Conformal Blocks and Operator Decomposition

Hello! Welcome back to our study of Conformal Field Theory.

In our last lesson, we introduced the Operator Product Expansion (OPE) and saw that it's a convergent series in CFT, allowing us to replace a product of operators at nearby points with a sum over a complete basis of local operators. This transformed the seemingly arbitrary function of cross-ratios in a four-point function, $G (u, v)$ , into a problem about operator algebra.

Today, we will build directly on that foundation. Our goal is to take the raw, infinite sum from the OPE and organize it into a more physically meaningful and computationally tractable form. We will decompose the four-point function into fundamental, universal building blocks known as conformal blocks. Each block will neatly package the contribution of a single primary operator and all its descendants, providing a structured way to understand the dynamics encoded in correlation functions.

From OPE to Conformal Blocks

Let's begin with a four-point function of identical scalar primary operators for simplicity:

⟨ ϕ (x_{1}) ϕ (x_{2}) ϕ (x_{3}) ϕ (x_{4})⟩

As we learned, we can apply the OPE to the pair of operators at $x_{1}$ and $x_{2}$ , assuming $∣ x_{12} ∣$ is small compared to other distances:

ϕ (x_{1}) ϕ (x_{2}) = O \sum C_{ϕϕ O} F_{O} (x_{1} - x_{2}, \partial_{2}) O (x_{2})

Here, the sum is over all primary operators $O$ in the theory. The function $F_{O}$ contains the primary $O$ and all its descendants, with a form completely fixed by conformal symmetry.

Inserting this into the four-point function gives:

⟨ ϕ (x_{1}) ϕ (x_{2}) ϕ (x_{3}) ϕ (x_{4})⟩ = O \sum C_{ϕϕ O} ⟨(F_{O} O (x_{2})) ϕ (x_{3}) ϕ (x_{4})⟩

This expression is a sum over contributions from each conformal family (a primary and its descendants). The central idea is to group all terms associated with a single exchanged primary $O$ into one object. This object is the conformal block.

Visually, we can think of this as "fusing" the operators $ϕ (x_{1})$ and $ϕ (x_{2})$ into a sum of intermediate operators $O$ , which then propagate and "split" into $ϕ (x_{3})$ and $ϕ (x_{4})$ .

A schematic representation of the conformal block decomposition. The four-point function is expressed as a sum over intermediate primary operators $O$ . Each term in the sum, known as a conformal partial wave, consists of the OPE coefficients (the vertices) and the conformal block $G_{O}$ (the internal line), which encapsulates the propagation of the entire conformal family of $O$ .

The Conformal Partial Wave Expansion

This decomposition is known as the conformal partial wave expansion. It systematically separates the theory-dependent data from the universal kinematics.

EPFL Lectures on Conformal Field Theory in D ⩾ 3

The following resource provides a clear and explicit discussion of this decomposition for d ≥ 3 dimensions. It formalizes the picture we just drew and defines the conformal block.

Please read section 4.1 "Recap" (pages 49-50) and the first part of section 4.3.2 "Tools for the bootstrap in D ⩾ 3 dimensions" (pages 55-56, up to the discussion of specific point configurations). Pay close attention to equation (4.18), which is the mathematical statement of the decomposition. The key insight is that the entire four-point function is written as a sum over primary operators, where each term involves two OPE coefficients and one universal function: the conformal block.

As you've seen in the reading, the four-point function of four identical scalar primaries $ϕ$ with dimension $Δ_{ϕ}$ can be written as:

⟨ ϕ (x_{1}) ϕ (x_{2}) ϕ (x_{3}) ϕ (x_{4})⟩ = \frac{1}{∣ x _{12} ∣ ^{2 Δ_{ϕ}} ∣ x _{34} ∣ ^{2 Δ_{ϕ}}} O \sum C_{ϕϕ O}^{2} G_{O} (u, v)

Let's break this down:

The prefactor $\frac{1}{∣ x _{12} ∣ ^{2 Δ_{ϕ}} ∣ x _{34} ∣ ^{2 Δ_{ϕ}}}$ is the kinematic part we found two lessons ago, which just accounts for the scaling dimension of the external operators.
The sum runs over all primary operators $O$ in the theory's spectrum.
$C_{ϕϕ O}$ are the OPE coefficients. This is dynamical data, specific to the particular CFT we are studying. They determine the "strength" of the coupling between $ϕ$ and $O$ . We have $C^{2}$ because the fusion $ϕϕ \to O$ happens at both the beginning and the end.
$G_{O} (u, v)$ is the conformal block. This is a universal, purely kinematic function. It depends only on the conformal cross-ratios $u$ and $v$ , the dimensions of the external operators ( $Δ_{ϕ}$ ), and the quantum numbers of the exchanged primary $O$ (its dimension $Δ_{O}$ and spin $ℓ_{O}$ ). It does not depend on the OPE coefficients or any other details of the theory.

The conformal block $G_{O}$ is the sum of contributions from the primary $O$ and all of its descendants.

What is a Conformal Block, Physically?

To gain a more concrete understanding of the conformal block, it's helpful to return to the picture of radial quantization on the cylinder $R \times S^{d - 1}$ .

EPFL Lectures on Conformal Field Theory in D ⩾ 3

The same resource provides an excellent physical interpretation of the conformal block using the cylinder picture. It reveals the block as a sum over states propagating through radial time.

Please read the remainder of section 4.3.2, starting from the discussion of the cylinder geometry (page 57, figure 4.3) to the end of the section on page 58. Focus on equation (4.22) and the surrounding text. This shows how the block is computed as a sum over the states in the conformal family of \mathcal{O}, propagating between two pairs of operators.

