Superized Leznov-Saveliev equations as the zero-curvature condition on a reduced connection

Abstract

The equations of open 2-dimensional Toda lattice (TL) correspond to Leznov-Saveliev equations (LSE) interpreted as zero-curvature Yang-Mills equations on the variety of $O(3)$-orbits on the Minkowski space when the gauge algebra is the image of $\mathfrak{sl}(2)$ under a principal embedding into a simple finite-dimensional Lie algebra $\mathfrak{g}(A)$ with Cartan matrix $A$. The known integrable super versions of TL equations correspond to matrices $A$ of two different types. I interpret the super LSE of one type 1 as zero-curvature equations for the \textit{reduced} connection on the non-integrable distribution on the supervariety of $OSp(1|2)$-orbits on the $N=1$-extended Minkowski superspace; the Leznov-Saveliev method of solution is applicable only to $\mathfrak{g}(A)$ finite-dimensional and admitting a superprincipal embedding $\mathfrak{osp}(1|2)\to\mathfrak{g}(A)$. The simplest LSE1 is the super Liouville equation; it can be also interpreted in terms of the superstring action. Olshanetsky introduced LSE2 -- another type of equations of super TL. Olshanetsky's equations, as well as LSE1 with infinite-dimensional $\mathfrak{g}(A)$, can be solved by the Inverse Scattering Method. To interpret these equations remains an open problem, except for the super Liouville equation -- the only case where these two types of LSE coincide. I also review related less known and less popular mathematical constructions involved.