Dimensional reduction in cohomological Donaldson-Thomas theory

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.

Automated summary

In this paper, we study perverse sheaves in oriented -1-shifted symplectic derived Artin stacks and prove the isomorphism between the hypercohomology of the perverse sheaf and the Borel-Moore homology of the base stack. We provide two applications of our main theorem.