Towards the theory of Yangians

Abstract

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-Heckmann/equivariant integrals between the fixed points in the 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.