no. 3
Twisted Quasimaps and Symplectic Duality for Hypertoric Spaces
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, Volume 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.

Received:
Revised:
Accepted:
Published online:
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)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{AIF_2025__75_3_1225_0,
     author = {McBreen, Michael and Sheshmani, Artan and Yau, Shing-Tung},
     title = {Twisted {Quasimaps} and {Symplectic} {Duality} for {Hypertoric} {Spaces}},
     journal = {Annales de l'Institut Fourier},
     pages = {1225--1269},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {75},
     number = {3},
     year = {2025},
     doi = {10.5802/aif.3681},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3681/}
}
TY  - JOUR
AU  - McBreen, Michael
AU  - Sheshmani, Artan
AU  - Yau, Shing-Tung
TI  - Twisted Quasimaps and Symplectic Duality for Hypertoric Spaces
JO  - Annales de l'Institut Fourier
PY  - 2025
SP  - 1225
EP  - 1269
VL  - 75
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3681/
DO  - 10.5802/aif.3681
LA  - en
ID  - AIF_2025__75_3_1225_0
ER  - 
%0 Journal Article
%A McBreen, Michael
%A Sheshmani, Artan
%A Yau, Shing-Tung
%T Twisted Quasimaps and Symplectic Duality for Hypertoric Spaces
%J Annales de l'Institut Fourier
%D 2025
%P 1225-1269
%V 75
%N 3
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3681/
%R 10.5802/aif.3681
%G en
%F AIF_2025__75_3_1225_0
McBreen, Michael; Sheshmani, Artan; Yau, Shing-Tung. Twisted Quasimaps and Symplectic Duality for Hypertoric Spaces. Annales de l'Institut Fourier, Volume 75 (2025) no. 3, pp. 1225-1269. doi : 10.5802/aif.3681. https://aif.centre-mersenne.org/articles/10.5802/aif.3681/

[1] Aganagic, M.; Frenkel, E.; Okounkov, A. Quantum q-Langlands correspondence, Trans. Mosc. Math. Soc., Volume 79 (2018), pp. 1-83 | DOI | MR | Zbl

[2] Anderson, Michael T.; Kronheimer, Peter B.; LeBrun, Claude Complete Ricci-flat Kähler manifolds of infinite topological type, Comm. Math. Phys., Volume 125 (1989) no. 4, pp. 637-642 http://projecteuclid.org/euclid.cmp/1104179632 | DOI | MR | Zbl

[3] Anno, Rina; Bezrukavnikov, Roman; Mirković, Ivan Stability conditions for Slodowy slices and real variations of stability, Mosc. Math. J., Volume 15 (2015) no. 2, pp. 187-203 | DOI | MR | Zbl

[4] Beilinson, A; Ginsburg, Victor Mixed categories, Ext-duality and representations (results and conjectures), 1986

[5] Beilinson, Alexander; Ginzburg, Victor; Soergel, Wolfgang Koszul duality patterns in representation theory, J. Amer. Math. Soc., Volume 9 (1996) no. 2, pp. 473-527 | DOI | MR | Zbl

[6] Bezrukavnikov, R.; Kaledin, D. Fedosov quantization in algebraic context, Mosc. Math. J., Volume 4 (2004) no. 3, pp. 559-592 | DOI | MR | Zbl

[7] Bezrukavnikov, R; Okounkov, A Monodromy and derived equivalences preparation (cit. on pp. 859, 860)

[8] Bielawski, Roger; Dancer, Andrew S. The geometry and topology of toric hyperkähler manifolds, Comm. Anal. Geom., Volume 8 (2000) no. 4, pp. 727-760 | DOI | MR | Zbl

[9] Braden, Tom; Licata, Anthony; Proudfoot, Nicholas; Webster, Ben Gale duality and Koszul duality, Adv. Math., Volume 225 (2010) no. 4, pp. 2002-2049 | DOI | MR | Zbl

[10] Braden, Tom; Licata, Anthony; Proudfoot, Nicholas; Webster, Ben Hypertoric category 𝒪, Adv. Math., Volume 231 (2012) no. 3-4, pp. 1487-1545 | DOI | MR | Zbl

[11] Braden, Tom; Licata, Anthony; Proudfoot, Nicholas; Webster, Ben Quantizations of conical symplectic resolutions II: category 𝒪 and symplectic duality. With an appendix by I. Losev, Astérisque, 384, Société Mathématique de France, 2016, pp. 75-179 | MR | Zbl

[12] Braverman, Alexander; Finkelberg, Michael; Nakajima, Hiraku Towards a mathematical definition of Coulomb branches of 3-dimensional 𝒩=4 gauge theories, II, Adv. Theor. Math. Phys., Volume 22 (2018) no. 5, pp. 1071-1147 | DOI | MR | Zbl

[13] Bruhn, Henning; Diestel, Reinhard; Kriesell, Matthias; Pendavingh, Rudi; Wollan, Paul Axioms for infinite matroids, Adv. Math., Volume 239 (2013), pp. 18-46 | DOI | MR | Zbl

[14] Bullimore, Mathew; Dimofte, Tudor; Gaiotto, Davide; Hilburn, Justin; Kim, Hee-Cheol Vortices and Vermas, Adv. Theor. Math. Phys., Volume 22 (2018) no. 4, pp. 803-917 | DOI | MR | Zbl

[15] Ciocan-Fontanine, Ionuţ; Kim, Bumsig; Maulik, Davesh Stable quasimaps to GIT quotients, J. Geom. Phys., Volume 75 (2014), pp. 17-47 | DOI | MR | Zbl

