Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 50 / 3 / Solution

Separate the constant embedding coordinate from its nonzero modes. Its integral gives the momentum-conservation delta function; below this is understood and absorbed into the amplitude normalization on the momentum-conserving locus. For the nonzero modes the Euclidean Polyakov action is quadratic, with worldsheet Green functionTreat the tachyon vertex operators as normal ordered, removing coincident self-contractions. The Gaussian functional integral, or the same calculation by Wick's theorem, givesAll remaining determinants are independent of the insertion points and momenta and enter the overall constant. This derives the Koba-Nielsen factor and the requested position integral.

For the Möbius transformation $f(z)=(az+b)/(cz+d)$, $ad-bc=1$, one hasThe transformed Koba-Nielsen factor contributes, at insertion $r$, the power $-{\alpha }^{\prime }{\sum }_{s\ne r}{k}_r\cdot k_s$ of $|c{z}_r+d|$. Four-momentum conservation makes this power ${\alpha }^{\prime }{k}_r^2$. The complete differential integrand therefore transforms byIt is invariant when ${\alpha }^{\prime }{k}_r^2=4$ for every external tachyon. The correlator alone is covariant; the integration measures are necessary for invariance. The group acting faithfully on the sphere is $PSL(2,\mathbb{C})$, with the two elements $\pm I$ of $SL(2,\mathbb{C})$ acting identically.

The apparent integral over all positions includes the residual conformal-group gauge volume. Divide by it and fix three distinct insertions. The Faddeev-Popov determinant, equivalently the three-point worldsheet ghost correlator and its conjugate, contributesSet $z_1=\Lambda$, $z_2=0$, $z_4=1$, then send $\Lambda$ to infinity on the sphere; the sign of the real direction is irrelevant. The factors involving $z_1$ behave as $|\Lambda {|}^{{\alpha }^{\prime }{k}_1\cdot {\sum }_{s\ne 1}{k}_s}=|\Lambda {|}^{-4}$, while $J\sim |\Lambda {|}^4$. They cancel, leaving $z_3=z$ as the one integration variable.

The remaining distance exponents areFor example $(k_2+k_3{)}^2=(k_1+k_4{)}^2=-t$ and each individual squared momentum is $4/{\alpha }^{\prime }$. AlsoConsequently the reduced amplitude is proportional toApply the supplied complex beta integral with $A=4+{\alpha }^{\prime }t/2$, $B=4+{\alpha }^{\prime }s/2$, and $C=4-A-B=4+{\alpha }^{\prime }u/2$. Its numerator arguments become $-1-{\alpha }^{\prime }t/4$, $-1-{\alpha }^{\prime }s/4$, $-1-{\alpha }^{\prime }u/4$, while its denominator arguments become $2+{\alpha }^{\prime }u/4$, $2+{\alpha }^{\prime }t/4$, $2+{\alpha }^{\prime }s/4$. Thus the four-point amplitude is the Virasoro–Shapiro amplitudewhere $C_0$ is momentum-independent. The integral converges initially when $\mathrm{Re}A<2$, $\mathrm{Re}B<2$, and $\mathrm{Re}(A+B)>2$; the displayed answer elsewhere is its analytic continuation.

The gamma function has simple poles at nonpositive integers and no zeros. At generic fixed $t$, the amplitude therefore has simple $s$-channel poles at $s=4(n-1)/{\alpha }^{\prime }$, $n=0,1,2,\dots$, with the same towers in $t$ and $u$ by crossing symmetry. They describe exchange of the closed string tachyon, massless states, and the infinitely many massive levels. The reciprocal denominator factors can give zeros at negative values $s=-4(\ell +2)/{\alpha }^{\prime }$; special kinematics can also make an exchange residue vanish.

To see the structure of a generic residue, set $a=-1-{\alpha }^{\prime }s/4$, $b=-1-{\alpha }^{\prime }t/4$, $c=-1-{\alpha }^{\prime }u/4$, so $a+b+c=1$. At $a=-n$, the Gamma function recurrence reduces the remaining factors to a constant times ${\prod }_{j=1}^n(b-j{)}^2$. The Virasoro–Shapiro amplitude pole residue is thus a polynomial of degree $2n$ in the other channel invariant, consistent with exchange up to spin $2n$. Crossed numerator poles must not simply be multiplied: near $a=-n+\delta$ and $b=-m+\epsilon$, $c=1+n+m-\delta -\epsilon$, and the zero of $1/\Gamma (1-c)$ cancels the putative double pole. The leading singular part is proportional toThe tree amplitude is meromorphic, with no loop threshold branch cuts; its channel singularities express factorization and string dual resonance.