In this talk I will report on a joint work in progress with E. Fatighenti, in which we study some special vector bundles on the Fano variety of lines of a cubic fourfold. We will see that these bundles are modular, but not atomic, and we will make some remarks on their infinitesimal deformations. In the first part of the talk I will introduce the main definitions, providing background and motivations. In the second part I will state the main results and we will look at what comes next; time permitting, we will comment on some parts of the proofs.