Abstract: The moduli space of Higgs bundles associates to a genus-g Riemann surface (possibly with punctures) the space of solutions to certain PDEs on that Riemann surface. Since this space is non-compact, its volume (and other integrals over it) are diverging. However, it also comes equipped with a universal circle action that shifts the phase of the Higgs field and that can be used to regularize integrals via equivariant integration. The resulting equivariant quantities, in particular, the equivariant index of the Dirac operator turn out to have an alternative formulation in terms of a very simple 2d TQFT on the Riemann surface, defined in terms of cutting and gluing a la Atiyah-Segal. The main goal of the talk will be to explain this new 2d TQFT and its surprising connections to other subjects, such as categorification of quantum group invariants, quantum K-theory of vortex moduli spaces, etc.