# Vertex operators in string theory

It is well known that operators in string theory are inserted at the origin. So, for example, the vertex operator for tachyon looks like $$:e^{ik.X(0, 0)}:$$ However, while computing the string amplitude we use an integrated version of the vertex operator i.e. $$\int d^{2}\sigma \sqrt{det(g)} e^{ik.X}$$ This seems like a contradiction to me. Please explain why are these two equivalent ?

• I think you mean $\sqrt{g}$ and not $\sqrt{\sigma}$ and that should help you find the answer... Apr 3, 2020 at 12:07

Under the state-operator mapping we have the following identification $$|k;0,0 \rangle \longleftrightarrow :e^{ik.X(0, 0)}:$$ where $$:e^{ik.X(0, 0)}:$$ is the normally ordered closed string tachyon vertex operator and $$|k;0,0 \rangle$$ is the closed string tachyon ground state.

Now, why is $$:e^{ik.X(0, 0)}:$$ not used in computing string S-matrix elements?

The answer is that any S-matrix element must be invariant under Weyl and diffeormphism transformations, something like $$S(k)= \sum_{compact \\ topologies} \int \frac{[dXdg]}{V_{diff \times Weyl}} exp(-S_{x}-\lambda\chi):e^{ik.X(0, 0)}:$$ can't work for obvious reasons, namely that $$S(k)$$ is not a scalar under diff and Weyl symmetries because $$:e^{ik.X(0, 0)}:$$ is an operator of conformal dimensions $$(\Delta=\frac{l_{S}^{2}k^{2}}{4} , \Delta=\frac{l_{S}^{2}k^{2}}{4})$$.

The solution is to integrate $$:e^{ik.X(0, 0)}:$$ over the relevant worldsheet as

$$\int d^{2}{\sigma} \sqrt{g} :e^{ikX}:.$$ Now is easy to see why this is the correct guess.

In locally flat coordinates we have the replacement $$\int d^{2}{\sigma} \sqrt{g} e^{ikX} \rightarrow \int d^{2}{z} e^{ikX}$$ where we can turn $$d^{2}{z}:e^{ikX}:$$ into a tensor of type $$(0,0)$$ by noticing that $$d^{2}{z}$$ is of type $$(1,1)$$ and choosing $$k^{2}=-m^{2}=\frac{4}{l_{S}}$$.

Sleeping beauty: Please notice how beautiful is the fact that the precise value of the mass of the tachyon is enforced by consistency.

We conclude that the correct expression for a stringy S-matrix element should be $$S(k)= \sum_{compact \\ topologies} \int \frac{[dXdg]}{V_{diff \times Weyl}} exp(-S_{x}-\lambda\chi)\prod_{a=1}^{n} \int d^{2}{\sigma} \sqrt{g} :e^{ik_{a}X_{a}}:$$ As desired.

• Actually you can get the integrated form naturally without having to guess if you use a ghost fields and ghost action as well. This is explained in witten's paper perturbative string theory revisited. Jun 4, 2020 at 3:56
• Well, my humble opinion is the argument is not exactly a guess. The only way to "cancel" the conformal dimension of the tachyon operator is to contract it with a tensor of type (1,1) and locally there is no other choice than $d^{2}z$. Anyway, if you have found a more satisfactory explanation, is ok. Jun 4, 2020 at 4:09
• Indeed, the argument interest me. Could you please point out the page in Witten's paper where the argument is elaborated? Jun 4, 2020 at 4:14
• See section 2.5 and 2.5.2 for a complete understanding. He talks of two kinds of vertex operators integrated and non-integrated and shows their equivalence. Jun 4, 2020 at 12:13
• Thanks :) I'm not too familiar with the theory of super Riemann surfaces, but now I see I should. Although the logic seems (at least to me) much more elaborated, I completely agree, the Witten's derivation is clear and much more powerful. Jun 5, 2020 at 4:05