Lagrangian Floer theory for trivalent graphs and homological mirror symmetry for curves

• Denis Auroux ^[1] ; Alexander I. Efimov ^[2] ; Ludmil Katzarkov ^[3]
  1. [1] Harvard University

     Harvard University

     City of Cambridge, Estados Unidos

     
  2. [2] Hebrew University of Jerusalem

     Hebrew University of Jerusalem

     Israel

     
  3. [3] Department of Mathematics, College of Arts and Sciences, University of Miami,USA

  Mostrar afiliaciones +
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 30, Nº. 5, 2024
• Idioma: inglés
• DOI: 10.1007/s00029-024-00988-6
• Enlaces
  • Texto completo
• Resumen

  • Mirror symmetry for higher genus curves is usually formulated and studied in terms of Landau–Ginzburg models; however the critical locus of the superpotential is arguably of greater intrinsic relevance to mirror symmetry than the whole Landau–Ginzburg model. Accordingly, we propose a new approach to the A-model of the mirror, viewed as a trivalent configuration of rational curves together with some extra data at the nodal points. In this context, we introduce a version of Lagrangian Floer theory and the Fukaya category for trivalent graphs, and show that homological mirror symmetry holds, namely, that the Fukaya category of a trivalent configuration of rational curves is equivalent to the derived category of a non-Archimedean generalized Tate curve. To illustrate the concrete nature of this equivalence, we show how explicit formulas for theta functions and for the canonical map of the curve arise naturally under mirror symmetry.

• Referencias bibliográficas
  • 1. Abouzaid, M.: A geometric criterion for generating the Fukaya category. Publ. Math. IHÉS 112, 191–240 (2010)
  • 2. Abouzaid, M., Auroux, D.: Homological mirror symmetry for hypersurfaces in (C∗)n, to appear in Geom. Topol., arXiv:2111.06543
  • 3. Abouzaid, M., Auroux, D., Efimov, A.I., Katzarkov, L., Orlov, D.: Homological mirror symmetry for punctured spheres. J. Am. Math. Soc....
  • 4. Abouzaid, M., Auroux, D., Katzarkov, L.: Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces. Publ....
  • 5. Abouzaid, M., Ganatra, S.: in preparation
  • 6. Abouzaid, M., Seidel, P.: Lefschetz fibration methods in wrapped Floer theory, in preparation
  • 7. Aganagic, M., Vafa, C.: Mirror symmetry, D-branes and counting holomorphic discs, arXiv:hep-th/0012041
  • 8. Auroux, D., Smith, I.: Fukaya categories of surfaces, spherical objects and mapping class groups. Forum Math. Sigma 9(e26), 1–50 (2021)
  • 9. Cannizzo, C.: Categorical mirror symmetry on cohomology for a complex genus 2 curve. Adv. Math. 375, 107392 (2020)
  • 10. Chan, K., Lau, S.-C., Leung, N.C.: SYZ mirror symmetry for toric Calabi-Yau manifolds. J. Differ. Geom. 90, 177–250 (2012)
  • 11. Chuang, J., Lazarev, A.: On the perturbation algebra. Journal of Algebra 519, 130–148 (2019)
  • 12. Clarke, P.: Duality for toric Landau-Ginzburg models, Adv. Theor. Mah. Phys. 21, 243-287 (2017), arXiv:0803.0447
  • 13. Efimov, A.I.: Homological mirror symmetry for curves of higher genus. Adv. Math. 230, 493–530 (2012)
  • 14. Etnyre, J.B., Ng, L.L., Sabloff, J.M.: Invariants of Legendrian knots and coherent orientations. J. Symplectic Geom. 1, 321–367 (2002)
  • 15. Faltings, G.: Semistable vector bundles on Mumford curves. Invent. Math. 74, 199–212 (1983)
  • 16. Gammage, B., Shende, V.: Mirror symmetry for very affine hypersurfaces, Acta Math. 229, 287-346 (2022), arXiv:1707.02959
  • 17. Gerritzen, L., van der Put, M.: Schottky groups and Mumford curves, Lect. Notes in Math. 817, Springer (1980)
  • 18. Gross, M., Katzarkov, L., Ruddat, H.: Towards mirror symmetry for varieties of general type. Adv. Math. 308, 208–275 (2017)
  • 19. Hanlon, A.: Monodromy of monomially admissible Fukaya-Seidel categories mirror to toric varieties. Adv. Math. 350, 662–746 (2019)
  • 20. Hicks, J.: Tropical Lagrangian hypersurfaces are unobstructed. J. Topol. 13, 1409–1454 (2020)
  • 21. Hori, K., Vafa, C.: Mirror symmetry, arXiv:hep-th/0002222
  • 22. Lee, H.: Homological mirror symmetry for open Riemann surfaces from pair-of-pants decompositions, arXiv:1608.04473
  • 23. Lekili, Y., Polishchuk, A.: Auslander orders over nodal stacky curves and partially wrapped Fukaya categories. J. Topol. 11, 615–644 (2018)
  • 24. Matessi, D.: Lagrangian pairs of pants. Int. Math. Res. Not. 2021, 11306–11356
  • 25. Mikhalkin, G.: Examples of tropical-to-Lagrangian correspondence. Eur. J. Math. 5, 1033–1066 (2019)
  • 26. Nadler, D.: Mirror symmetry for the Landau-Ginzburg A-model M = Cn, W = z1 ...zn. Duke Math. J. 168, 1–84 (2019)
  • 27. Orlov, D.: Triangulated categories of singularities and equivalences between Landau-Ginzburg models. Sb. Math. 197, 1827–1840 (2006)
  • 28. Polishchuk, A., Zaslow, E.: Categorical mirror symmetry: the elliptic curve. Adv. Theor. Math. Phys. 2, 443–470 (1998)
  • 29. Ruddat, H.: Perverse curves and mirror symmetry. J. Algebr. Geom. 26, 17–42 (2017)
  • 30. Seidel, P.: Fukaya categories and Picard-Lefschetz theory, Zurich Lect. in Adv. Math., European Math. Soc., Zürich (2008)
  • 31. Seidel, P.: Homological mirror symmetry for the genus two curve. J. Algebr. Geom. 20, 727–769 (2011)
  • 32. Sibilla, N., Treumann, D., Zaslow, E.: Ribbon graphs and mirror symmetry. Sel. Math. 20, 979–1002 (2014)