Conformal Primary Operators and Descendants

Hello! Welcome to the fourth lesson of our course on Conformal Field Theory.

In our last lesson, we established a powerful and elegant framework for understanding the conformal group. By using the embedding space formalism, we showed that the conformal group in $d$ dimensions is isomorphic to $S O (d + 1, 1)$ and that its Lie algebra, $so (d + 1, 1)$ , contains the Poincaré algebra as a subalgebra. We now have the complete set of generators for the conformal algebra: translations ( $P_{μ}$ ), Lorentz transformations ( $M_{μν}$ ), dilatations ( $D$ ), and special conformal transformations ( $K_{μ}$ ).

Today, we will use this algebraic structure to classify the fundamental objects in a Conformal Field Theory: the operators. Just as your work with the Poincaré group involved classifying particles and fields by how they transform, we will classify CFT operators based on their transformation properties under the full conformal group. This will lead us to define primary operators, the cornerstone of any CFT, and their associated descendants.

Defining Primary Operators

In any quantum field theory, operators (or fields) are the basic observables and building blocks. In a CFT, we are interested in operators that have simple, well-defined properties under conformal transformations. The "simplest" possible objects are called primary operators.

Intuitively, a primary operator is one that transforms covariantly under a conformal transformation $x \to x^{'}$ by just picking up a local scaling factor. For a scalar operator $ϕ (x)$ , this is:
$ϕ (x) \to ϕ^{'} (x^{'}) = Ω (x)^{- Δ} ϕ (x)$
Here, $Ω (x)$ is the local scale factor of the transformation at point $x$ , and $Δ$ is a number called the scaling dimension of the operator. For operators with intrinsic spin (i.e., that transform non-trivially under rotations), this is slightly modified to include a rotation.

EPFL Lectures on Conformal Field Theory in D ⩾ 3 ...

The following text from Slava Rychkov's 'EPFL Lectures on Conformal Field Theory' introduces this concept clearly. It defines primary operators based on this simple transformation rule and contrasts them with descendants.

Please read section 1.3.4, 'Transformation rule for operators'. Focus on the definitions of primary operators (Eq. 1.58) and operators with spin (Eq. 1.59). Note the introduction of descendants as operators (like derivatives of primaries) that have a more complex, inhomogeneous transformation rule.

This definition is intuitive, but for practical calculations and a deeper structural understanding, it's more powerful to work with the Lie algebra generators.

The Algebraic Definition of Primaries

Let's translate the finite transformation law into the language of infinitesimal transformations, expressed as commutation relations between the conformal generators and an operator $O (x)$ . From your QFT background, you'll recognize that the action of a generator $G$ on a field $O$ is given by the commutator $[G, O (x)]$ .

The following reading gives the complete set of these commutation relations and then reveals the beautifully simple and powerful algebraic definition of a primary operator.

EPFL Lectures on Conformal Field Theory in D ⩾ 3 ...

This section from the same lecture notes by Rychkov provides the infinitesimal version of the operator transformation rules. It is the key to a rigorous definition.

Please read the subsection 'Algebra action on quantum operators' (pages 36-37). Pay close attention to: The list of commutators in Eq. (3.17). The crucial simplification that occurs when these commutators are evaluated at the origin, x=0. The statement in Eq. (3.18) and the subsequent text, which gives the defining property of a primary operator.

As you've just read, while the action of most generators on an operator at a generic point $x$ is quite involved, the structure simplifies immensely at the origin. An operator $O (0)$ is defined as primary if it satisfies:

Eigenstate of Dilatations: It has a definite scaling dimension $Δ$ .
$[D, O (0)] = - i Δ O (0)$
Transforms in a representation of the rotation group: It has a definite spin.
$[M_{μν}, O (0)] = - i Σ_{μν} O (0)$
where $Σ_{μν}$ are the generators of the Lorentz group in the appropriate representation (for a scalar, $Σ_{μν} = 0$ ).
Annihilated by Special Conformal Transformations:
$[K_{μ}, O (0)] = 0$

This last condition is the most important one. It singles out the primaries from all other operators. In the language of representation theory, primary operators are the "highest-weight" vectors of the conformal algebra, where $K_{μ}$ act as lowering operators.

Descendant Operators

If primary operators are defined by being annihilated by $K_{μ}$ at the origin, what about operators that are not? These are the descendants.

A descendant operator is, in essence, a derivative of a primary operator. We can generate the entire family of descendants by acting on a primary operator with the momentum operator $P_{μ}$ , which acts as a raising operator.

Recall the commutation relation $[D, P_{μ}] = - i P_{μ}$ . If we have a primary operator $O$ with dimension $Δ$ , we can create a new operator, let's call it $O^{'}$ , by acting with $P_{μ}$ . The state corresponding to $O^{'}$ is $P_{μ} ∣ O ⟩$ . Let's see its dimension:

D (P_{μ} ∣ O ⟩) = ([D, P_{μ}] + P_{μ} D) ∣ O ⟩ = (- i P_{μ} + P_{μ} (i Δ)) ∣ O ⟩ = i (Δ + 1) (P_{μ} ∣ O ⟩)

So, acting with $P_{μ}$ (which corresponds to taking a derivative, since $[P_{μ}, O (x)] = - i \partial_{μ} O (x)$ ) increases the scaling dimension by 1.

The set containing one primary operator and all of its descendants (created by applying $P_{μ_{1}}, P_{μ_{2}}, \dots$ ) forms a complete, irreducible representation of the conformal algebra, known as a conformal multiplet or conformal family.

Test your understanding!

Let $O (x)$ be a scalar primary operator with scaling dimension $Δ$ . Show that its first descendant, $O^{'} (x) = \partial_{ν} O (x)$ , is not a primary operator. You can do this by showing that $[K_{μ}, O^{'} (0)] \neq = 0$ .

