Vertex Operators and the State-Operator Correspondence

Hello! Welcome back to our final module on the application of CFT to bosonic string theory.

In our last lesson, we reached a pivotal conclusion: the quantum consistency of the bosonic string (specifically, the absence of negative-norm "ghost" states) requires the spacetime it lives in to have a critical dimension of D=26. We also found that the normal-ordering constant must be a=1. This led us to a definite prediction for the string's mass spectrum, starting with a tachyonic ground state.

So far, we have described the string's quantum states abstractly, as elements of a Fock space built by creation operators acting on a vacuum. But to describe interactions—string scattering—we need a way to represent these asymptotic "in" and "out" states as local objects within our 2D worldsheet CFT. Today, we will bridge this gap.

Our goal is to construct vertex operators for string states using the state-operator correspondence. This powerful CFT tool provides the dictionary we need, translating the abstract Hilbert space of states into a concrete set of local operators that we can insert onto the worldsheet.

1. The State-Operator Correspondence

At the heart of our construction is a profound property of conformal field theories: the state-operator correspondence (or state-operator map). It establishes a one-to-one relationship between the quantum states of the theory and the local operators that can be defined within it.

The key idea is to use radial quantization. Instead of quantizing on surfaces of constant worldsheet time $τ$ , we quantize on concentric circles of constant radius $r = ∣ z ∣$ on the complex plane. In this picture, the radial coordinate $r$ plays the role of time. The origin, $r = 0$ , corresponds to the infinite past ( $τ \to - \infty$ ), and $r \to \infty$ corresponds to the infinite future ( $τ \to + \infty$ ).

Radial Quantization: Plane to Cylinder Mapping — This image shows the conformal map $z = e^{\tau + i\sigma}$ which transforms the Euclidean plane (left) into an infinite cylinder (right). Concentric circles of constant radius $|z| = e^\tau$ on the plane become circles of constant Euclidean time $\tau$ on the cylinder. An operator inserted at the origin $z=0$ of the plane corresponds to an initial state prepared in the infinite past $\tau \to -\infty$ on the cylinder.

Inserting a local operator $O (0)$ at the origin is equivalent to preparing a specific state $∣ O ⟩$ in the infinite past. The path integral over the disk of radius $R$ , with the operator at its center, defines a wavefunctional on the boundary circle $∣ z ∣ = R$ , which is the state $∣ O ⟩$ .

Lectures on String Theory - CLASSE (Cornell)

The following section from the Cornell string theory notes provides a more formal and intuitive explanation of this correspondence, using the path integral formalism you're familiar with.

Please read section 5.6, 'The state-operator map' (pages 39-41). Focus on: The relationship between operators acting on states (like the string mode oscillators \alpha_{-n}^\mu) and local fields in the CFT (like derivatives of X^\mu(0)). The path integral argument: how integrating up to a time slice (or a radial slice) defines a state (a wavefunctional). The final mapping between specific string states (e.g., (|0,0;k angle)) and their corresponding local operators (e.g., e^{ik\cdot X(0)}).

This correspondence is the linchpin. It tells us that instead of talking about an incoming string state with some momentum and excitation level, we can talk about inserting a corresponding local "vertex operator" onto the worldsheet.

2. Physical Vertex Operators

To calculate a physical amplitude, like a string scattering process, we will eventually need to compute correlation functions of these vertex operators, e.g., $⟨ V_{1} (z_{1}) V_{2} (z_{2}) \dots ⟩$ .

Since the specific location $z$ where we insert an operator is just a choice of coordinates on the worldsheet, a physical quantity should not depend on it. To obtain a coordinate-independent object, we must integrate the operator over the entire worldsheet. The full vertex operator for a state $∣ ψ ⟩$ is therefore:
$V_{ψ} = g_{s} \int d^{2} z V_{ψ} (z, z ˉ)$
where $V_{ψ} (z, z ˉ)$ is the local vertex operator corresponding to the state $∣ ψ ⟩$ , and $g_{s}$ is the string coupling constant.

Now, a crucial constraint arises. The worldsheet theory has conformal symmetry. For the integral $\int d^{2} z V_{ψ}$ to be conformally invariant, the integrand itself, $d^{2} z V_{ψ}$ , must transform as a scalar. Under a scaling $z \to λ z$ , we have $d^{2} z \to ∣ λ ∣^{2} d^{2} z$ . If the operator $V_{ψ}$ has conformal weights $(h, h ˉ)$ , it transforms as $V_{ψ} \to λ^{- h} λ ˉ^{- h ˉ} V_{ψ}$ . For the whole expression to be invariant, the scaling factors must cancel. This requires:
$h = 1 and h ˉ = 1$
Therefore, a local operator corresponds to a physical string state if and only if it is a primary field of conformal weight (1, 1).

