Closed bosonic four-tachyon Virasoro–Shapiro amplitude

Given: The adopted critical flat model and its common source-declared external normalization, tachyon mass shell and momentum conservation, formal free-boson Gaussian correlator, and the closed genus coupling convention. The integral uses $d\mu =2dxdy$ on $\mathbb{C}$.

Bosonic Polyakov Model and String Tension Alpha Prime fix the adopted flat model and ${\alpha }^{\mathrm{\prime }}>0$.

Critical Dimension Flat Bosonic String and Closed Bosonic Lightcone Spectrum conditionally give $D=26$, intercept $1$, and ${\alpha }^{\mathrm{\prime }}{M}^2\mathrm{/}4=N-1$; both current readiness records are escalated through the unresolved quantum Lorentz-closure prerequisite.

Exact chiral free-boson stress tensor and vertex OPE proves its one-component, real-charge chiral coefficient formula by oscillator commutation. The complex-charge, Lorentzian and nonchiral extension used here is established in [L1], not supplied by that example.

Bosonic bc ghost and Beltrami moduli pairing fixes the local antighost and conformal-Killing treatment; Bosonic Bc Ghost System gives the local $b,c$ mode algebra and the vector-field transformation law for $c$. The global sphere zero-mode basis is supplied by [F5].

Sphere and boundary conformal-Killing modes for tree gauge fixing: the sphere has exactly three complex conformal-Killing vector modes.

Complex Euler beta integral for a four-point sphere correlator: for $a=-1-{\alpha }^{\mathrm{\prime }}s\mathrm{/}4$, $b=-1-{\alpha }^{\mathrm{\prime }}t\mathrm{/}4$ in its convergence region, ${\int }_{\mathbb{C}}\mathrm{\mid }z{\mathrm{\mid }}^{2a-2}\mathrm{\mid }1-z{\mathrm{\mid }}^{2b-2}d\mu =2\pi \mathrm{\Gamma }(a)\mathrm{\Gamma }(b)\mathrm{\Gamma }(1-a-b)\mathrm{/}[\mathrm{\Gamma }(1-a)\mathrm{\Gamma }(1-b)\mathrm{\Gamma }(a+b)]$.

Closed string genus-n-point coupling power gives $g_s^2$ for a sphere with four canonically sphere-normalized closed vertices; this topology factor is separate from any external-state conversion in another source convention.

The external-state normalization is a source input. For the isolated first tachyon pole the boundary sign is $s+4\mathrm{/}{\alpha }^{\mathrm{\prime }}+i0$, because $s=-k^2$ and Weigand §5.4.1 Eq. (5.73), printed p. 106 (PDF p. 109), writes $-k^2+4\mathrm{/}{\alpha }^{\mathrm{\prime }}+i\epsilon$. This local pole convention is adopted separately from the Gamma quotient; no global multi-channel contour or residue positivity is derived. The tachyon amplitude's universal Gamma ratio is independent of that common prefactor.

The formal Gaussian input in Given uses two independent chiral copies of 26 polynomial Heisenberg systems. To specify its Lorentzian and complex-momentum convention, replace the real charge labels in [F3] by $\gamma \in {\mathbb{C}}^{26}$, take the algebraic direct sum of the corresponding polynomial sectors, and use coordinatewise charge shifts and the same normally ordered exponentials as [F3]. Put $r_0=i$, $r_j=1$ for $1\le j\le 25$, and assign to momentum $k_i$ the charge vector $q_i^{\mu }=\sqrt{{\alpha }^{\mathrm{\prime }}\mathrm{/}2}{r}_{\mu }{k}_i^{\mu }$. Then ${\sum }_{\mu }{q}_i^{\mu }{q}_j^{\mu }=({\alpha }^{\mathrm{\prime }}\mathrm{/}2)k_i\cdot k_j$. The central-commutator calculation in [F3], Verification 1.4 and 2.1, works unchanged for complex charges: its coefficients use only addition and multiplication of charges, and each fixed oscillator-degree coefficient is finite. Applying it in each of the 26 factors gives ${\prod }_{i<j}(z_i-z_j{)}^{({\alpha }^{\mathrm{\prime }}\mathrm{/}2)k_i\cdot k_j}$ when ${\sum }_i{k}_i=0$. Use a second, independent copy with the same charges, rather than their conjugates, for the barred coordinate. Choose opposite logarithm branches so $\log (z_i-z_j)+\log ({\bar{z}}_i-{\bar{z}}_j)=2\log \mathrm{\mid }{z}_i-z_j\mathrm{\mid }$ on the Euclidean configuration domain. The product is therefore ${\prod }_{i<j}\exp ({\alpha }^{\mathrm{\prime }}{k}_i\cdot k_j\log \mathrm{\mid }{z}_i-z_j\mathrm{\mid })$, holomorphic in the complex momentum components on that branch. This is the specified formal nonchiral Gaussian correlator; the imaginary time-component charges do not define a positive Euclidean target-time measure.