Hint: Use the Jacobi identity $[A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0$ with $A = K_{μ}$ , $B = P_{ν}$ , and $C = O (0)$ . You will need the commutator $[K_{μ}, P_{ν}] = 2 i (g_{μν} D - M_{μν})$ .

Show answer

We want to compute $[K_{μ}, \partial_{ν} O (0)]$ . First, note that $\partial_{ν} O (0)$ is proportional to $[P_{ν}, O (0)]$ . Specifically, $[P_{ν}, O (x)] = - i \partial_{ν} O (x)$ , so at $x = 0$ , $\partial_{ν} O (0) = i [P_{ν}, O (0)]$ .
So we need to compute $[K_{μ}, i [P_{ν}, O (0)]]$ .

Let's use the Jacobi identity:
$[K_{μ}, [P_{ν}, O (0)]] + [P_{ν}, [O (0), K_{μ}]] + [O (0), [K_{μ}, P_{ν}]] = 0$

Let's analyze each term:

$[K_{μ}, [P_{ν}, O (0)]]$ : This is the term we want to find.
$[P_{ν}, [O (0), K_{μ}]] = - [P_{ν}, [K_{μ}, O (0)]]$ . Since $O (0)$ is primary, $[K_{μ}, O (0)] = 0$ . So this entire term is zero.
$[O (0), [K_{μ}, P_{ν}]] = [O (0), 2 i (g_{μν} D - M_{μν})] = - 2 i [g_{μν} D - M_{μν}, O (0)]$
Using the fact that $O (0)$ is a primary scalar ( $M_{μν}$ acts as 0) with dimension $Δ$ , this becomes:
$- 2 i (g_{μν} [D, O (0)] - [M_{μν}, O (0)]) = - 2 i (g_{μν} (- i Δ O (0)) - 0) = - 2 g_{μν} Δ O (0)$

Plugging back into the Jacobi identity:
$[K_{μ}, [P_{ν}, O (0)]] + 0 - 2 g_{μν} Δ O (0) = 0$
$[K_{μ}, [P_{ν}, O (0)]] = 2 g_{μν} Δ O (0)$

Finally, substituting back $\partial_{ν} O (0) = i [P_{ν}, O (0)]$ , we get:
$[K_{μ}, \partial_{ν} O (0)] = i [K_{μ}, [P_{ν}, O (0)]] = 2 i g_{μν} Δ O (0)$
Since this is non-zero (for $Δ \neq = 0$ ), $\partial_{ν} O (0)$ is not annihilated by $K_{μ}$ and is therefore a descendant, not a primary.

The 2D Perspective: OPEs and Primary Fields

In 2D CFT, which is central to string theory, it is often more convenient to work in complex coordinates $z = x^{1} + i x^{2}$ and $z ˉ = x^{1} - i x^{2}$ . The conformal algebra splits into two independent copies (the Virasoro algebra), one acting on $z$ (holomorphic) and one on $z ˉ$ (anti-holomorphic).

In this context, the transformation properties of operators are elegantly encoded in the Operator Product Expansion (OPE). The OPE describes the behavior of a product of two operators as their separation goes to zero. A primary operator is then defined by the form of its OPE with the stress-energy tensor $T (z)$ , which is the generator of conformal transformations.

4. Introducing Conformal Field Theory

David Tong's lecture notes on String Theory provide an excellent introduction to this 2D perspective. The following reading defines primary operators via their OPE with the stress tensor.

Please read section 4.2.3, 'Primary Operators' (pages 73-75). Focus on: The definition of an operator's 'weight' (h, ˜h), which are the 2D analogs of scaling dimension and spin. The OPE of the stress tensor T(z) with a generic operator of weight (h, ˜h). The crucial definition: a primary operator is one where this OPE has no singularities worse than (z-w)⁻². The resulting simple transformation law for a primary field under a finite conformal map z → ˜z(z) (Eq. 4.18).

This OPE-based definition is equivalent to the one using $K_{μ}$ . The absence of higher-order poles like $(z - w)^{- 3}$ in the $T (z) O (w)$ OPE is the 2D manifestation of the operator being annihilated by the "lowering" generators of the conformal algebra. Operators whose OPE with $T (z)$ do contain higher-order poles are descendants. This perspective will become indispensable when we begin computing correlation functions.

Conclusion

Today we have defined the elementary particles of the CFT world: the primary operators and their descendants. This classification scheme, based on transformation properties under the conformal group, is fundamental to the entire subject.

Key Takeaways:

Primary operators are the fundamental operators in a CFT. They transform covariantly under conformal transformations, picking up a simple scaling factor.
In the language of the conformal algebra, a primary operator placed at the origin is an eigenstate of dilatations ( $D$ ) and rotations ( $M_{μν}$ ) and is annihilated by the special conformal generators ( $K_{μ}$ ).
Descendant operators are created by taking derivatives of primary operators (i.e., by acting with the momentum generator $P_{μ}$ ). They have more complicated, inhomogeneous transformation laws.
A primary operator and its tower of descendants form a conformal family, which is an irreducible representation of the conformal algebra.
In 2D CFT, these properties are encoded in the Operator Product Expansion (OPE) with the stress-energy tensor $T (z)$ .

We have defined operators as abstract entities transforming in a certain way. But what is their physical meaning? In our next lesson, we will introduce radial quantization and uncover one of the most profound and beautiful ideas in CFT: the state-operator correspondence. This principle establishes a one-to-one mapping between the operators we have just defined and the physical states of the theory.

Recommended resources

Defining Primary Operators

The Algebraic Definition of Primaries

Descendant Operators

The 2D Perspective: OPEs and Primary Fields

Conclusion