The BPS decomposition theorem
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arXiv
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| Format: | Preprint |
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2025
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| author | Hennecart, Lucien Kinjo, Tasuki |
| author_facet | Hennecart, Lucien Kinjo, Tasuki |
| contents | 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--Ibáñez Nuñez--Kinjo--Pădurariu 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_21298 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The BPS decomposition theorem Hennecart, Lucien Kinjo, Tasuki Algebraic Geometry Geometric Topology Representation Theory 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--Ibáñez Nuñez--Kinjo--Pădurariu 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. |
| title | The BPS decomposition theorem |
| topic | Algebraic Geometry Geometric Topology Representation Theory |
| url | https://arxiv.org/abs/2509.21298 |