Two-loop superstring five-point amplitudes I: Construction via chiral splitting and pure spinors

Understand

The full two-loop amplitudes for five massless states in Type~II and Heterotic superstrings are constructed in terms of convergent integrals over the genus-two moduli space of compact Riemann surfaces and integrals of Green functions and Abelian differentials on the surface.

• The construction combines elements from the BRST cohomology of the pure spinor formulation and from chiral splitting with the help of loop momenta and homology invariance.
• The $\alpha' \to 0$ limit of the resulting superstring amplitude is shown to be in perfect agreement with the previously known amplitude computed in Type~II supergravity.
• Investigations of the $\alpha'$ expansion of the Type~II amplitude and comparisons with predictions from S-duality are relegated to a first companion paper.