# Mobius invariance of the worldsheet 3-point function

Consider the CFT that corresponds to a gauge-fixed closed bosonic string.

Ground level string states are described by vertex operators such as $$V(p) = :\exp(i p_{\mu} X^{\mu}(z, \bar{z})):$$ which are conformal primaries with weight $$h = \bar{h} = \frac{\alpha'}{4} p^2.$$

The physical states of the strings must have $$h = \bar{h} = 1$$, therefore, the physical ground state is the tachyon.

Consider the 3-point function of three ground state operators:

$$G_{p_1, p_2, p_3}(z_1, z_2, z_3; \bar{z}_1, \bar{z}_2, \bar{z}_3) = \left< V(p_1)(z_1, \bar{z}_1) V(p_2)(z_2, \bar{z}_2) V(p_3)(z_3, \bar{z}_3) \right>.$$

Because we're dealing with a free quantum field theory, it isn't hard to calculate this function exactly. Afaik this is called the "Coulomb gas" representation, and the expression is

$$G_{p_1, p_2, p_3}(z_1, z_2, z_3; \bar{z}_1, \bar{z}_2, \bar{z}_3) = |z_{12}|^{\alpha' p_1 \cdot p_2} \cdot |z_{13}|^{\alpha' p_1 \cdot p_3} \cdot |z_{23}|^{\alpha' p_2 \cdot p_3},$$

where $$z_{ij} = z_i - z_j = -z_{ji}$$, and $$|z_{ij}| = (z_{ij} \cdot \bar{z}_{ij})^{1/2} = |z_{ji}|$$.

However, I expect a general 3-point function to be completely fixed by global conformal symmetries – the Mobius group $$SL(2, \mathbb{C})$$. The general form for three primary fields with weights $$h_i, \bar{h}_i$$ is:

$$\left< \phi_1(z_1, \bar{z}_1) \phi_2(z_2, \bar{z}_2) \phi_3(z_3, \bar{z}_3) \right> = C_{123} z_{12}^{h_3-h_1-h_2} z_{13}^{h_2-h_1-h_3} z_{23}^{h_1-h_2-h_3} \bar{z}_{12}^{\bar{h}_3-\bar{h}_1-\bar{h}_2} \bar{z}_{13}^{\bar{h}_2-\bar{h}_1-\bar{h}_3} \bar{z}_{23}^{\bar{h}_1-\bar{h}_2-\bar{h}_3},$$

where $$C_{123}$$ is the 3-point structure constant of the CFT.

For any physical string state, $$h = \bar{h} = 1$$, therefore Mobius invariance requires that

$$\left< \phi_1(z_1, \bar{z}_1) \phi_2(z_2, \bar{z}_2) \phi_3(z_3, \bar{z}_3) \right> = C_{123} z_{12}^{-1} z_{13}^{-1} z_{23}^{-1} \bar{z}_{12}^{-1} \bar{z}_{13}^{-1} \bar{z}_{23}^{-1} = C_{123} |z_{12}|^{-2} |z_{13}|^{-2} |z_{23}|^{-2}.$$

This seems to suggest that for the physical ground states participating in scattering (tachyons),

$$\forall i, j, i \neq j: \quad \alpha' p_i \cdot p_j = -2.$$

This restriction seems very odd to me, I've never seen anything like it before.

My questions are:

1. Have I missed something crucial?
2. If not, what is the origin of the constraint on the values of tachyon momenta?
• hint: momentum conservation .. – Wakabaloola May 24 at 14:18
• @Wakabaloola of course, that’s what I was missing! Three spacelike vectors of the same length that sum to zero can only be arranged such that the angle between any two is $2 \pi /3$. That totally makes sense, thank you. Now how about writing this up as an actual answer so that I can upvote and accept it? :) – Prof. Legolasov May 24 at 14:33

Using momentum conservation and onshell conditions, $$p_1+p_2+p_3=0, \qquad p_1^2=p_2^2=p_3^2=\frac{4}{\alpha'},$$ it follows that for any $$i\neq j\neq k$$, \begin{aligned} 2p_i\cdot p_j &= (p_i+p_j)^2-p_i^2-p_j^2 \\ &=p_k^2-p_i^2-p_j^2\\ &=-\frac{4}{\alpha'}, \end{aligned} so this relation is really just kinematics and onshell conditions.