operator correspondence, this becomes an operator equation, $χ (z) = 0$ , inside any correlation function.
This operator equation translates into a partial differential equation for the correlation function, known as the BPZ equation.
In minimal models, which have a finite number of (degenerate) primary fields, BPZ equations allow us to solve for correlation functions and determine the OPE structure constants.

You have now seen the complete chain of logic, from abstract algebra to a numerical prediction for a structure constant. In our next lesson, we will formalize the structure of minimal models by introducing the Kac table, which provides a systematic way to classify the operator content and central charges of these special theories.

Recommended resources

1. Null Vectors and Degenerate Fields

2. From Null Fields to Differential Equations

3. The Calculation: A Summary

Conclusion