The failure of Atkin-Lehner symmetry for lattice compactified strings

We present a general discussion on the possible existence of an Atkin-Lehner symmetry leading to a zero cosmological constant in lattice compactified heterotic string theories. We explicitly find such a symmetry to be impossible for four dimensional strings (with the simplest world-sheet supersymmetry) by enumerating all the possible partition functions. We also show, in more generality, that an Atkin-Lehner symmetry is incompatible with general physical constraints on the singularity structure of the string amplitude in moduli space expect for a unique partition function (related to the Leech lattice) in two space-time dimensions. Therefore those theories with space-time supersymmetry remain the only known examples with vanishing cosmological constant in greater than two space-time dimensions.