The integrality conjecture and the cohomology of preprojective stacks

We study the Borel-Moore homology of stacks of representations of preprojective algebras Pi(Q) , via the study of the DT theory of the undeformed 3-Calabi-Yau completion Pi(Q)[x] . Via a result on the supports of the BPS sheaves for Pi(Q)[x]-mod , we prove purity of the BPS cohomology for the stack of Pi(Q)[x] -modules and define BPS sheaves for stacks of Pi(Q)-modules. These are mixed Hodge modules on the coarse moduli space of Pi(Q)-modules that control the Borel-Moore homology and geometric representation theory associated to these stacks. We show that the hypercohomology of these objects is pure and thus that the Borel-Moore homology of stacks of Pi(Q) -modules is also pure. We transport the cohomological wall-crossing and integrality theorems from DT theory to the category of Pi(Q )-modules. We use our results to prove positivity of a number of "restricted" Kac polynomials, determine the critical cohomology of Hilb(n)(A(3)) , and the Borel-Moore homology of genus one character stacks, as well as providing various applications to the cohomological Hall algebras associated to Borel-Moore homology of stacks of modules over preprojective algebras, including the PBW theorem, and torsion-freeness.