Two-loop superstring five-point amplitudes II: Low energy expansion and S-duality

Authors: Eric D'Hoker, Carlos R. Mafra, Boris Pioline, Oliver Schlotterer

Preprint number: UUITP-29/20

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 alpha' 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 II 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 alpha', 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 D² R⁵ and D⁴ R⁵ are found to match those of D⁴ R⁴ and D⁶ R⁴, respectively, as required by non-linear supersymmetry. To the next order, a D⁶ R⁵ effective interaction arises, which is independent of the supersymmetric completion of D⁸ R⁴, and already arose at genus one. A novel identity on genus-two modular graph functions, which we prove, ensures that up to order D⁶ R⁵, 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.