P=W conjectures for character varieties with symplectic resolution

Felisetti, Camilla; Mauri, Mirko

doi:10.5802/jep.196

Cited by 13 publications

(6 citation statements)

References 81 publications

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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

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show abstract

“…Though the conjecture is formulated for smooth moduli spaces, it would be interesting to see whether an analogue of the conjecture exists in the singular case of moduli of Higgs bundles of non-co-prime rank and degree and the corresponding character varieties. Indeed, for moduli spaces with a symplectic resolution, the conjecture has been proved by the author and Mauri in [15], relying on the results of this article.…”

Section: Introductionmentioning

confidence: 99%

Intersection Cohomology of the Moduli Space of Higgs Bundles on a Genus 2 Curve

Felisetti

2021

J. Inst. Math. Jussieu

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Let C be a smooth projective curve of genus $2$ . Following a method by O’Grady, we construct a semismall desingularisation $\tilde {\mathcal {M}}_{Dol}^G$ of the moduli space $\mathcal {M}_{Dol}^G$ of semistable G-Higgs bundles of degree 0 for $G=\mathrm {GL}(2,\mathbb {C}), \mathrm {SL}(2,\mathbb {C})$ . By the decomposition theorem of Beilinson, Bernstein and Deligne, one can write the cohomology of $\tilde {\mathcal {M}}_{Dol}^G$ as a direct sum of the intersection cohomology of $\mathcal {M}_{Dol}^G$ plus other summands supported on the singular locus. We use this splitting to compute the intersection cohomology of $\mathcal {M}_{Dol}^G$ and prove that the mixed Hodge structure on it is pure, in analogy with what happens to ordinary cohomology in the smooth case of coprime rank and degree.

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“…Mauri [Mau21a] generalized the computation to rank n = 2 and arbitrary genus. Felisetti-Mauri [FM22] proved the P = W conjecture for intersection cohomology in genus g = 1 and arbitrary rank n, and in genus g = 2 and rank n = 2. Mauri [Mau21b] also studied topological mirror symmetry for these varieties, in rank n = 2.…”

Section: Introductionmentioning

confidence: 99%

Intersection cohomology of character varieties for punctured Riemann surfaces

Mathieu¹

2023

Journal De L’École Polytechnique — Mathématiques

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We study intersection cohomology of character varieties for punctured Riemann surfaces with prescribed monodromies around the punctures. Relying on a previous result from Mellit [Mel20a] for semisimple monodromies we compute the intersection cohomology of character varieties with monodromies of any Jordan type. This proves the Poincaré polynomial specialization of a conjecture from Letellier [Let15]. Résumé (Cohomologie d'intersection des variétés de caractères des surfaces de Riemann épointées)Nous étudions la cohomologie d'intersection des variétés de caractères des surfaces de Riemann épointées, la monodromie autour des points enlevés étant fixée. En nous appuyant sur un résultat de Mellit [Mel20a] pour des monodromies semi-simples, nous calculons la cohomologie d'intersection des variétés de caractères avec des monodromies ayant un type de Jordan quelconque. Ceci prouve la spécialisation au polynôme de Poincaré d'une conjecture de Letellier [Let15].

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P=W conjectures for character varieties with symplectic resolution

Cited by 13 publications

References 81 publications

A Panorama Of Physical Mathematics c. 2022

A Panorama Of Physical Mathematics c. 2022

Intersection Cohomology of the Moduli Space of Higgs Bundles on a Genus 2 Curve

Intersection cohomology of character varieties for punctured Riemann surfaces

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