The BPS decomposition theorem

We prove the BPS decomposition theorem (a.k.a. cohomological integrality theorem) decomposing the cohomology of smooth symmetric stacks into the Weyl-invariant part of the cohomological Hall induction of the intersection cohomology of good moduli spaces. As a consequence, we establish the BPS decomposition theorem for the Borel--Moore homology of $0$-shifted symplectic stacks and for the critical cohomology of symmetric $(-1)$-shifted symplectic stacks, thereby generalizing the main result of Bu--Davison--Ib\'a\~nez Nu\~nez--Kinjo--P\u{a}durariu to the non-orthogonal setting. We will present three applications of our main result. First, we confirm Halpern-Leistner's conjecture on the purity of the Borel--Moore homology of $0$-shifted symplectic stacks admitting proper good moduli spaces, extending Davison's work on the moduli stack of objects in $2$-Calabi--Yau categories. Second, we prove versions of Kirwan surjectivity for the critical cohomology of symmetric $(-1)$-shifted symplectic stacks and for the Borel--Moore homology of $0$-shifted symplectic stacks. Finally, by applying our main result to the character stacks associated with compact oriented $3$-manifolds, we reduce the quantum geometric Langlands duality conjecture for $3$-manifolds, as formulated by Safronov, from an isomorphism between infinite-dimensional critical cohomologies to a comparison of finite-dimensional BPS cohomologies.