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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 ?

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

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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.

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  • $\begingroup$ 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. $\endgroup$ Jun 4, 2020 at 3:56
  • $\begingroup$ 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. $\endgroup$ Jun 4, 2020 at 4:09
  • $\begingroup$ Indeed, the argument interest me. Could you please point out the page in Witten's paper where the argument is elaborated? $\endgroup$ Jun 4, 2020 at 4:14
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    $\begingroup$ 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. $\endgroup$ Jun 4, 2020 at 12:13
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    $\begingroup$ 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. $\endgroup$ Jun 5, 2020 at 4:05

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