On the P = W conjecture for $$SL_n$$

Cataldo, Mark Andrea de; Maulik, Davesh; Shen, Junliang

doi:10.1007/s00029-022-00803-0

Cited by 7 publications

(2 citation statements)

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“…The topology -specifically the homology and cohomology -of Hitchin moduli spaces and character varieties has been the subject of much work beginning with a study of SYZ-type [1028] mirror symmetry and its relation to Langlands duality [1029]. In another direction, a very general conjecture known as the "P = W " conjecture has been extensively studied in [1030][1031][1032][1033][1034][1035][1036][1037][1038][1039][1040][1041][1042]. The results can be interpreted very nicely using BPS states associated with string theory compactification on local Calabi-Yau manifolds [1043][1044][1045][1046].…”

Section: Hyperkähler and Quaternionic Kähler Geometrymentioning

confidence: 99%

A Panorama Of Physical Mathematics c. 2022

Bah¹,

Freed²,

Moore³

et al. 2022

Preprint

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What follows is a broad-brush overview of the recent synergistic interactions between mathematics and theoretical physics of quantum field theory and string theory. The discussion is forwardlooking, suggesting potentially useful and fruitful directions and problems, some old, some new, for further development of the subject. This paper is a much extended version of the Snowmass whitepaper on physical mathematics [1].

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Section: Hyperkähler and Quaternionic Kähler Geometrymentioning

confidence: 99%

A Panorama Of Physical Mathematics c. 2022

Bah¹,

Freed²,

Moore³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Subsequently, de Cataldo, Maulik, and Shen [7] proved the conjecture for

\mathrm{GL}(n)

‐Hitchin systems on genus two curves by using an analogue of Donagi, Ein, and Lazarsfeld's deformation coming from the deformation of an abelian surface to the normal cone of an embedded curve. The Debarre integrable system on the generalized Kummer variety [9] deforms to a compactification of the

\mathrm{SL}

‐Hitchin system on a genus two curve; however, de Cataldo, Maulik, and Shen [8] showed that there are obstructions to using this deformation to prove the P = W conjecture in this case. Note that these results are for twisted Hitchin systems, where the moduli spaces are smooth.…”

Section: Introductionmentioning

confidence: 99%

Deformations of compact Prym fibrations to Hitchin systems

Sawon

Shen

2022

Bulletin of London Math Soc

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Purity and 2-Calabi–Yau categories

Davison

2024

Invent. math.

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For various 2-Calabi–Yau categories $\mathscr{C}$ C for which the classical stack of objects $\mathfrak{M}$ M has a good moduli space $p\colon \mathfrak{M}\rightarrow \mathcal{M}$ p : M → M , we establish purity of the mixed Hodge module complex $p_{!}\underline{{\mathbb{Q}}}_{{\mathfrak {M}}}$ p ! Q _ M . We do this by using formality in 2CY categories, along with étale neighbourhood theorems for stacks, to prove that the morphism $p$ p is modelled étale-locally by the semisimplification morphism from the stack of modules of a preprojective algebra. Via the integrality theorem in cohomological Donaldson–Thomas theory we then prove purity of $p_{!}\underline{{\mathbb{Q}}}_{{\mathfrak {M}}}$ p ! Q _ M . It follows that the Beilinson–Bernstein–Deligne–Gabber decomposition theorem for the constant sheaf holds for the morphism $p$ p , despite the possibly singular and stacky nature of ${\mathfrak {M}}$ M , and the fact that $p$ p is not proper. We use this to define cuspidal cohomology for ${\mathfrak {M}}$ M , which conjecturally provides a complete space of generators for the BPS algebra associated to $\mathscr{C}$ C . We prove purity of the Borel–Moore homology of the moduli stack $\mathfrak{M}$ M , provided its good moduli space ℳ is projective, or admits a suitable contracting ${\mathbb{C}}^{*}$ C ∗ -action. In particular, when $\mathfrak{M}$ M is the moduli stack of Gieseker semistable sheaves on a K3 surface, this proves a conjecture of Halpern-Leistner. We use these results to moreover prove purity for several stacks of coherent sheaves that do not admit a good moduli space. Without the usual assumption that $r$ r and $d$ d are coprime, we prove that the Borel–Moore homology of the stack of semistable degree $d$ d rank $r$ r Higgs sheaves is pure and carries a perverse filtration with respect to the Hitchin base, generalising the usual perverse filtration for the Hitchin system to the case of singular stacks of Higgs sheaves.

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On the P = W conjecture for $$SL_n$$

Cited by 7 publications

References 22 publications

A Panorama Of Physical Mathematics c. 2022

A Panorama Of Physical Mathematics c. 2022

Deformations of compact Prym fibrations to Hitchin systems

Purity and 2-Calabi–Yau categories

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