[16] Dancer, Andrew; Swann, Andrew Hypertoric manifolds and hyperkähler moment maps, Special metrics and group actions in geometry (Springer INdAM Ser.), Volume 23, Springer, Cham, 2017, pp. 107-127 | DOI | MR | Zbl

[17] Dancso, Zuszsanna; McBreen, Michael; Shende, Vivek Deletion-contraction triangles for Hausel–Proudfoot varieties, J. Eur. Math. Soc., 2023 | DOI | MR | Zbl

[18] De Wilde, Marc; Lecomte, Pierre B. A. Existence of star-products and of formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds, Lett. Math. Phys., Volume 7 (1983) no. 6, pp. 487-496 | DOI | MR | Zbl

[19] Etingof, Pavel; Ginzburg, Victor Symplectic reflection algebras, Calogero–Moser space, and deformed Harish–Chandra homomorphism, Invent. Math., Volume 147 (2002) no. 2, pp. 243-348 | DOI | MR | Zbl

[20] Fedosov, Boris V A simple geometrical construction of deformation quantization, J. Differ. Geom., Volume 40 (1994) no. 2, pp. 213-238 http://projecteuclid.org/euclid.jdg/1214455536 | DOI | MR | Zbl

[21] Goto, Ryushi On toric hyper-Kähler manifolds given by the hyper-Kähler quotient method, Infinite analysis, Part A, B (Kyoto, 1991) (Adv. Ser. Math. Phys.), Volume 16, World Sci. Publ., River Edge, NJ, 1992, pp. 317-338 | DOI | MR | Zbl

[22] Hattori, Kota The volume growth of hyper-Kähler manifolds of type A ∞ , J. Geom. Anal., Volume 21 (2011) no. 4, pp. 920-949 | DOI | MR | Zbl

[23] Hausel, Tamás; Proudfoot, Nicholas Abelianization for hyperkähler quotients, Topology, Volume 44 (2005) no. 1, pp. 231-248 | DOI | MR | Zbl

[24] Hikita, Tatsuyuki An algebro-geometric realization of the cohomology ring of Hilbert scheme of points in the affine plane, Int. Math. Res. Not. IMRN (2017) no. 8, pp. 2538-2561 | DOI | MR | Zbl

[25] Hilburn, Justin; Kamnitzer, Joel; Weekes, Alex BFN Springer theory, Comm. Math. Phys., Volume 402 (2023) no. 1, pp. 765-832 | DOI | MR | Zbl

[26] Intriligator, K.; Seiberg, N. Mirror symmetry in three-dimensional gauge theories, Phys. Lett. B, Volume 387 (1996) no. 3, pp. 513-519 | DOI | MR | Zbl

[27] Kamnitzer, Joel; McBreen, Michael; Proudfoot, Nicholas The quantum Hikita conjecture, Adv. Math., Volume 390 (2021), 107947, 53 pages | DOI | MR | Zbl

[28] Kamnitzer, Joel; Tingley, Peter; Webster, Ben; Weekes, Alex; Yacobi, Oded Highest weights for truncated shifted Yangians and product monomial crystals (2015) (https://arxiv.org/abs/1511.09131)

[29] Kim, Bumsig Stable quasimaps to holomorphic symplectic quotients, Schubert calculus—Osaka 2012 (Adv. Stud. Pure Math.), Volume 71, Math. Soc. Japan, Tokyo (2016), pp. 139-160 | DOI | MR | Zbl

[30] Losev, Ivan Isomorphisms of quantizations via quantization of resolutions, Adv. Math., Volume 231 (2012) no. 3-4, pp. 1216-1270 | DOI | MR | Zbl

[31] Maulik, Davesh; Okounkov, Andrei Quantum groups and quantum cohomology, Astérisque, Société Mathématique de France, 2019 no. 408, ix+209 pages | DOI | MR | Zbl

[32] Mazorchuk, Volodymyr; Ovsienko, Serge; Stroppel, Catharina Quadratic duals, Koszul dual functors, and applications, Trans. Amer. Math. Soc., Volume 361 (2009) no. 3, pp. 1129-1172 | DOI | MR | Zbl

[33] McBreen, Michael; Webster, Ben Homological Mirror Symmetry for Hypertoric Varieties I (2018) (https://arxiv.org/abs/1804.10646)

[34] Musson, Ian M.; Van den Bergh, Michel Invariants under tori of rings of differential operators and related topics, Mem. Amer. Math. Soc., 136, American Mathematical Society, Providence, RI, 1998 no. 650, viii+85 pages | DOI | MR | Zbl

[35] Nakajima, Hiraku Towards a mathematical definition of Coulomb branches of 3-dimensional 𝒩=4 gauge theories, I, Adv. Theor. Math. Phys., Volume 20 (2016) no. 3, pp. 595-669 | DOI | MR | Zbl

[36] Premet, Alexander Special transverse slices and their enveloping algebras, Adv. Math., Volume 170 (2002) no. 1, pp. 1-55 (With an appendix by Serge Skryabin) | DOI | MR | Zbl

[37] Proudfoot, Nicholas J. A survey of hypertoric geometry and topology, Toric topology. International conference, Osaka, Japan, May 28–June 3, 2006 (Contemp. Math.), Volume 460, American Mathematical Society, Providence, RI (2008), pp. 323-338 | DOI | MR | Zbl

[38] Pushkar, Petr P.; Smirnov, Andrey V.; Zeitlin, Anton M. Baxter Q-operator from quantum K-theory, Adv. Math., Volume 360 (2020), p. 106919, 63 | DOI | MR | Zbl

[39] Soergel, Wolfgang Kategorie 𝒪, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc., Volume 3 (1990) no. 2, pp. 421-445 | DOI | MR | Zbl

Cited by Sources: