Duke Mathematical Journal

Mirror symmetry for the Landau–Ginzburg A-model M=Cn, W=z1zn

David Nadler

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We calculate the category of branes in the Landau–Ginzburg A-model with background M=Cn and superpotential W=z1zn in the form of microlocal sheaves along a natural Lagrangian skeleton. Our arguments employ the framework of perverse schobers, and our results confirm expectations from mirror symmetry.

Article information

Source
Duke Math. J., Volume 168, Number 1 (2019), 1-84.

Dates
Received: 5 January 2017
Revised: 6 July 2018
First available in Project Euclid: 17 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1545037298

Digital Object Identifier
doi:10.1215/00127094-2018-0036

Mathematical Reviews number (MathSciNet)
MR3909893

Zentralblatt MATH identifier
07036279

Subjects
Primary: 53D37: Mirror symmetry, symplectic aspects; homological mirror symmetry; Fukaya category [See also 14J33]
Secondary: 32S30: Deformations of singularities; vanishing cycles [See also 14B07]

Keywords
microlocal sheaves mirror symmetry perverse schobers

Citation

Nadler, David. Mirror symmetry for the Landau–Ginzburg $A$ -model $M=\mathbb{C}^{n}$ , $W=z_{1}\cdots z_{n}$. Duke Math. J. 168 (2019), no. 1, 1--84. doi:10.1215/00127094-2018-0036. https://projecteuclid.org/euclid.dmj/1545037298


