Comments on DT(4) invariants of (graded) quivers and local Calabi-Yau varieties

I will discuss some recent and ongoing works on DT invariants of quivers associated to local Calabi-Yau 3-folds, and on conjectural DT4 invariants of local Calabi-Yau 4-folds, in the spirit of "physical mathematics" --- physics computations leading to potentially interesting mathematics. In the CY3 case, I will explain a recently proposed covering formula for quiver DT invariants [arXiv:2603.15334], wherein the DT invariants of some quiver Q are expressed as a sum of DT invariants of a "larger" Galois-covering quiver. I will aim to explain our partial, physics-based derivation of the covering formula. In the CY4 case, I will look at graded quivers associated to exceptional collections of coherent sheaves on local CY 4-folds and discuss what their "DT4 invariants" should look like according to our current physics intuition. These DT4 invariants are generally rational functions of various equivariant parameters of the local geometry.