Moduli spaces of Higgs bundles have appeared for different reasons in mathematical physics, topology and number theory. Recently, Marc de Cataldo, Tamas Hausel and Luca Migliorini observed a hitherto unexplained identity between invariants related to singularitiesof the integrable system defined by these spaces and arithmetic invariants of a closely related, non-singular space. They conjecture that this identity is a general phenomenon. In the article we survey the origins of this conjecture and show how they have connected very different areas of mathematics. In the process we indicate some of the unexpected applications that resulted from this passage between the different fields.