Global Geometric Deformations of the Virasoro algebra, current and affine algebras by Krichever-Novikov type algebra

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.

arXiv comment17 pages

References in corpus (4)

• The Wess-Zumino-Witten-Novikov theory, Knizhnik-Zamolodchikov equations, and Krichever-Novikov algebras, I
• Degenerations of Generalized Krichever Novikov algebras on Tori
• Global geometric deformations of current algebras as Krichever-Novikov type algebras
• Differential Operator Algebras on compact Riemann Surfaces

Cited by in corpus (5)

• Realizations of the three point algebra $\mathfrak{sl}(2,\mathcal{R})\oplus \left({\Omega }_{\mathcal{R}}/d\mathcal{R}\right)$
• The Low-Dimensional Algebraic Cohomology of the Virasoro Algebra
• $N$-point Virasoro algebras are multi-point Krichever--Novikov type algebras
• The Vanishing of the Low-Dimensional Cohomology of the Witt and the Virasoro algebra
• An elementary proof of the vanishing of the second cohomology of the Witt and Virasoro algebra with values in the adjoint module