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ABSTRACT: Moduli of Higgs pairs on algebraic curves are of central importance in mirror symmetry and mathematical physics. Recently, moduli of Higgs pairs on algebraic surfaces have also emerged as a subject of active research. Tanaka-Thomas use moduli spaces of stable Higgs pairs on surfaces to formulate an algebro-geometric definition of Vafa-Witten invariants. The invariants are defined by integration over a virtual fundamental class and are formed from contributions of components; the physical notion of S-duality translates to conjectural relationships between these contributions. One component, the so-called ``vertical component,’’ can be realized as a nested Hilbert scheme of curves and points on a surface. I will explain work in progress with M. Kool and T. Laarakker, in which we express refined invariants of the vertical component in terms of invariants of a well-studied quiver variety, the instanton moduli space of torsion-free framed sheaves on $\mathbb{CP}^2$. We then use a recent blow-up formula of Kuhn-Leigh-Tanaka to obtain formulas conjectured by Göttsche-Kool-Laarakker. One consequence is a formula for vertical refined Vafa-Witten invariants in rank 2, verifying a conjecture of Göttsche-Kool.