no. 3
Twisted Quasimaps and Symplectic Duality for Hypertoric Spaces
[Quasi-applications tordues et dualité symplectique pour espaces hypertoriques]
McBreen, Michael1 ; Sheshmani, Artan2, 3 ; Yau, Shing-Tung4
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, Tome 75 (2025) no. 3, pp. 1225-1269.

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.

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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
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

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)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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McBreen, Michael; Sheshmani, Artan; Yau, Shing-Tung. Twisted Quasimaps and Symplectic Duality for Hypertoric Spaces. Annales de l'Institut Fourier, Tome 75 (2025) no. 3, pp. 1225-1269. doi : 10.5802/aif.3681. https://aif.centre-mersenne.org/articles/10.5802/aif.3681/

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