The sl(2n|2n)^(1) Super-Toda Lattices and the Heavenly Equations as Continuum Limit

TL;DR

This paper derives new integrable (super)heavenly equations from the continuum limit of super-Toda models related to affine superalgebras, revealing supersymmetry properties and reductions to lower dimensions.

Contribution

It introduces four classes of integrable heavenly equations from super-Toda models, generalizing previous results and exploring supersymmetry and dimensional reduction.

Findings

Derived four classes of integrable heavenly equations.

Identified N=1 and hidden N=2 supersymmetries.

Constructed supersymmetric models in (1+1) dimensions.

Abstract

The $n\to\infty$ continuum limit of super-Toda models associated with the affine $sl(2n|2n)^{(1)}$ (super)algebra series produces $(2+1)$-dimensional integrable equations in the ${\bf S}^{1}\times {\bf R}^2$ spacetimes. The equations of motion of the (super)Toda hierarchies depend not only on the chosen (super)algebras but also on the specific presentation of their Cartan matrices. Four distinct series of integrable hierarchies in relation with symmetric-versus-antisymmetric, null-versus-nonnull presentations of the corresponding Cartan matrices are investigated. In the continuum limit we derive four classes of integrable equations of heavenly type, generalizing the results previously obtained in the literature. The systems are manifestly N=1 supersymmetric and, for specific choices of the Cartan matrix preserving the complex structure, admit a hidden N=2 supersymmetry. The coset reduction of the (super)-heavenly equation to the ${\bf I}\times{\bf R}^{(2)}=({\bf S}^{1}/{\bf Z}_2)\times {\bf R}^2$ spacetime (with ${\bf I}$ a line segment) is illustrated. Finally, integrable $N=2,4$ supersymmetrically extended models in $(1+1)$ dimensions are constructed through dimensional reduction of the previous systems.