Stable Ulrich bundles on cubic fourfolds

In this paper, we examine the presence of Ulrich bundles on cubic fourfolds. We establish necessary and sufficient conditions for the existence of Ulrich bundles of a specific rank r. As a consequence, we show the existence of a family of non-decomposable Ulrich bundles of rank r on certain cubic fourfolds, which are dependent on approximately r parameters and have wild representation type. Our study also encompasses examples of arithmetically Cohen-Macaulay smooth surfaces that are not intersections of cubic fourfolds and codimension two subvarieties.