Title: Superized Leznov-Saveliev equations as the zero-curvature condition on a reduced connection Show Chinese title

Title: 超对称Leznov-Saveliev方程作为约化连接上的零曲率条件

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. Show Chinese abstract

Abstract: 开放二维Toda格子（TL）的方程对应于Leznov-Saveliev方程（LSE），这些方程被解释为在闵可夫斯基空间上$O(3)$-轨道的流形上的零曲率Yang-Mills方程，当规范代数是$\mathfrak{sl}(2)$在简单有限维李代数$\mathfrak{g}(A)$中的主嵌入的像时，该李代数的Cartan矩阵为$A$。已知的TL方程的可积超版本对应于两种不同类型的矩阵$A$。 我将一种类型的超LSE解释为在超流形的非可积分布上的\textit{减少的}连接的零曲率方程，该超流形是$OSp(1|2)$-轨道在$N=1$扩展的闵可夫斯基超空间上的；Leznov-Saveliev求解方法仅适用于$\mathfrak{g}(A)$有限维且允许超主嵌入的$\mathfrak{osp}(1|2)\to\mathfrak{g}(A)$。 最简单的 LSE1 是超黎曼方程；也可以用超弦作用量来解释。 Olshanetsky 引入了 LSE2——另一种超TL类型的方程。 Olshanetsky 的方程以及具有无限维$\mathfrak{g}(A)$的 LSE1 可以通过反散射方法求解。 除了超黎曼方程——这两种类型的 LSE 相重合的唯一情况外，解释这些方程仍然是一个开放问题。 我还回顾了涉及的相关较少为人知和不太流行的数学构造。