Categorified Hall products for local surfaces and wall-crossing

The two dimensional categorified Hall algebra is introduced by Porta-Sala as a categorification of cohomological Hall algebra for surfaces by Kapranov-Vasserot. In this talk I will show that categorified Hall products induce actions to Donaldson-Thomas categories for local surfaces. Then I propose the existence of semi-orthogonal decompositions of DT categories under wall-crossing described by categorified Hall products, which in particular implies the d-critical analogue of D/K conjecture. I will focus on the example of MNOP/PT wall-crossing, and prove the conjecture for reduced curve classes.