Bosonizations of $\widehat{\mathfrak{sl}}_2$ and Integrable Hierarchies

[摘要] We construct embeddings of $\widehat{\mathfrak{sl}}_2$ in lattice vertex algebras by composing the Wakimoto realization with the Friedan-Martinec-Shenker bosonization. The Kac-Wakimoto hierarchy then gives rise to two new hierarchies of integrable, non-autonomous, non-linear partial differential equations. A new feature of our construction is that it works for any value of the central element of $\widehat{\mathfrak{sl}}_2$; that is, the level becomes a parameter in the equations.

[关键词] af fine Kac–Moody algebra;Casimir element;Friedan–Martinec–Shenker bosonization;lattice vertex algebra;Virasoro algebra;Wakimoto realization