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 (∇Φ)⁴ coupling, we propose that the ∇²(∇Φ)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 ∇²(∇Φ)⁴ coupling reduces to the ∇²F⁴ and R²F² couplings in D=4, along with an infinite series of corrections of order e^−R, from four-dimensional 1/4-BPS dyons whose worldline 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.