  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 $Y_{\ge0}$ 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, $Y_{\ge0}$ is
homeomorphic to the cube $[0,1]^{\dim_{\mathbb{C}}Y}$. In this paper, we give a
proof of Rietsch's conjecture on $Y_{\ge0}$ in Lie type A by using toric
geometry which is closely related to the Peterson variety.