As the text explains, by mapping the operator positions to a cylinder, the conformal block for an exchanged operator $O$ with dimension $Δ$ and spin $ℓ$ can be expressed as:

G_{Δ, ℓ} = ∣ n ⟩ \in family of O \sum ⟨ 0∣ ϕ_{1} ϕ_{2} ∣ n ⟩ e^{- E_{n} τ} ⟨ n ∣ ϕ_{3} ϕ_{4} ∣0 ⟩

Here, the sum is over all states $∣ n ⟩$ in the conformal family of $O$ (the primary state $∣ O ⟩$ and all its descendant states). $E_{n}$ is the energy (scaling dimension) of the state $∣ n ⟩$ , and $τ$ is the "time" separation on the cylinder, which is a function of the cross-ratios.

This formula makes the physics clear: the conformal block sums up the contributions of a single primary field and all its descendants as they propagate as intermediate states. Conformal symmetry completely fixes the matrix elements $⟨ ϕϕ ∣ n ⟩$ and the energies $E_{n}$ relative to the primary, which is why the block is a universal function.

The 2D Case: Holomorphic Factorization

In 2D CFT, which is central to your interest in string theory, the structure becomes even richer due to the infinite-dimensional conformal algebra. The conformal blocks factorize into a product of a holomorphic part and an anti-holomorphic part:

G_{Δ, ℓ} (u, v) \to F_{h, h ˉ} (z) F ˉ_{h, h ˉ} (z ˉ)

where $z$ and $z ˉ$ are complex coordinates related to the cross-ratios ( $u = z z ˉ$ , $v = (1 - z) (1 - z ˉ)$ ), and $h, h ˉ$ are the conformal weights of the exchanged operator. The representation of the Virasoro algebra built on a primary state is called a Verma module. The holomorphic conformal block $F (z)$ is therefore the contribution from a single Verma module.

Introduction to two-dimensional conformal field theory

Let's look at the 2D definition. This resource defines the conformal block as the contribution from a Verma module and provides a hands-on exercise to compute its first few terms.

Please read section 3.2, starting from the OPE of primary fields (equation 3.7) and continuing to Definition 3.8, "Conformal block". This section succinctly defines the block in 2D and shows how it appears in the decomposition of the 4-point function.

The definition of the conformal block as the normalized contribution of a Verma module is the precise 2D analogue of the more pictorial sum-over-states on the cylinder.

Test your understanding!

The resource you just read presents an exercise (3.9) to compute the conformal block $F_{Δ}^{(s)} (z)$ up to order $z$ . The result is:

F_{Δ}^{(s)} (z) = z^{Δ - Δ_{1} - Δ_{2}} [1 + \frac{( Δ + Δ _{1} - Δ _{2} ) ( Δ + Δ _{4} - Δ _{3} )}{2Δ} z + O (z^{2})]

Let's analyze this. The leading term $z^{Δ - Δ_{1} - Δ_{2}}$ comes from the exchange of the primary operator $O_{Δ}$ itself. The term linear in $z$ comes from the exchange of descendant operators. Which level of descendants contributes to this $O (z)$ term? (Hint: In 2D, a descendant at level $N$ is created by acting with operators like $L_{- k_{1}} \dots L_{- k_{m}}$ where $\sum k_{i} = N$ . The dimension of such a state is $Δ + N$ .)

Show answer

The power of $z$ in the expansion of the block is related to the dimension of the exchanged state. The full block is a sum of terms of the form $z^{Δ_{n} - Δ_{1} - Δ_{2}}$ , where $Δ_{n}$ is the dimension of an exchanged state.
The primary contributes the term with $Δ_{n} = Δ$ .
A descendant at level $N$ has dimension $Δ_{n} = Δ + N$ . Its contribution to the block would be proportional to $z^{Δ + N - Δ_{1} - Δ_{2}} = z^{Δ - Δ_{1} - Δ_{2}} \cdot z^{N}$ .
Therefore, the term of order $O (z)$ inside the square brackets corresponds to the contribution from descendants at level N=1. The only level-1 descendant of a primary $O_{Δ}$ is $L_{- 1} O_{Δ}$ , which has dimension $Δ + 1$ .

Conclusion

We have taken a crucial step today, moving from the abstract OPE to a concrete decomposition of correlation functions. This framework is one of the pillars of modern theoretical physics, from statistical mechanics to quantum gravity.

Key Takeaways:

Any CFT four-point function can be decomposed into a sum over conformal blocks. This is called the conformal partial wave expansion.
Each conformal block is a universal kinematic function of the cross-ratios, representing the complete contribution of one primary operator and all its descendants being exchanged.
The expansion separates the universal kinematics (the blocks, $G_{O}$ ) from the theory's specific dynamics, which are encoded in two pieces of data: the spectrum of primary operators ${Δ_{O}, ℓ_{O}}$ and the OPE coefficients ${C_{ij k}}$ .
In 2D, the blocks factorize into holomorphic and anti-holomorphic parts, corresponding to the contributions of Verma modules.

We now have a representation of the four-point function as a sum of known functions. But how does this help us determine the unknown CFT data? The key is that the OPE is associative. We can choose to expand $⟨ ϕ_{1} ϕ_{2} ϕ_{3} ϕ_{4} ⟩$ by first fusing $ϕ_{1}$ and $ϕ_{2}$ , or we could just as well have fused $ϕ_{1}$ and $ϕ_{4}$ . Both must give the same answer. In our next lesson, we will see that demanding this equality leads to a powerful set of self-consistency equations—the conformal bootstrap equations—which profoundly constrain the allowed CFT data.

Recommended resources

From OPE to Conformal Blocks

The Conformal Partial Wave Expansion

What is a Conformal Block, Physically?

The 2D Case: Holomorphic Factorization

Conclusion