Multiplicative dimensional reduction

TL;DR

This paper proves a multiplicative version of the dimensional reduction theorem in cohomological Donaldson--Thomas theory, providing new insights into BPS cohomology, mirror symmetry, and applications to Seifert-fibred 3-manifolds.

Contribution

It introduces a multiplicative dimensional reduction theorem in cohomological Donaldson--Thomas theory and applies it to mirror symmetry and Seifert-fibred 3-manifolds.

Findings

BPS cohomology admits an orbifold-like description for certain stacks.

A new two-dimensional formulation of the topological mirror symmetry conjecture is proposed.

The twisted version applies to cohomological DT theory for S^1-bundles and Seifert-fibred 3-manifolds.

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.