String duality, automorphic forms, and generalized Kac-Moody algebras

TL;DR

This review surveys deep connections among string duality, automorphic forms, and generalized Kac–Moody algebras via BPS states in extended SUSY vacua. It highlights three recurring themes: (i) BPS state spaces form algebras that may realize GKMs; (ii) automorphic forms arising from these algebras encode threshold corrections and nonperturbative prepotentials; and (iii) automorphic denominator products potentially count black-hole degeneracies. The authors discuss concrete constructions, such as BRST-based OPEs for heterotic BPS states, Theta-transforms of Narain lattices yielding automorphic products, and duality-driven relations between heterotic and Type II prepotentials, illustrating how these structures might underpin moduli geometry and nonperturbative effects. While the connections are compelling, a precise, computation-ready synthesis remains elusive, though gravitational corrections and threshold phenomena point toward a common algebraic origin rooted in GKMs and their automorphic forms.

Abstract

This is a summary of a review talk at the Trieste conference on duality symmetries in string theory, April 1997. We review some work that has been done on the relation of BPS states, automorphic forms and the geometry of the quantum ground states of string compactifications with extended supersymmetry, emphasizing the (hypothetically) central role of BPS algebras.