The refined local Donaldson-Thomas theory of curves

We solve the $K$-theoretically refined Donaldson-Thomas theory of local curves. Our results avoid degeneration techniques, but rather exploit direct localisation methods to reduce the refined Donaldson-Thomas partition function to the equivariant intersection theory of skew nested Hilbert schemes on smooth projective curves. We show that the latter is determined, for every Young diagram, by three universal series, which we compute in terms of the 1-leg $K$-theoretic equivariant vertex. In the refined limit, our results establish a formula for the refined topological string partition function of local curves proposed by Aganagic-Schaeffer. In the second part, we show that analogous structural results hold for the refined Pandharipande-Thomas theory of local curves. As an application, we deduce the K-theoretic DT/PT correspondence for local curves in arbitrary genus, as conjectured by Nekrasov-Okounkov. Thanks to the recent machinery developed by Pardon, we expect our explicit results on local curves to play a key role towards the proof of the refined GW/PT conjectural correspondence of Brini-Schuler for all smooth Calabi-Yau threefolds.