This simple condition is incredibly powerful. It will allow us to rediscover the mass-shell conditions for physical string states directly from the worldsheet CFT.

3. Constructing Vertex Operators for String States

Let's now use this machinery to build the vertex operators for the states we found in the previous lesson.

The Tachyon Vertex Operator

The simplest string state is the ground state with momentum $k^{μ}$ , which we denote $∣0; k ⟩$ . Based on the state-operator map, the natural candidate for its local vertex operator is:
$V_{k} (z, z ˉ) =: e^{ik \cdot X (z, z ˉ)} :$
The colons denote normal ordering, which is standard for defining composite operators in QFT. This operator is a primary field. Let's find its conformal weight.

Introduction to String Theory, Lecture Notes

The following lecture notes by Prof. Beisert provide a very clear and direct calculation of the properties of this operator and its connection to the tachyon.

Please read the first part of section 8.1 'Vertex Operators' (up to the 'Excited Strings' subsection on page 8.2). Focus on: Why :e^{ik\cdot X}:, as a momentum eigenstate, is the correct operator for the string vacuum. The OPE of this operator with the stress-energy tensor T(z), which reveals its conformal weight to be h = ar{h} = rac{\alpha' k^2}{4}. (Note that Beisert uses \kappa^2 = \alpha'.) How imposing the physical operator condition h=ar{h}=1 immediately gives the mass-shell condition k^2 = 4/\alpha', which corresponds to the tachyon mass M^2 = -k^2 = -4/\alpha'.

This is a beautiful result. The abstract requirement from canonical quantization ( $a = 1$ , $N = 0$ ) and the CFT requirement for a physical vertex operator ( $h = h ˉ = 1$ ) give the exact same physical condition for the ground state.

The full, integrated vertex operator for the tachyon is:
$V_{k}^{tachyon} = g_{s} \int d^{2} z : e^{ik \cdot X (z, z ˉ)} : with k^{2} = 4/ α^{'}$

Vertex Operators for Massless States

Now let's move to the first excited level, $N = 1$ . These are the massless states. A generic state at this level is created by acting with one left-moving and one right-moving oscillator on the vacuum:
$∣ ζ, k ⟩ = ζ_{μν} α_{- 1}^{μ} α ~_{- 1}^{ν} ∣0; k ⟩$
where $ζ_{μν}$ is a polarization tensor. From the previous lesson, we know that the physical state conditions require this state to be massless, $k^{2} = 0$ , and the polarization to be transverse, $k^{μ} ζ_{μν} = k^{ν} ζ_{μν} = 0$ .

To construct the corresponding vertex operator, we use the following dictionary from the state-operator map:

The vacuum $∣0; k ⟩$ corresponds to $: e^{ik \cdot X} :$ .
The creation operator $α_{- 1}^{μ}$ corresponds to $i 2 α^{'} \partial X^{μ}$ .
The creation operator $α ~_{- 1}^{ν}$ corresponds to $i 2 α^{'} \partial ˉ X^{ν}$ .

Ignoring the normalization constants for a moment, the local vertex operator is:
$V_{ζ, k} (z, z ˉ) = ζ_{μν} : \partial X^{μ} (z) \partial ˉ X^{ν} (z ˉ) e^{ik \cdot X (z, z ˉ)} :$

Let's check its conformal weight. We are multiplying three primary fields:

$\partial X^{μ}$ , with weight $(1, 0)$ .
$\partial ˉ X^{ν}$ , with weight $(0, 1)$ .
$: e^{ik \cdot X} :$ , with weight $(\frac{α ^{'} k ^{2}}{4}, \frac{α ^{'} k ^{2}}{4})$ .

The weights add, so the total weight of $V_{ζ, k}$ is $(1 + \frac{α ^{'} k ^{2}}{4}, 1 + \frac{α ^{'} k ^{2}}{4})$ .
For this to be a physical vertex operator, we need weight $(1, 1)$ . This implies $\frac{α ^{'} k ^{2}}{4} = 0$ , which means $k^{2} = 0$ .
Once again, the CFT condition perfectly reproduces the mass-shell condition derived from canonical quantization! The local operator for any massless state must have $k^{2} = 0$ to attain the required $(1, 1)$ conformal weight.

