Four-derivative couplings and BPS dyons in heterotic CHL orbifolds

Three-dimensional string models with half-maximal supersymmetry are believedto be invariant under a large U-duality group which unifies the S and Tdualities in four dimensions. We propose an exact, U-duality invariant formulafor four-derivative scalar couplings of the form $F(\Phi) (\nabla\Phi)^4$ in aclass of string vacua known as CHL $\mathbb{Z}_N$ heterotic orbifolds with $N$prime, generalizing our previous work which dealt with the case of heteroticstring on $T^6$. We derive the Ward identities that $F(\Phi)$ must satisfy, andcheck that our formula obeys them. We analyze the weak coupling expansion of$F(\Phi)$, and show that it reproduces the correct tree-level and one-loopcontributions, plus an infinite series of non-perturbative contributions.Similarly, the large radius expansion reproduces the exact $F^4$ coupling infour dimensions, including both supersymmetric invariants, plus infinite seriesof instanton corrections from half-BPS dyons winding around the large circle,and from Taub-NUT instantons. The summation measure for dyonic instantonsagrees with the helicity supertrace for half-BPS dyons in 4 dimensions in allcharge sectors. In the process we clarify several subtleties about CHL modelsin $D=4$ and $D=3$, in particular we obtain the exact helicity supertraces for1/2-BPS dyonic states in all duality orbits.