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 $\mathrm{GL}_n$-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 $\mathrm{SL}_n$ and $\mathrm{PGL}_n$ for prime $n$. Finally, for prime $n$, we prove a Langlands duality statement for the $\mathrm{SL}_n$ and $\mathrm{PGL}_n$ cohomological Donaldson-Thomas invariants.