Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 005, 19 pages      arXiv:1407.5335      https://doi.org/10.3842/SIGMA.2015.005
Contribution to the Special Issue on New Directions in Lie Theory

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

Bojko Bakalov a and Daniel Fleisher b
a) Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA
b) Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, Rehovot 76100, Israel

Received July 22, 2014, in final form January 09, 2015; Published online January 14, 2015

Abstract
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.

Key words: affine Kac-Moody algebra; Casimir element; Friedan-Martinec-Shenker bosonization; lattice vertex algebra; Virasoro algebra; Wakimoto realization.

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