The Fano of lines, the Kuznetsov component, and a flop

The Kuznetsov component of the derived category of a cubic fourfold is a non-commutative K3 surface'. Its symmetric square is hence a non-commutative hyperkaehler fourfold’. We prove that this category is equivalent to the derived category of an actual hyperkaehler fourfold: the Fano of lines in the cubic. This verifies a conjecture of Galkin. One of the key steps in our proof is a new derived equivalence for a specific 12-dimensional flop.