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.
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Notes
More precisely Porta–Sala’s construction work in a dg-categorical setting, where not only the stacks of exact sequences but also higher parts of the Waldhausen construction are required in order to control the higher associativity. Porta and Sala pointed out that the same would apply to our situation, see Remark 3.9.
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Acknowledgements
The author is grateful to Francesco Sala and Mauro Porta for asking a question whether DT categories admit Hall-type algebra structures when the previous paper [32] was posted on arXiv, and also several useful comments for the first draft of this paper. The author is also grateful to Andrei Negut for useful comments, and Tasuki Kinjo for valuable discussions. The author is supported by World Premier International Research Center Initiative (WPI initiative), MEXT, Japan, and Grant-in Aid for Scientific Research Grant (No. 19H01779) from MEXT, Japan.
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