Toward a theory of Yangians

We review the main ideas underlying the emerging theory of Yangians—the new type of hidden symmetry in string-inspired models. Their classification by quivers is a far-going generalization of simple Lie algebras classification by Dynkin diagrams. However, this is still a kind of project, while a more constructive approach goes through toric Calabi-Yau spaces, related supersymmetric systems, and the Duistermaat-Heckman or equivariant integrals between the fixed points in the Atiyah-Drinfeld-Hitchin-Manin (ADHM)-like moduli spaces. These fixed points are classified by crystals (Young-type diagrams), and Yangian generators describe “instanton” transitions between them. Detailed examples will be presented elsewhere.