Twisted Quasimaps and Symplectic Duality for Hypertoric Spaces

McBreen, Michael; Sheshmani, Artan; Yau, Shing-Tung

doi:10.5802/aif.3681

Twisted Quasimaps and Symplectic Duality for Hypertoric Spaces
[Quasi-applications tordues et dualité symplectique pour espaces hypertoriques]

McBreen, Michael ¹ ; Sheshmani, Artan ^2,³ ; Yau, Shing-Tung ⁴

¹ Chinese University of Hong Kong, Sha Tin, N.T., Hong Kong (China)
² Massachusetts Institute of Technology (MIT), IAiFi Institute, 182 Memorial Drive, Cambridge, MA 02139 (USA)
³ Yanqi Lake Beijing Institute for Mathematical Sciences and Applications (BIMSA). Huairou, Beijing, 101408 National Research University Higher School of Economics, Russian Federation, Laboratory of Mirror Symmetry, NRU HSE, 6 Usacheva str., Moscow (Russia), 119048
⁴ Yau center of mathematics, Tsinghua university, Beijing (China)

Annales de l'Institut Fourier, Online first, 45 p.

Résumé
Abstract

Nous étudions l’espace de modules des quasi-applications tordues vers une variété hypertorique $X$ , branche de Higgs d’une théorie de jauge supersymétrique abélienne en dimension 3. Ces variétés paramétrisent des systèmes stables d’applications entre faisceaux de rang 1 sur $¶^{1}$ . Nous identifions la cohomologie de ces espaces avec le groupe Ext d’une paire de modules holonomes d’un “espace de lacets quantique” de $X$ , qui apparaît comme branche de Higgs d’une théorie avec un nombre infini de champs de matière. Sa branche de Coulomb est un analogue périodique de la branche de Coulomb associée à $X$ . La dualité symplectique nous permet d’obtenir une formule pour la fonction génératrice des invariants des quasi-applications tordues, utilisant le caractère d’un module basculant sur la branche de Coulomb périodique. Nous donnons une formule close lorsque $X$ est l’abélianisation du carquois associé au cotangent d’une variété de drapeaux.

We study moduli spaces of twisted quasimaps to a hypertoric variety $X$ , arising as the Higgs branch of an abelian supersymmetric 3D gauge theory. These parametrize systems of maps between rank one sheaves on $¶^{1}$ , subject to a stability condition. We identify the singular cohomology of these moduli spaces with the Ext group of a pair of holonomic modules over the “quantized loop space” of $X$ , which we view as a Higgs branch for a related theory with infinitely many matter fields. We construct the coulomb branch of this theory, as a periodic analogue of the coulomb branch associated to $X$ . Using the formalism of symplectic duality, we derive an expression for the generating function of twisted quasimap invariants in terms of the character of a certain tilting module on the periodic coulomb branch. We give a closed formula when $X$ arises as the abelianisation of the $N$ -step flag quiver.

Reçu le : 2021-08-05
Révisé le : 2023-04-20
Accepté le : 2023-08-04
Première publication : 2025-02-10

DOI : 10.5802/aif.3681

Classification : 14N35, 14M25, 14J33, 53D30, 53D55
Keywords: Hypertoric variety, Quasimaps, Symplectic Duality, 3D mirror symmetry
Mots-clés : Variétés hypertoriques, quasi-applications, dualité symplectique, symétrie miroir 3D

Affiliations des auteurs :

McBreen, Michael ¹ ; Sheshmani, Artan ^{2, 3} ; Yau, Shing-Tung ⁴

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     title = {Twisted {Quasimaps} and {Symplectic} {Duality} for {Hypertoric} {Spaces}},
     journal = {Annales de l'Institut Fourier},
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McBreen, Michael; Sheshmani, Artan; Yau, Shing-Tung. Twisted Quasimaps and Symplectic Duality for Hypertoric Spaces. Annales de l'Institut Fourier, Online first, 45 p.

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