Cohomological Donaldson-Thomas theory for local systems on the $3$-torus

This paper studies the Cohomological Donaldson-Thomas theory of $G$-local systems on the topological three torus. Using an exponential map we prove cohomological integrality for ${\mathrm{G}\mathrm{L}}_n$-local systems using the statement of cohomological integrality for the tripled Jordan quiver from Davison-Meinhardt (2020). Using this result we prove a version of cohomological integrality for ${\mathrm{S}\mathrm{L}}_n$ and ${\mathrm{P}\mathrm{G}\mathrm{L}}_n$ for prime $n$. Finally, for prime $n$, we prove a Langlands duality statement for the ${\mathrm{S}\mathrm{L}}_n$ and ${\mathrm{P}\mathrm{G}\mathrm{L}}_n$ cohomological Donaldson-Thomas invariants.