Homological mirror symmetry for hypertoric varieties, I: Conic equivariant sheaves

McBreen, Michael¹ ; Webster, Ben²

¹ Department of Mathematics, Chinese University of Hong Kong, Hong Kong
² Department of Pure Mathematics, University of Waterloo, Waterloo, ON, Canada, Perimeter Institute for Theoretical Physics, Waterloo, ON, Canada

Geometry & topology, Tome 28 (2024) no. 3, pp. 1005-1063.

Résumé

We consider homological mirror symmetry in the context of hypertoric varieties, showing that an appropriate category of $B$–branes (that is, coherent sheaves) on an additive hypertoric variety matches a category of $A$–branes on a Dolbeault hypertoric manifold for the same underlying combinatorial data. For technical reasons, the $A$–branes we consider are modules over a deformation quantization (that is, DQ–modules). We consider objects in this category equipped with an analogue of a Hodge structure, which corresponds to a ${\mathbb{𝔾}}_m$–action on the dual side of the mirror symmetry.

This result is based on hands-on calculations in both categories. We analyze coherent sheaves by constructing a tilting generator, using the characteristic $p$ approach of Kaledin; the result is a sum of line bundles, which can be described using a simple combinatorial rule. The endomorphism algebra $H$ of this tilting generator has a simple quadratic presentation in the grading induced by ${\mathbb{𝔾}}_m$–equivariance. In fact, we can confirm it is Koszul, and compute its Koszul dual $H^{!}$.

We then show that this same algebra appears as an Ext–algebra of simple $A$–branes in a Dolbeault hypertoric manifold. The ${\mathbb{𝔾}}_m$–equivariant grading on coherent sheaves matches a Hodge grading in this category.

Détail

DOI : 10.2140/gt.2024.28.1005

Keywords: mirror symmetry, hypertoric, characteristic $p$, deformation quantization

Bibliographie

[1] M Abouzaid, On homological mirror symmetry for toric varieties, PhD thesis, University of Chicago (2007)

[2] R Anno, R Bezrukavnikov, I Mirković, Stability conditions for Slodowy slices and real variations of stability, Mosc. Math. J. 15 (2015) 187 | DOI

[3] D Arapura, Mixed Hodge structures associated to geometric variations, from: "Cycles, motives and Shimura varieties" (editor V Srinivas), Tata Inst. Fund. Res. Stud. Math. 21, Narosa (2010) 1

[4] A Beilinson, V Ginzburg, W Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996) 473 | DOI

[5] R Bezrukavnikov, Noncommutative counterparts of the Springer resolution, from: "International Congress of Mathematicians, II" (editors M Sanz-Solé, J Soria, J L Varona, J Verdera), Eur. Math. Soc. (2006) 1119

[6] R Bezrukavnikov, D Kaledin, Fedosov quantization in positive characteristic, J. Amer. Math. Soc. 21 (2008) 409 | DOI

[7] T Braden, A Licata, N Proudfoot, B Webster, Gale duality and Koszul duality, Adv. Math. 225 (2010) 2002 | DOI

[8] T Braden, A Licata, N Proudfoot, B Webster, Hypertoric category O, Adv. Math. 231 (2012) 1487 | DOI

[9] T Braden, A Licata, N Proudfoot, B Webster, Quantizations of conical symplectic resolutions, II : Category O and symplectic duality, Astérisque 384, Soc. Math. France (2016) 75 | DOI

[10] T Braden, N Proudfoot, B Webster, Quantizations of conical symplectic resolutions, I : Local and global structure, Astérisque 384, Soc. Math. France (2016) 1 | DOI

[11] A Braverman, M Finkelberg, H Nakajima, Towards a mathematical definition of Coulomb branches of 3–dimensional N = 4 gauge theories, II, Adv. Theor. Math. Phys. 22 (2018) 1071 | DOI

[12] A Braverman, D Maulik, A Okounkov, Quantum cohomology of the Springer resolution, Adv. Math. 227 (2011) 421 | DOI

[13] J Brundan, N Davidson, Categorical actions and crystals, from: "Categorification and higher representation theory" (editors A Beliakova, A D Lauda), Contemp. Math. 683, Amer. Math. Soc. (2017) 105 | DOI

