Modular invariant gluon-graviton scattering in AdS at one loop

We consider mixed gluon-graviton scattering in Type IIB string theory on AdS₅ × S⁵/ℤ₂ in the presence of D7 branes, which is dual to a mixed correlator of the SO(8) and SU(2)_L flavor multiplets of a certain 4d N = 2 USp(2N) gauge theory with complexified coupling τ. We compute this holographic correlator in the large N and finite τ expansion using constraints from derivatives of the mass deformed sphere free energy, which we compute using supersymmetric localization at large N and finite τ in terms of modular invariant non-holomorphic Eisenstein series. In particular, we combine this constraint with the known flat space limit to fix the R²F² higher derivative correction to the correlator in terms of the weight one non-holomorphic Eisenstein series, and also to fix the logarithmic threshold. We also compute the one-loop correction to the correlator, and match it to the expected flat space limit result.