On the algebras of BPS states

TL;DR

This paper constructs an algebraic framework for BPS states in supersymmetric theories, revealing deep connections with generalized Kac-Moody algebras, string dualities, and geometric structures on Calabi-Yau manifolds.

Contribution

It introduces a novel algebraic structure on BPS states, linking string theory, algebra, and geometry, and explores dualities and mirror symmetry implications.

Findings

BPS state algebra relates to generalized Kac-Moody algebra

Duality maps BPS algebras between heterotic and type II theories

Mirror symmetry exchanges associated algebras on Calabi-Yau 3-folds

Abstract

We define an algebra on the space of BPS states in theories with extended supersymmetry. We show that the algebra of perturbative BPS states in toroidal compactification of the heterotic string is closely related to a generalized Kac-Moody algebra. We use D-brane theory to compare the formulation of RR-charged BPS algebras in type II compactification with the requirements of string/string duality and find that the RR charged BPS states should be regarded as cohomology classes on moduli spaces of coherent sheaves. The equivalence of the algebra of BPS states in heterotic/IIA dual pairs elucidates certain results and conjectures of Nakajima and Gritsenko & Nikulin, on geometrically defined algebras and furthermore suggests nontrivial generalizations of these algebras. In particular, to any Calabi-Yau 3-fold there are two canonically associated algebras exchanged by mirror symmetry.