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

Abstract : In type II string theory compactified on a $d$-dimensional torus $T^d$ down to $D=10-d$ dimensions, the $R^4$ and $\nabla^4 R^4$ four-graviton couplings are known exactly, for all values of the moduli, in terms of certain Eisenstein series of the U-duality group $E_{d}(\mathbb{Z})$. 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 $\nabla^4 R^4$ coupling are proportional to the appropriate helicity supertrace. Mathematically, our results provide the complete Fourier expansion of the next-to-minimal theta series of $E_{d+1}(\mathbb{Z})$ with respect to the maximal parabolic subgroup with Levi component $E_{d}$ for $d\leq 6$, and the complete Abelian part of the Fourier expansion of the same for $d=7$.
Document type :
Journal articles

https://hal.archives-ouvertes.fr/hal-01395652
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Submitted on : Friday, November 11, 2016 - 11:19:05 AM
Last modification on : Friday, December 3, 2021 - 12:07:55 PM

### Citation

Guillaume Bossard, Boris Pioline. Exact $\nabla^4 R^4$ couplings and helicity supertraces. Journal of High Energy Physics, Springer Verlag (Germany), 2017, 1701 (050), ⟨10.1007/JHEP01(2017)050⟩. ⟨hal-01395652⟩

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