Multiplicative dimensional reduction
Tasuki Kinjo
TL;DR
The paper proves a multiplicative analogue of the dimensional reduction for cohomological Donaldson–Thomas theory, showing that the 0-th BPS cohomology of the loop stack of a 0-shifted symplectic stack is controlled by torsion-loop data via a natural isomorphism. This yields a decomposition of cohomology for moduli spaces of $G$-local systems and $G$-Higgs bundles into contributions from torsion elements, with quasi-isolated elements governing the nonzero terms, and connects to a topological mirror symmetry framework, including logarithmic variants. The work also develops a twisted multiplicative reduction applicable to $S^1$-bundles over surfaces and Seifert-fibred 3-manifolds, providing a 3D perspective and laying groundwork for Langlands-dual and parabolic extensions.
Abstract
We prove the multiplicative version of the dimensional reduction theorem in cohomological Donaldson--Thomas theory. More precisely, we show that the BPS cohomology associated with the loop stack of a $0$-shifted symplectic stack admits a description analogous to orbifold cohomology, even though our stacks are not necessarily Deligne--Mumford. As an application, we propose a new, purely two-dimensional formulation of the topological mirror symmetry conjecture for the moduli space of $G$-Higgs bundles, which in turn leads to a formulation of the conjecture for logarithmic $G$-Higgs bundles. We also investigate a twisted version of the multiplicative dimensional reduction, which applies, in particular, to the cohomological Donaldson--Thomas theory for $S^1$-bundles over compact oriented surfaces, and more generally to Seifert-fibred $3$-manifolds.