[14] M Bullimore, T Dimofte, D Gaiotto, J Hilburn, Boundaries, mirror symmetry, and symplectic duality in 3D N = 4 gauge theory, J. High Energy Phys. (2016) 108 | DOI

[15] A Cannas Da Silva, Lectures on symplectic geometry, 1764, Springer (2001) | DOI

[16] Z Dancso, M Mcbreen, V Shende, Deletion-contraction triangles for Hausel–Proudfoot varieties, preprint (2019)

[17] P Deligne, Un théorème de finitude pour la monodromie, from: "Discrete groups in geometry and analysis" (editor R Howe), Progr. Math. 67, Birkhäuser (1987) 1 | DOI

[18] P Freyd, Representations in abelian categories, from: "Proceedings of the Conference on Categorical Algebra" (editors S Eilenberg, D K Harrison, H Röhrl), Springer (1966) 95 | DOI

[19] K Fukaya, Y G Oh, H Ohta, K Ono, Lagrangian intersection Floer theory : anomaly and obstruction, I, 46.1, Amer. Math. Soc. (2009) | DOI

[20] K Fukaya, Y G Oh, H Ohta, K Ono, Lagrangian intersection Floer theory : anomaly and obstruction, II, 46.2, Amer. Math. Soc. (2009) | DOI

[21] B Gammage, M Mcbreen, B Webster, Homological mirror symmetry for hypertoric varieties, II, preprint (2019)

[22] I Ganev, Quantizations of multiplicative hypertoric varieties at a root of unity, J. Algebra 506 (2018) 92 | DOI

[23] M Groechenig, M Mcbreen, Hypertoric Hitchin systems and Kirchhoff polynomials, Int. Math. Res. Not. 2022 (2022) 15271 | DOI

[24] M Gross, P M H Wilson, Large complex structure limits of K3 surfaces, J. Differential Geom. 55 (2000) 475

[25] D Kaledin, Derived equivalences by quantization, Geom. Funct. Anal. 17 (2008) 1968 | DOI

[26] M Kashiwara, R Rouquier, Microlocalization of rational Cherednik algebras, Duke Math. J. 144 (2008) 525 | DOI

[27] S C Lau, X Zheng, SYZ mirror symmetry for hypertoric varieties, Comm. Math. Phys. 373 (2020) 1133 | DOI

[28] D Maulik, A Okounkov, Quantum groups and quantum cohomology, 408, Soc. Math. France (2019) | DOI

[29] V Mazorchuk, S Ovsienko, C Stroppel, Quadratic duals, Koszul dual functors, and applications, Trans. Amer. Math. Soc. 361 (2009) 1129 | DOI

[30] M B Mcbreen, D K Shenfeld, Quantum cohomology of hypertoric varieties, Lett. Math. Phys. 103 (2013) 1273 | DOI

[31] I M Musson, M Van Den Bergh, Invariants under tori of rings of differential operators and related topics, 650, Amer. Math. Soc. (1998) | DOI

[32] H Nakajima, Towards a mathematical definition of Coulomb branches of 3–dimensional N = 4 gauge theories, I, Adv. Theor. Math. Phys. 20 (2016) 595 | DOI

[33] N J Proudfoot, A survey of hypertoric geometry and topology, from: "Toric topology" (editors M Harada, Y Karshon, M Masuda, T Panov), Contemp. Math. 460, Amer. Math. Soc. (2008) 323 | DOI

[34] M Saito, A young person’s guide to mixed Hodge modules, from: "Hodge theory and –analysis" (editor L Ji), Adv. Lect. Math. 39, International (2017) 517

[35] T J Stadnik Jr., Étale splittings of certain Azumaya algebras on toric and hypertoric varieties in positive characteristic, Adv. Math. 233 (2013) 268 | DOI

[36] B Webster, Coherent sheaves and quantum Coulomb branches, I: Tilting bundles from integrable systems, preprint (2019)

[37] B Webster, Coherent sheaves and quantum Coulomb branches, II: Quiver gauge theories and knot homology, preprint (2022)