Exact $\nabla^4 R^4$ couplings and helicity supertraces

In type II string theory compactified on a d-dimensional torus T$^{d}$ down to D = 10−ddimensions,the ℛ$^{4}$ and ∇$^{4}$ℛ$^{4}$ four-gravitoncouplingsareknownexactly,forall 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 ∇$^{4}$ℛ$^{4}$ 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.