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

TL;DR

This paper advances the understanding of cohomological Donaldson-Thomas theory for local systems on the 3-torus, establishing integrality results and Langlands duality for certain groups, with implications for algebraic geometry and mathematical physics.

Contribution

It proves cohomological integrality for $ ext{GL}_n$-local systems and extends this to $ ext{SL}_n$ and $ ext{PGL}_n$, also establishing Langlands duality for prime n.

Findings

Cohomological integrality for $ ext{GL}_n$-local systems on the 3-torus.

Extension of integrality results to $ ext{SL}_n$ and $ ext{PGL}_n$ for prime n.

Proof of Langlands duality for $ ext{SL}_n$ and $ ext{PGL}_n$ cohomological DT invariants.

Abstract

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.