Refined sheaf counting on local K3 surfaces

We compute all refined sheaf counting invariants – Vafa-Witten, reduced DT, stable pairs and Gopakumar-Vafa – for all classes on local $K3$ surfaces. Along the way we develop rank 0 Vafa-Witten theory on $K3$ surfaces. An important feature of the calculation is that the ``instanton contribution" – of sheaves supported scheme theoretically on $S$ – to any of the invariants depends only on the square of the class, not its divisibility.