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

TL;DR

This paper computes exact non-perturbative corrections to certain four-graviton couplings in string theory compactified on tori, linking them to BPS black hole contributions and mathematical automorphic forms.

Contribution

It provides the exact exponential corrections to $R^4$ and $ abla^4 R^4$ couplings in string theory, connecting physical BPS states to the Fourier expansion of Eisenstein series.

Findings

Exact Fourier expansion of next-to-minimal theta series for $E_{d+1}(Z)$.

Explicit determination of BPS black hole contributions to couplings.

Mathematical characterization of automorphic forms related to string amplitudes.

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 $d=7$.