Abstract
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 α′ → 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 α′ expansion of the Type II amplitude and comparisons with predictions from S-duality are relegated to a first companion paper. A construction from first principles in the RNS formulation of the genus-two amplitude with five external NS states is relegated to a second companion paper.
Highlights
The perturbative evaluation of superstring amplitudes in the Ramond-Neveu-Schwarz (RNS) formulation proceeds systematically from first principles
Various identities for the Clifford-Dirac algebra and pure spinors are collected in appendix A; basics ingredients of Riemann surfaces and their function theory are summarized in appendix B; a detailed derivation of the chiral splitting procedure suitable for the pure spinor formulation is presented in appendix C; and the operator product expansions of the pure spinor worldsheet fields are gathered in appendix D
We have proposed a spacetime supersymmetric expression for the chiral twoloop five-point amplitude relevant to massless states of Type II, Heterotic, and Type I superstring theories
Summary
The perturbative evaluation of superstring amplitudes in the Ramond-Neveu-Schwarz (RNS) formulation proceeds systematically from first principles (see for example [1,2,3,4] and references therein). Our key result is the construction of the chiral amplitude K(5) which is a function of external momenta, chiral polarization vectors and spinors, loop momenta, and a complex analytic dependence on vertex operator points and moduli of the underlying compact Riemann surface Σ. The integration of the pairing of left and right chiral amplitudes over loop momenta, vertex operator points, and moduli gives the physical amplitude for five external states in the supergravity multiplet. The kinematic factors T1m,2,3|4,5, T23,1|4,5, S2;4|5|1,2 in pure spinor superspace will be developed below, giving access to arbitrary combinations of external states from the massless supersymmetry multiplets These kinematic factors are independent of moduli, vertex points, and loop momenta. Various identities for the Clifford-Dirac algebra and pure spinors are collected in appendix A; basics ingredients of Riemann surfaces and their function theory are summarized in appendix B; a detailed derivation of the chiral splitting procedure suitable for the pure spinor formulation is presented in appendix C; and the operator product expansions of the pure spinor worldsheet fields are gathered in appendix D
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