Lines, Twisted Cubics on Cubic Fourfolds, and the Monodromy of the Voisin Map

For a general cubic fourfold $Y$ with associated Fano variety of lines $F$, we show that the monodromy group of the finite degree 16 rational Voisin self-map $\psi \text{\hspace{0.17em}⁣}:F⇢F$ is maximal. To achieve this, we investigate the intriguing interplay between $\psi$ and the fixed locus of the antisymplectic involution on the LLSvS variety $Z$, examined via the degree 6 Voisin map $F\times F⇢Z$.