Protected couplings and BPS dyons in half-maximal supersymmetric string vacua

We analyze four- and six-derivative couplings in the low energy effective action of $D=3$ string vacua with half-maximal supersymmetry. In analogy with an earlier proposal for the $(\mathrm{\nabla }\mathrm{\Phi }{)}^4$ coupling, we propose that the ${\mathrm{\nabla }}^2(\mathrm{\nabla }\mathrm{\Phi }{)}^4$ coupling is given exactly by a manifestly U-duality invariant genus-two modular integral. In the limit where a circle in the internal torus decompactifies, the ${\mathrm{\nabla }}^2(\mathrm{\nabla }\mathrm{\Phi }{)}^4$ coupling reduces to the $D^2{F}^4$ and $R^2{F}^2$ couplings in $D=4$, along with an infinite series of corrections of order $e^{-R}$, from four-dimensional 1/4-BPS dyons whose wordline winds around the circle. Each of these contributions is weighted by a Fourier coefficient of a meromorphic Siegel modular form, explaining and extending standard results for the BPS index of 1/4-BPS dyons.