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 $d\le 6$, and the complete Abelian part of the Fourier expansion of the same for $d=7$.

40 pages

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Cited by in corpus (8)

• Cancellation of divergences up to three loops in exceptional field theory
• Four-derivative couplings and BPS dyons in heterotic CHL orbifolds
• Emergence of $R^4$-terms in M-theory
• Exact effective interactions and 1/4-BPS dyons in heterotic CHL orbifolds
• 1/8-BPS Couplings and Exceptional Automorphic Functions
• Perturbative higher-derivative terms in N=6 asymmetric orbifolds
• Mock modularity at work, or black holes in a forest
• Reductions of Exceptional Field Theories