Momentum lattice for CHL string

We propose some analogue of the Narain lattice for CHL string. The symmetries of this lattice are the symmetries of the perturbative spectrum. We explain in this language the known results about the possible gauge groups in compactified theory. For the four-dimensional theory, we explicitly describe the action of S-duality on the background fields. We show that the moduli spaces of the six-, seven- and eight-dimensional compactifications coincide with the moduli spaces of the conjectured type IIA, M-theory and F-theory duals. We classify the rational components of the boundary of the moduli space in seven, eight and nine dimensions.