Dimensional reduction in cohomological Donaldson–Thomas theory

Abstract

For oriented $-1$-shifted symplectic derived Artin stacks, Ben-Bassat, Brav, Bussi and Joyce introduced certain perverse sheaves on them which can be regarded as sheaf-theoretic categorifications of the Donaldson–Thomas invariants. In this paper, we prove that the hypercohomology of the above perverse sheaf on the $-1$-shifted cotangent stack over a quasi-smooth derived Artin stack is isomorphic to the Borel–Moore homology of the base stack up to a certain shift of degree. This is a global version of the dimensional reduction theorem due to Davison. We give two applications of our main theorem. Firstly, we apply it to the study of the cohomological Donaldson–Thomas invariants for local surfaces. Secondly, regarding our main theorem as a version of the Thom isomorphism theorem for dual obstruction cones, we propose a sheaf-theoretic construction of the virtual fundamental classes for quasi-smooth derived Artin stacks.