Enhanced gauge groups in N=4 topological amplitudes and Lorentzian Borcherds algebras

Journal article, 2011

We continue our study of algebraic properties of N = 4 topological amplitudes in heterotic string theory compactified on T(2), initiated in arXiv:1102.1821. In this work we evaluate a particular one-loop amplitude for any enhanced gauge group h subset of e(8) circle plus e(8), i.e. for arbitrary choice of Wilson line moduli. We show that a certain analytic part of the result has an infinite product representation, where the product is taken over the positive roots of a Lorentzian Kac-Moody algebra g(++). The latter is obtained through double extension of the complement g = (e(8) circle plus e(8))/h. The infinite product is automorphic with respect to a finite index subgroup of the full T-duality group SO(2, 18; Z) and, through the philosophy of Borcherds-Gritsenko-Nikulin, this defines the denominator formula of a generalized Kac-Moody algebra G(g(++)), which is an 'automorphic correction' of g(++). We explicitly give the root multiplicities of G(g(++)) for a number of examples.