Serre-invariant stability conditions and Ulrich bundles on cubic threefolds

We prove a general criterion which ensures that a fractional Calabi-Yau category of dimension at most 2 admits a unique Serre-invariant stability condition, up to the action of the universal cover of GL+2 (R). We apply this result to the Kuznetsov component Ku(X) of a cubic threefold X. In particular, we show that all the known stability conditions on Ku(X) are invariant with respect to the action of the Serre functor and thus lie in the same orbit with respect to the action of the universal cover of GL+2 (R). As an application, we show that the moduli space of Ulrich bundles of rank at least 2 on X is irreducible, answering a question asked by Lahoz, Macrì and Stellari in [LMS15].