  We clarify the role played by BPS states in the calculation of threshold
corrections of D=4, N=2 heterotic string compactifications. We evaluate these
corrections for some classes of compactifications and show that they are sums
of logarithmic functions over the positive roots of generalized Kac-Moody
algebras. Moreover, a certain limit of the formulae suggests a reformulation of
heterotic string in terms of a gauge theory based on hyperbolic algebras such
as $E_{10}$. We define a generalized Kac-Moody Lie superalgebra associated to
the BPS states. Finally we discuss the relation of our results with string
duality.
