Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 306 / 3

The three unintegrated vertices fix the residual Möbius symmetry of the sphere and saturate its three holomorphic and antiholomorphic ghost zero modes. Moving a fixed insertion changes the amplitude by a BRST-exact operator, whose expectation with physical BRST-closed vertices vanishes. The remaining integrated positions are dummy variables. Thus the amplitude is independent of the chosen coordinates $(z_i,{\bar{z}}_i)$.

The zero mode of the free embedding field enforces momentum conservation, while contractions of normal-ordered exponentials give the Koba-Nielsen factor:Combining this with the supplied ghost correlator yields, up to the conventional overall normalization,Thereforewith the factor $(2\pi {)}^{26}$ absorbed into the amplitude normalization used by the question.

Fix $z_1=0$, $z_2=1$, and $z_3=\infty$, and write $z_4=z$. For the closed-string tachyon, $k_i^2=4/{\alpha }^{\prime }$. Hencewhere momentum conservation was used in the second identity. Taking the limit at infinity together with the ghost factor gives the Virasoro–Shapiro amplitudeChoosing a different set of three punctures to fix merely applies a Möbius transformation and relabels identical external tachyons. Exchanging the roles of the relevant punctures interchanges $s$ and $t$ without changing the integral, proving crossing symmetry under $s\leftrightarrow t$.