String Interactions and the Free Boson CFT

We have quantized the free bosonic string and uncovered several unmistakably stringy features: Regge trajectories, massless spin-two states, and Hagedorn growth. But a theory is not only a spectrum. We also need interactions.

For point particles, interactions are usually described by local vertices in a spacetime Feynman diagram. A cubic scalar interaction, for instance, is drawn as three worldlines meeting at a point. Strings behave differently. A closed string splits into two closed strings by sweeping out a smooth pair-of-pants worldsheet; two closed strings join by the same surface viewed with the opposite time orientation. The interaction is not placed at a point on the worldsheet. It is encoded in the topology and geometry of the worldsheet itself.

A string Feynman diagram is a two-dimensional surface. A long thin tube represents propagation of an intermediate string state and gives the pole structure familiar from field theory.

Schematically, perturbative string amplitudes have the form

$$\mathcal A = \sum_{\text{topologies}} g_s^{-\chi} \int \frac{[DX\,Dh]}{\text{Diff}\times\text{Weyl}} \,e^{-S_E[X,h]} \, \prod_i \mathcal V_i .$$

Here $\chi$ is the Euler characteristic of the worldsheet, $g_s$ is the string coupling, and the operators $\mathcal V_i$ describe the external string states. This formula is not yet a complete prescription: later we must gauge-fix the metric, introduce ghosts, integrate over moduli, and impose BRST invariance. But the conceptual point is already visible:

$$\text{particle perturbation theory: graphs}, \qquad \text{string perturbation theory: surfaces}.$$

In this page we prepare the CFT tools needed to make this precise.

Euclidean worldsheets and the complex plane

In Lorentzian conformal gauge the free string action is

$$S = -\frac{1}{4\pi\alpha'} \int d\tau\,d\sigma\, \eta^{\alpha\beta} \partial_\alpha X^\mu \partial_\beta X_\mu .$$

For path integrals and operator products, it is usually better to rotate to Euclidean time,

$$\tau=-i\tau_E,$$

so that

$$S_E = \frac{1}{4\pi\alpha'} \int d\tau_E\,d\sigma\, \left[ (\partial_{\tau_E}X)^2+(\partial_\sigma X)^2 \right].$$

Introduce the complex cylinder coordinate

$$w=\tau_E+i\sigma, \qquad \bar w=\tau_E-i\sigma .$$

For a closed string, $\sigma\sim\sigma+2\pi$, so the free Euclidean worldsheet is a cylinder. The exponential map

$$z=e^w=e^{\tau_E+i\sigma}$$

sends the cylinder to the punctured complex plane. Constant Euclidean time slices become circles,

$$\tau_E=\text{constant} \quad \Longleftrightarrow \quad |z|=e^{\tau_E}.$$

Thus Euclidean time evolution on the cylinder becomes radial evolution on the plane.

The map $z=e^w$ turns time ordering on the Euclidean cylinder into radial ordering on the plane. The remote past and future of the cylinder become $z=0$ and $z=\infty$.

This is the beginning of radial quantization. It will become essential when we identify local operators with string states.

Radial ordering

On the Euclidean plane the analogue of time ordering is radial ordering. For two local operators,

$$R\!\left[ \mathcal O_1(z_1,\bar z_1) \mathcal O_2(z_2,\bar z_2) \right] = \begin{cases} \mathcal O_1(z_1,\bar z_1)\mathcal O_2(z_2,\bar z_2), & |z_1|>|z_2|, \\ \mathcal O_2(z_2,\bar z_2)\mathcal O_1(z_1,\bar z_1), & |z_2|>|z_1|. \end{cases}$$

The outer operator is later in Euclidean cylinder time. This simple dictionary is the reason contour integrals in the $z$-plane reproduce commutators and symmetry transformations.

Radial ordering is time ordering after the cylinder-plane map. An operator at larger $|z|$ is later in Euclidean time.

For correlation functions we write

$$\left\langle R\{\mathcal O_1(z_1,\bar z_1)\cdots \mathcal O_n(z_n,\bar z_n)\} \right\rangle .$$

In most CFT notation the $R$ is suppressed, but it is conceptually always present.

