Automorphic products, generalized Kac-Moody algebras and string amplitudes

Abstract

We review automorphic products and generalized Kac-Moody algebras from a physics point of view. We discuss the appearance of automorphic products in BPS-saturated one-loop quantities in heterotic string theory. At particular points in moduli space, with enhanced gauge symmetry \U0001d525, these products can be used to define a generalized Kac-Moody algebra \U0001d50a(\U0001d524++) as an automorphic correction of the Lorentzian Kac-Moody algebra \U0001d524++, which is obtained through double extension of the complement \U0001d524 = (\U0001d5228 ⊕ \U0001d5228)/\U0001d525. The root multiplicities of \U0001d50a(\U0001d524++) are then encoded in the Fourier coefficients of certain modular forms, which appear directly in the integrand of the one-loop quantities. We review particular examples of this extension for compactifications with \U0001d4a9 = 2 and \U0001d4a9 = 4 space-time supersymmetry.