Enhanced gauge groups inN=4topological amplitudes and Lorentzian Borcherds algebras

Abstract

We continue our study of algebraic properties of $\mathcal{N}=4$ topological amplitudes in heterotic string theory compactified on ${\mathbb{T}}^{2}$, initiated in arXiv:1102.1821. In this work we evaluate a particular one-loop amplitude for any enhanced gauge group $\mathfrak{h}\ensuremath{\subset}{\mathfrak{e}}_{8}\ensuremath{\bigoplus}{\mathfrak{e}}_{8}$, 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 ${\mathfrak{g}}^{++}$. The latter is obtained through double extension of the complement $\mathfrak{g}=({\mathfrak{e}}_{8}\ensuremath{\bigoplus}{\mathfrak{e}}_{8})/\mathfrak{h}$. The infinite product is automorphic with respect to a finite index subgroup of the full $T$-duality group $SO(2,18;\mathbb{Z})$ and, through the philosophy of Borcherds-Gritsenko-Nikulin, this defines the denominator formula of a generalized Kac-Moody algebra $\mathcal{G}({\mathfrak{g}}^{++})$, which is an 'automorphic correction' of ${\mathfrak{g}}^{++}$. We explicitly give the root multiplicities of $\mathcal{G}({\mathfrak{g}}^{++})$ for a number of examples.