Totally nonnegative part of the Peterson variety in Lie type A

The Peterson variety (which we denote by $Y$) is a subvariety of the flag variety, introduced by Dale Peterson to describe the quantum cohomology rings of all the partial flag varieties. Motivated by the mirror symmetry for partial flag varieties, Rietsch studied the totally nonnegative part $Y_{\ge 0}$ and its cell decomposition. Based on the structure of those cells, Rietsch gave the following conjecture in Lie type A; as a cell decomposed space, $Y_{\ge 0}$ is homeomorphic to the cube $[0,1{]}^{{\dim }_{\mathbb{C}}Y}$. In this paper, we give a proof of Rietsch’s conjecture on $Y_{\ge 0}$ in Lie type A by using toric geometry which is closely related to the Peterson variety.