The different polarizations $ζ_{μν}$ correspond to the different massless particles:

Graviton: $ζ_{μν}$ is symmetric and traceless.
Kalb-Ramond B-field: $ζ_{μν}$ is antisymmetric.
Dilaton: $ζ_{μν}$ is proportional to the transverse projector, $η_{μν} - k_{μ} k ˉ_{ν} - k_{ν} k ˉ_{μ}$ .

Lectures on String Theory - CLASSE (Cornell)

The Cornell notes provide a concise summary of the vertex operators for these excited states. It's a good way to consolidate what we've just derived.

Please read from equation (5.27) to the end of the section on page 42. Note the explicit forms for the vertex operators corresponding to one- and two-tensor excitations, and the crucial step of integrating over the worldsheet to restore diffeomorphism invariance.

Test your understanding!

Consider a state at the second massive level, $N = 2$ , for an open string. A simple such state is $∣ ψ ⟩ = ζ_{μ} α_{- 2}^{μ} ∣0; k ⟩$ . What is the corresponding local vertex operator $V_{ψ} (z)$ for this state? (For an open string, we only have one set of modes, and the operator is inserted on the boundary of the worldsheet, so it depends only on one coordinate $z$ ). What condition does the conformal weight requirement impose on the mass of this state?

Show answer

The dictionary tells us that the operator $α_{- 2}^{μ}$ corresponds to a second derivative of $X^{μ}$ . So, the local vertex operator is $V_{ψ} (z) = ζ_{μ} : \partial^{2} X^{μ} (z) e^{ik \cdot X (z)} :$ .

The conformal weight of $\partial^{2} X^{μ}$ is $h = 2$ , and the weight of $: e^{ik \cdot X} :$ , which has both left- and right-moving parts, is $h = h ˉ = α^{'} k^{2} /4$ . For an open string operator living on the boundary, we just care about the single holomorphic weight, which is the sum of the weights of its components. So, the total weight is $h = 2 + α^{'} k^{2} /4$ .

The physical condition for a vertex operator on the boundary of the worldsheet is $h = 1$ . Therefore, we must have:
$2 + \frac{α ^{'} k ^{2}}{4} = 1 ⟹ \frac{α ^{'} k ^{2}}{4} = - 1 ⟹ k^{2} = - 4/ α^{'}$
The mass-squared is $M^{2} = - k^{2} = 4/ α^{'}$ .
Let's check this against the canonical quantization result: $M^{2} = \frac{1}{α ^{'}} (N - a)$ . For $N = 2$ and $a = 1$ , we get $M^{2} = \frac{1}{α ^{'}} (2 - 1) = 1/ α^{'}$ . The results don't match!

What went wrong? The state $α_{- 2}^{μ} ∣0; k ⟩$ is not a primary state of the Virasoro algebra, and its operator counterpart $: \partial^{2} X^{μ} e^{ik \cdot X} :$ , is not a primary field. The state-operator correspondence maps primary states to primary fields. Our simple dictionary ( $α_{- n} \to \partial^{n} X$ ) is a bit too naive for descendants. A more careful construction is needed for non-primary states, which we won't detail here, but this highlights the central importance of primary fields in CFT.

Conclusion

Today, we have established the crucial link between the abstract Hilbert space of string states and the practical language of the worldsheet CFT. This bridge is built upon the state-operator correspondence.

Key Takeaways:

The state-operator correspondence in CFT provides a one-to-one map between quantum states and local operators. In radial quantization, inserting an operator at the origin is equivalent to preparing a state in the infinite past.
Physical, coordinate-independent quantities in string theory are obtained by integrating local operators over the worldsheet. This leads to a crucial constraint: the local operator corresponding to a physical string state must be a primary field of conformal weight (1, 1).
Applying this condition to the candidate vertex operators $: e^{ik \cdot X} :$ and $: \partial X^{μ} \partial ˉ X^{ν} e^{ik \cdot X} :$ correctly reproduces the mass-shell conditions for the tachyon ( $M^{2} = - 4/ α^{'}$ ) and the massless states ( $M^{2} = 0$ ), respectively. This provides a powerful consistency check between the canonical quantization and the worldsheet CFT approaches.

We now have all the tools to describe not just free strings, but their interactions. The scattering amplitude for several strings is given by the worldsheet correlation function of their corresponding vertex operators.

Preview of the Next Lesson:

We have successfully quantized the string and identified its physical states and vertex operators using "old covariant quantization." However, this method has some lingering awkwardness, particularly in proving the no-ghost theorem for all levels. Next, we will introduce a more powerful and elegant method: BRST quantization. We will introduce the $b c$ -ghost system to make the conformal symmetry manifest at the quantum level and define physical states as elements of a BRST cohomology. This is the modern foundation for string quantization.

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