Hall-type algebras for categorical Donaldson-Thomas theories on local surfaces

Abstract: We show that the categorified cohomological Hall algebra structures on surfaces constructed by Porta-Sala descend to those on Donaldson-Thomas categories on local surfaces introduced in the author's previous paper. A similar argument also shows that Pandharipande-Thomas categories on local surfaces admit actions of categorified COHA for zero dimensional sheaves on surfaces. We also construct annihilator actions of its simple operators, and show that their commutator in the K-theory satisfies the relation similar to the one of Weyl algebras. This result may be regarded as a categorification of Weyl algebra action on homologies of Hilbert schemes of points on locally planar curves due to Rennemo, which is relevant for Gopakumar-Vafa formula of generating series of PT invariants.