Abstract
In an earlier paper, we constructed the genus-two amplitudes for five external massless states in Type II and Heterotic string theory, and showed that the α′ expansion of the Type II amplitude reproduces the corresponding supergravity amplitude to leading order. In this paper, we analyze the effective interactions induced by Type IIB superstrings beyond supergravity, both for U(1)R-preserving amplitudes such as for five gravitons, and for U(1)R-violating amplitudes such as for one dilaton and four gravitons. At each order in α′, the coefficients of the effective interactions are given by integrals over moduli space of genus-two modular graph functions, generalizing those already encountered for four external massless states. To leading and sub-leading orders, the coefficients of the effective interactions D2ℛ5 and D4ℛ5 are found to match those of D4ℛ4 and D6ℛ4, respectively, as required by non-linear supersymmetry. To the next order, a D6ℛ5 effective interaction arises, which is independent of the supersymmetric completion of D8ℛ4, and already arose at genus one. A novel identity on genus-two modular graph functions, which we prove, ensures that up to order D6ℛ5, the five-point amplitudes require only a single new modular graph function in addition to those needed for the four-point amplitude. We check that the supergravity limit of U(1)R-violating amplitudes is free of UV divergences to this order, consistently with the known structure of divergences in Type IIB supergravity. Our results give strong consistency tests on the full five-point amplitude, and pave the way for understanding S-duality beyond the BPS-protected sector.
Highlights
Scattering amplitudes of massless states are the basic observables in string theory and, in principle, are well-defined at arbitrary order in perturbation theory
At each order in α, the coefficients of the effective interactions are given by integrals over moduli space of genus-two modular graph functions, generalizing those already encountered for four external massless states
A novel identity on genus-two modular graph functions, which we prove, ensures that up to order D6R5, the five-point amplitudes require only a single new modular graph function in addition to those needed for the four-point amplitude
Summary
Scattering amplitudes of massless states are the basic observables in string theory and, in principle, are well-defined at arbitrary order in perturbation theory (for reviews see [1,2,3,4]). Combining perturbative results at tree-level and genus-one orders for the four-graviton scattering amplitude with requirements of spacetime supersymmetry and S-duality invariance [16,17,18,19,20], the axion-dilaton dependence of the coefficients of the effective interactions of the form R4, D4R4 and D6R4 were determined in terms of non-holomorphic modular functions of SL(2, Z). This has been accomplished in ten dimensions and after compactification on a torus, in terms of certain automorphic functions of the U-duality group An overview of the function theory on Riemann surfaces of genus two is presented in appendix A; the detailed calculations of the α‘ expansion of the genus-two integrals is given in section B; the analysis of the non-separating, separating, and tropical degenerations of the integrals is given in appendix C; the identity (1.1) is proved in appendix D and details on the overall normalization of the genus-two amplitude are given in appendix E
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