Gluon scattering in AdS at finite string coupling from localization

We consider gluons scattering in Type IIB string theory on AdS₅ × S⁵/ℤ₂ in the presence of D7 branes, which is dual to the flavor multiplet correlator in a certain 4d N = 2 USp(2N) gauge theory with SO(8) flavor symmetry and 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 to all orders in 1/N and finite τ using supersymmetric localization. In particular, we fix the F⁴ higher derivative correction to gluon scattering on AdS at finite string coupling τ_s = τ in terms of Jacobi theta functions, which feature the expected relations between the SL(2, ℤ) duality and the SO(8) triality of the CFT, and match it to the known flat space term. We also use the flat space limit to compute D²F⁴ corrections of the correlator at finite τ in terms of a non-holomorphic Eisenstein series. At weak string coupling, we find that the AdS correlator takes a form which is remarkably similar to that of the flat space Veneziano amplitude.