Global Geometric Deformations of the Virasoro Algebra, Current and Affine Algebras by Krichever–Novikov Type Algebras

Abstract

In two earlier articles we constructed algebraic-geometric families of genus one (i.e. elliptic) Lie algebras of Krichever–Novikov type. The considered algebras are vector fields, current and affine Lie algebras. These families deform the Witt algebra, the Virasoro algebra, the classical current, and the affine Kac–Moody Lie algebras respectively. The constructed families are not equivalent (not even locally) to the trivial families, despite the fact that the classical algebras are formally rigid. This effect is due to the fact that the algebras are infinite dimensional. In this article the results are reviewed and developed further. The constructions are induced by the geometric process of degenerating the elliptic curves to singular cubics. The algebras are of relevance in the global operator approach to the Wess–Zumino–Witten–Novikov models appearing in the quantization of Conformal Field Theory.