The free boson path integral

In complex coordinates,

$$z=\sigma^1+i\sigma^2, \qquad \bar z=\sigma^1-i\sigma^2, \qquad \partial=\partial_z, \qquad \bar\partial=\partial_{\bar z},$$

the Euclidean free-boson action is

$$S_E = \frac{1}{2\pi\alpha'} \int d^2z\, \partial X^\mu \bar\partial X_\mu .$$

The path integral is Gaussian. The two-point function is therefore the Green function of the two-dimensional Laplacian. With the convention

$$\partial\bar\partial \ln |z-w|^2 = \pi\,\delta^{(2)}(z-w),$$

one obtains

$$\boxed{ \left\langle X^\mu(z,\bar z)X^\nu(w,\bar w) \right\rangle = -\frac{\alpha'}{2}\eta^{\mu\nu}\ln |z-w|^2 . }$$

This logarithm is the fundamental local singularity from which the free-boson CFT is built.

In two dimensions the scalar Green function is logarithmic. Derivatives of the logarithm produce the poles that appear in operator product expansions.

Normalization checkpoint

The formulas here keep $\alpha'$ explicit. In the common convention $\alpha'=2$, the two-point function becomes simply

$$\left\langle X^\mu(z,\bar z)X^\nu(w,\bar w)\right\rangle = -\eta^{\mu\nu}\ln |z-w|^2 .$$

We will keep the general $\alpha'$ form in the main text.

Operator products from short distances

The operator product expansion is the statement that as two insertions approach each other, their product can be replaced inside correlators by a sum of local operators at one point:

$$\mathcal O_A(z,\bar z)\mathcal O_B(w,\bar w) \sim \sum_C C_{AB}^{\;\;C}(z-w,\bar z-\bar w)\, \mathcal O_C(w,\bar w).$$

The symbol $\sim$ means equality of singular terms as $z\to w$ inside correlation functions. Terms regular at $z=w$ are usually omitted.

For the free boson, differentiating the logarithmic Green function gives

$$\partial X^\mu(z)\,X^\nu(w,\bar w) \sim -\frac{\alpha'}{2} \frac{\eta^{\mu\nu}}{z-w},$$

and

$$\partial X^\mu(z)\,\partial X^\nu(w) \sim -\frac{\alpha'}{2} \frac{\eta^{\mu\nu}}{(z-w)^2}.$$

Similarly,

$$\bar\partial X^\mu(\bar z)\,\bar\partial X^\nu(\bar w) \sim -\frac{\alpha'}{2} \frac{\eta^{\mu\nu}}{(\bar z-\bar w)^2},$$

while holomorphic and antiholomorphic derivatives have no local singularity with each other:

$$\partial X^\mu(z)\,\bar\partial X^\nu(\bar w) \sim 0.$$

These four OPEs are the workhorses of perturbative string theory.

Normal ordering

The field $X^\mu$ has a singular self-contraction, so composite operators must be normal ordered. A useful definition is

$$:X^\mu(z,\bar z)X^\nu(w,\bar w): = X^\mu(z,\bar z)X^\nu(w,\bar w) + \frac{\alpha'}{2}\eta^{\mu\nu}\ln |z-w|^2 .$$

For local composites, one first subtracts the singular part and then takes the coincident limit. For example,

$$:\partial X^\mu\partial X_\mu:(w) = \lim_{z\to w} \left[ \partial X^\mu(z)\partial X_\mu(w) + \frac{\alpha'}{2}\frac{D}{(z-w)^2} \right].$$

Normal ordering separates singular short-distance physics from finite operator data.

As $z\to w$, the product of fields splits into singular contractions plus a normal-ordered local operator. Wick contractions are the practical way to compute free-boson OPEs.

Exponential operators

String vertex operators are built from exponentials of $X^\mu$. The basic normal-ordered exponential is

$$:e^{ik\cdot X(z,\bar z)}: .$$

Using Wick’s theorem,

$$:e^{ik_1\cdot X(z,\bar z)}: :e^{ik_2\cdot X(w,\bar w)}: \sim |z-w|^{\alpha' k_1\cdot k_2} :e^{i(k_1+k_2)\cdot X(w,\bar w)}: .$$

Differentiated fields act on exponentials by a simple pole:

$$\partial X^\mu(z) :e^{ik\cdot X(w,\bar w)}: \sim -i\frac{\alpha'}{2}\frac{k^\mu}{z-w} :e^{ik\cdot X(w,\bar w)}: .$$

These formulas already contain the seed of string scattering amplitudes: products of many plane-wave vertices generate powers of $|z_i-z_j|^{\alpha' k_i\cdot k_j}$. Later, after adding the correct ghost factors and integrating over punctures, these powers become Veneziano and Virasoro-Shapiro amplitudes.

