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The two D6R4 type invariants and their higher order generalisation

We show that there are two distinct classes of D6R4 type supersymmetry invariants in maximal supergravity. The second class includes a coupling in F2D4R4 that generalises to 1/8 BPS protected F2kD4R4 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>0, 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 D6R4 threshold function is the sum of an Eisenstein function and a solution to an inhomogeneous Poisson equation in string theory.