Export citation

References

  • [1] M. Abouzaid and D. Auroux, Homological mirror symmetry for hypersurfaces in $(\mathbb{C}^{*})^{n}$, in preparation.
  • [2] M. Abouzaid, D. Auroux, A. I. Efimov, L. Katzarkov, and D. Orlov, Homological mirror symmetry for punctured spheres, J. Amer. Math. Soc. 26 (2013), 1051–1083.
  • [3] M. Abouzaid, D. Auroux, and L. Katzarkov, Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces, Publ. Math. Inst. Hautes Études Sci. 123 (2016), 199–282.
  • [4] M. Abouzaid and P. Seidel, An open string analogue of Viterbo functoriality, Geom. Topol. 14 (2010), 627–718.
  • [5] R. Anno and T. Logvinenko. Spherical DG-functors, J. Eur. Math. Soc. (JEMS) 19 (2017), 2577–2656.
  • [6] D. Auroux, “Fukaya categories and bordered Heegaard-Floer homology” in Proceedings of the International Congress of Mathematicians, Volume II, Hindustan Book Agency, New Delhi, 2010, 917–941.
  • [7] A. Beilinson and V. Drinfeld, Quantization of Hitchin Hamiltonians and Hecke eigensheaves, preprint, https://www.math.uchicago.edu/mitya/langlands/hitchin/BD-hitchin.pdf (accessed 16 October 2018).
  • [8] D. Ben-Zvi, D. Nadler, and A. Preygel, Integral transforms for coherent sheaves, J. Eur. Math. Soc. (JEMS) 19 (2017), 3763–3812.
  • [9] J. Bernstein, Algebraic theory of $\mathcal{D}$-modules, preprint, https://www.math.uchicago.edu/mitya/langlands/Bernstein/Bernstein-dmod.ps (accessed 16 October 2018).
  • [10] A. Bondal, “Derived categories of toric varieties” in Convex and Algebraic Geometry, Oberwolfach Conf. Rep. 3, Eur. Math. Soc., Zürich, 2006, 284–286.
  • [11] T. Braden, N. Proudfoot, B. Webster, Quantizations of conical symplectic resolutions, I: Local and global structure, Astérisque 384 (2016), 1–73.
  • [12] T. Dyckerhoff and M. Kapranov, Triangulated surfaces in triangulated categories, J. Eur. Math. Soc. (JEMS) 20 (2018), 1473–1524.
  • [13] Y. Eliashberg, D. Nadler, and L. Starkston, Arboreal Weinstein skeleta, in preparation.
  • [14] B. Fang, Homological mirror symmetry is $T$-duality for $\mathbb{P}^{n}$, Commun. Number Theory Phys. 2 (2008), 719–742.
  • [15] B. Fang, C.-C. Liu, D. Treumann, and E. Zaslow, A categorification of Morelli’s theorem, Invent. Math. 186 (2011), 79–114.
  • [16] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Lagrangian Intersection Floer Theory: Anomaly and Obstruction, AMS/IP Stud. Adv. Math. 46, Amer. Math. Soc., Providence, 2009.
  • [17] D. Gaitsgory, Notes on geometric Langlands: Generalities on dg categories, preprint, 2012, http://abel.math.harvard.edu/~gaitsgde/GL/textDG.pdf.
  • [18] S. Ganatra, J. Pardon, and V. Shende, Localizing the Fukaya category of a Weinstein manifold, in preparation.
  • [19] V. Ginzburg, Perverse sheaves on a loop group and Langlands’ duality, preprint, arXiv:alg-geom/9511007v4 [math.AG].
  • [20] M. Kapranov and V. Schechtman, Perverse schobers, preprint, arXiv:1411.2772 [math.AG].
  • [21] M. Kashiwara and T. Kawai, On the holonomic systems of microdifferential equations, III, Publ. Res. Inst. Math. Sci. 17 (1981), 813–979.
  • [22] M. Kashiwara and P. Schapira, Sheaves on Manifolds, Grundlehren Math. Wiss. 292, Springer, Berlin, 1994.
  • [23] D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165–184.
  • [24] B. Keller, “On differential graded categories” in International Congress of Mathematicians, Vol. II, Eur. Math. Soc. (EMS), Zürich, 2006, 151–190.
  • [25] M. Kontsevich, Symplectic geometry of homological algebra, lecture notes, 2009, www.ihes.fr/~maxim/TEXTS/Symplectic_AT2009.pdf.
  • [26] T. Kuwagaki, The nonequivariant coherent-constructible correspondence for toric stacks, preprint, arXiv:1610.03214 [math.SG].
  • [27] G. Laumon, Faisceaux caractères (d’après Lusztig), Astérisque 177–178 (1989), 231–260, Séminaire Bourbaki 1988/1989, no. 709.
  • [28] G. Lusztig, Character sheaves, I, Adv. Math 56 (1985), 193–237.
  • [29] J. Lurie, Higher algebra, preprint, http://www.math.harvard.edu/~lurie/HA.pdf (accessed 16 October 2018).
  • [30] G. Mikhalkin, Decomposition into pairs-of-pants for complex algebraic hypersurfaces, Topology 43 (2004), 1035–1065.
  • [31] I. Mirković and K. Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. of Math. (2) 166 (2007), 95–143.
  • [32] D. Nadler, Microlocal branes are constructible sheaves, Selecta Math. (N.S.) 15 (2009), 563–619.
  • [33] D. Nadler, Cyclic symmetries of $A_{n}$-quiver representations, Adv. Math. 269 (2015), 346–363.
  • [34] D. Nadler, Arboreal singularities, Geom. Topol. 21 (2017), 1231–1274.
  • [35] D. Nadler, A combinatorial calculation of the Landau–Ginzburg model $M=\mathbb{C}^{3}$, $W=z_{1}z_{2}z_{3}$, Selecta Math. (N.S.) 23 (2017), 519–532.
  • [36] D. Nadler, Non-characteristic expansions of Legendrian singularities, preprint, arXiv:1507.01513v2 [math.SG].
  • [37] D. Nadler, Wrapped microlocal sheaves on pairs of pants, preprint, arXiv:1604.00114 [math.SG].
  • [38] D. Nadler and E. Zaslow, Constructible sheaves and the Fukaya category, J. Amer. Math. Soc. 22 (2009), 233–286.
  • [39] H. Ruddat, N. Sibilla, D. Treumann, and E. Zaslow, Skeleta of affine hypersurfaces, Geom. Topol. 18 (2014), 1343–1395.
  • [40] S. Scherotzke and N. Sibilla, The non-equivariant coherent-constructible correspondence and a conjecture of King, Selecta Math. (N.S.) 22 (2016), 389–416.
  • [41] P. Seidel, Graded Lagrangian submanifolds, Bull. Soc. Math. France 128 (2000), 103–149.
  • [42] P. Seidel, Fukaya Categories and Picard-Lefschetz Theory, Zurich Lect. Adv. Math., Eur. Math. Soc. (EMS), Zürich, 2008.
  • [43] P. Seidel, Homological mirror symmetry for the genus two curve, J. Algebraic Geom. 20 (2011), 727–769.
  • [44] V. Shende, D. Treumann, H. Williams, and E. Zaslow, Cluster varieties from Legendrian knots, preprint, arXiv:1512.08942 [math.SG].
  • [45] V. Shende, D. Treumann, and E. Zaslow, Legendrian knots and constructible sheaves, Invent. Math. 207 (2017), 1031–1133.
  • [46] N. Sheridan, On the homological mirror symmetry conjecture for pairs of pants, J. Differential Geom. 89 (2011), 271–367.
  • [47] D. Treumann, Remarks on the nonequivariant coherent-constructible correspondence for toric varieties, preprint, arXiv:1006.5756 [math.AG].
  • [48] D. Treumann and E. Zaslow, Cubic planar graphs and Legendrian surface theory, preprint, arXiv:1609.04892v2 [math.SG].
  • [49] D. Vaintrob, Microlocal mirror symmetry on the torus, preprint, 2017, http://www.math.ias.edu/~mitkav/just_torus.pdf.
  • [50] I. Waschkies, The stack of microlocal perverse sheaves, Bull. Soc. Math. France 132 (2004), 397–462.