  We give an interpretation of the Fano variety of lines on a cubic fourfold
and of the hyperkahler eightfold, constructed by Lehn, Lehn, Sorger and van
Straten from twisted cubic curves in a cubic fourfold non containing a plane,
as moduli spaces of Bridgeland stable objects in the Kuznetsov component. As a
consequence, we reprove the categorical version of Torelli Theorem for cubic
fourfolds, we obtain the identification of the period point of LLSvS eightfold
with that of the Fano variety, and we discuss derived Torelli Theorem for cubic
fourfolds.
