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We show that for each   k   and   n  , the cyclic shift map on the GrassmannianGr   k  ,  n(  C  )has exactlyn  kfixed points. There is a unique totally nonnegative fixed point, given by taking   n   equally spaced points on the trigonometric moment curve (if   k   is odd) or the symmetric moment curve (if   k   is even). We introduce a parameter    q  ∈   C  ×    , and show that the fixed points of a   q  ‐deformation of the cyclic shift map are precisely the critical points of the mirror‐symmetric superpotential    F  q    onGr   k  ,  n(  C  )    . This follows from results of Rietsch about the quantum cohomology ring ofGr   k  ,  n(  C  )    . We survey many other diverse contexts which feature moment curves and the cyclic shift map.

Moment curves and cyclic symmetry for positive Grassmannians