The two ∇⁶R⁴ type invariants and their higher order generalisation

Abstract: We show that there are two distinct classes of ∇⁶R⁴ type supersymmetry invariants in maximal supergravity. The second class includes a coupling in F²∇⁴R⁴ that generalises to 1/8 BPS protected F^2k∇⁴R⁴ couplings. We work out the supersymmetry constraints on the corresponding threshold functions, and argue that the functions in the second class satisfy to homogeneous differential equations for arbitrary k ≥ 1, such that the corresponding exact threshold functions in type II string theory should be proportional to Eisenstein series, which we identify. This analysis explains in particular that the exact ∇⁶R⁴ threshold function is the sum of an Eisenstein function and a solution to an inhomogeneous Poisson equation in string theory.