What to remember

The most important takeaways are:

$$\left\langle X^\mu(z,\bar z)X^\nu(w,\bar w)\right\rangle = -\frac{\alpha'}{2}\eta^{\mu\nu}\ln |z-w|^2,$$ $$\partial X^\mu(z)\partial X^\nu(w) \sim -\frac{\alpha'}{2} \frac{\eta^{\mu\nu}}{(z-w)^2},$$

and

$$:e^{ik_1\cdot X(z,\bar z)}: :e^{ik_2\cdot X(w,\bar w)}: \sim |z-w|^{\alpha' k_1\cdot k_2} :e^{i(k_1+k_2)\cdot X(w,\bar w)}: .$$

The next page introduces the stress tensor $T(z)$, which turns these local OPEs into the full machinery of conformal symmetry.

Exercises

Exercise 1. Cylinder time and radial ordering

Let $z=e^{\tau_E+i\sigma}$. Show that $\tau_E$ ordering on the cylinder is equivalent to radial ordering on the plane.

Solution

Taking the absolute value gives

$$|z|=|e^{\tau_E+i\sigma}|=e^{\tau_E}.$$

Thus

$$\tau_{E,1}>\tau_{E,2} \quad \Longleftrightarrow \quad e^{\tau_{E,1}}>e^{\tau_{E,2}} \quad \Longleftrightarrow \quad |z_1|>|z_2|.$$

Later Euclidean time corresponds to larger radius on the plane.

Exercise 2. Derivative OPEs

Starting from

$$\langle X^\mu(z,\bar z)X^\nu(w,\bar w)\rangle = -\frac{\alpha'}{2}\eta^{\mu\nu}\ln |z-w|^2,$$

derive

$$\partial X^\mu(z)\partial X^\nu(w) \sim -\frac{\alpha'}{2} \frac{\eta^{\mu\nu}}{(z-w)^2}.$$

Solution

The holomorphic part of the logarithm is $\ln(z-w)$. Therefore

$$\partial_z \ln(z-w)=\frac{1}{z-w}.$$

Differentiating with respect to $w$ gives

$$\partial_w\frac{1}{z-w} = \frac{1}{(z-w)^2}.$$

Hence

$$\partial_z\partial_w \left[ -\frac{\alpha'}{2}\eta^{\mu\nu}\ln(z-w) \right] = -\frac{\alpha'}{2} \frac{\eta^{\mu\nu}}{(z-w)^2}.$$

This is the singular OPE.

Exercise 3. Exponential OPE

Use Wick’s theorem to show that

$$:e^{ik_1\cdot X(z,\bar z)}: :e^{ik_2\cdot X(w,\bar w)}: \sim |z-w|^{\alpha' k_1\cdot k_2} :e^{i(k_1+k_2)\cdot X(w,\bar w)}: .$$

Solution

For two normal-ordered exponentials, Wick’s theorem gives the cross-contraction factor

$$\exp\left[ \left\langle i k_1\cdot X(z,\bar z)\, i k_2\cdot X(w,\bar w) \right\rangle \right].$$

The contraction is

$$\left\langle i k_1\cdot X(z,\bar z)\, i k_2\cdot X(w,\bar w) \right\rangle = -k_1\cdot k_2 \left( -\frac{\alpha'}{2}\ln |z-w|^2 \right).$$

Thus the factor is

$$\exp\left[ \frac{\alpha'}{2}k_1\cdot k_2\ln |z-w|^2 \right] = |z-w|^{\alpha' k_1\cdot k_2}.$$

The regular part of the product can then be expanded around $w$, giving the displayed OPE.

Exercise 4. OPE with a plane wave

Show that

$$\partial X^\mu(z) :e^{ik\cdot X(w,\bar w)}: \sim -i\frac{\alpha'}{2}\frac{k^\mu}{z-w} :e^{ik\cdot X(w,\bar w)}: .$$

Solution

Contract $\partial X^\mu(z)$ with the exponent:

$$\partial X^\mu(z)\, i k_\nu X^\nu(w,\bar w) \sim i k_\nu \left( -\frac{\alpha'}{2} \frac{\eta^{\mu\nu}}{z-w} \right).$$

This equals

$$-i\frac{\alpha'}{2}\frac{k^\mu}{z-w}.$$

Multiplying by the uncontracted normal-ordered exponential gives

$$\partial X^\mu(z) :e^{ik\cdot X(w,\bar w)}: \sim -i\frac{\alpha'}{2}\frac{k^\mu}{z-w} :e^{ik\cdot X(w,\bar w)}: .$$