The refined local Donaldson-Thomas theory of curves

We solve the K-theoretically refined Donaldson-Thomas theory of local curves in arbitrary genus and degree.

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.

In the refined limit, our results establish a formula for the refined topological string partition function of local curves conjecturally 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.