Exact ∇⁴R⁴ couplings and helicity supertraces

In type II string theory compactified on a d-dimensional torus T^d down to D=10−d dimensions, the R⁴ and ∇⁴R⁴ four-graviton couplings are known exactly, for all values of the moduli, in terms of certain Eisenstein series of the U-duality group E_d(ℤ). In the limit where one circle in the torus becomes large, these couplings are expected to reduce to their counterpart in dimension D+1, plus threshold effects and exponentially suppressed corrections corresponding to BPS black holes in dimension D+1 whose worldline winds around the circle. By combining the weak coupling and large radius limits, we determine these exponentially suppressed corrections exactly, and demonstrate that the contributions of 1/4-BPS black holes to the ∇⁴R⁴ coupling are proportional to the appropriate helicity supertrace. Mathematically, our results provide the complete Fourier expansion of the next-to-minimal theta series of E_d+1(ℤ) with respect to the maximal parabolic subgroup with Levi component E_d for d≤6, and the complete Abelian part of the Fourier expansion of the same for d=7.