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. This seminar is part of the activities of the Dipartimento di Eccellenza CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.