Universality of Descendent Integrals over Moduli Spaces of Stable Sheaves on $K3$ Surfaces

We interprete results of Markman on monodromy operators as a universality statement for descendent integrals over moduli spaces of stable sheaves on $K3$ surfaces. This yields effective methods to reduce these descendent integrals to integrals over the punctual Hilbert scheme of the $K3$ surface. As an application we establish the higher rank Segre–Verlinde correspondence for $K3$ surfaces as conjectured by Göttsche and Kool.