Cohomological Hall algebras for 3-Calabi-Yau categories
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909999290122240 |
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| author | Kinjo, Tasuki Park, Hyeonjun Safronov, Pavel |
| author_facet | Kinjo, Tasuki Park, Hyeonjun Safronov, Pavel |
| contents | The aim of this paper is to construct the cohomological Hall algebras for $3$-Calabi--Yau categories admitting a strong orientation data. This can be regarded as a mathematical definition of the algebra of BPS states, whose existence was first mathematically conjectured by Kontsevich and Soibelman. Along the way, we prove Joyce's conjecture on the functorial behaviour of the Donaldson--Thomas perverse sheaves for the attractor Lagrangian correspondence of $(-1)$-shifted symplectic stacks. This result allows us to construct a parabolic induction map for cohomological Donaldson--Thomas invariants of $G$-local systems on $3$-manifolds for a reductive group $G$, which can be regarded as a $3$-manifold analogue of the Eisenstein series functor in the geometric Langlands program. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_12838 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cohomological Hall algebras for 3-Calabi-Yau categories Kinjo, Tasuki Park, Hyeonjun Safronov, Pavel Algebraic Geometry High Energy Physics - Theory Geometric Topology Representation Theory The aim of this paper is to construct the cohomological Hall algebras for $3$-Calabi--Yau categories admitting a strong orientation data. This can be regarded as a mathematical definition of the algebra of BPS states, whose existence was first mathematically conjectured by Kontsevich and Soibelman. Along the way, we prove Joyce's conjecture on the functorial behaviour of the Donaldson--Thomas perverse sheaves for the attractor Lagrangian correspondence of $(-1)$-shifted symplectic stacks. This result allows us to construct a parabolic induction map for cohomological Donaldson--Thomas invariants of $G$-local systems on $3$-manifolds for a reductive group $G$, which can be regarded as a $3$-manifold analogue of the Eisenstein series functor in the geometric Langlands program. |
| title | Cohomological Hall algebras for 3-Calabi-Yau categories |
| topic | Algebraic Geometry High Energy Physics - Theory Geometric Topology Representation Theory |
| url | https://arxiv.org/abs/2406.12838 |