  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 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 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 D^2 R^5 and
D^4 R^5 are found to match those of D^4 R^4 and D^6 R^4, respectively, as
required by non-linear supersymmetry. To the next order, a D^6 R^5 effective
interaction arises, which is independent of the supersymmetric completion of
D^8 R^4, and already arose at genus one. A novel identity on genus-two modular
graph functions, which we prove, ensures that up to order D^6 R^5, 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.
