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

Bojko Bakalov ; Daniel Fleisher

Symmetry, integrability and geometry: methods and applications, Tome 11 (2015)

Résumé

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.

Détail

Keywords: affine Kac–Moody algebra; Casimir element; Friedan–Martinec–Shenker bosonization; lattice vertex algebra; Virasoro algebra; Wakimoto realization.

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