Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-22T21:58:56.513Z Has data issue: false hasContentIssue false
  • English
  • Français

Quantum mirrors of log Calabi–Yau surfaces and higher-genus curve counting

Part of: Projective and enumerative geometry Surfaces and higher-dimensional varieties Symplectic geometry, contact geometry

Published online by Cambridge University Press:  07 January 2020

Pierrick Bousseau
Pierrick Bousseau*
Affiliation:
Department of Mathematics, Imperial College London, LondonSW7 2AZ, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

Gross, Hacking and Keel have constructed mirrors of log Calabi–Yau surfaces in terms of counts of rational curves. Using $q$-deformed scattering diagrams defined in terms of higher-genus log Gromov–Witten invariants, we construct deformation quantizations of these mirrors and we produce canonical bases of the corresponding non-commutative algebras of functions.

Keywords

mirror symmetrydeformation quantizationGromov–Witten invariants

MSC classification

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants 14J33: Mirror symmetry 14J81: Relationships with physics 53D55: Deformation quantization, star products
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 2 , February 2020 , pp. 360 - 411
Copyright
© The Author 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

1

Current address: Institute for Theoretical Studies, ETH Zurich, 8092 Zurich, Switzerland email [email protected]

This work is supported by EPSRC award 1513338, ‘Counting curves in algebraic geometry’, Imperial College London, and has benefited from the EPRSC [EP/L015234/1], EPSRC Centre for Doctoral Training in Geometry and Number Theory (London School of Geometry and Number Theory), University College London.

References

Abouzaid, M., Auroux, D. and Katzarkov, L., Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces, Publ. Math. Inst. Hautes Études Sci. 123 (2016), 199–282; MR 3502098.CrossRefGoogle Scholar
Abramovich, D. and Chen, Q., Stable logarithmic maps to Deligne-Faltings pairs II, Asian J. Math. 18 (2014), 465–488; MR 3257836.CrossRefGoogle Scholar
Abramovich, D., Chen, Q., Gross, M. and Siebert, B., Punctured logarithmic curves, Preprint (2017), available on the webpage of Mark Gross.Google Scholar
Abramovich, D. and Wise, J., Birational invariance in logarithmic Gromov–Witten theory, Compos. Math. 154 (2018), 595–620; MR 3778185.CrossRefGoogle Scholar
Aganagic, M., Dijkgraaf, R., Klemm, A., Mariño, M. and Vafa, C., Topological strings and integrable hierarchies, Comm. Math. Phys. 261 (2006), 451–516; MR 2191887.CrossRefGoogle Scholar
Aganagic, M. and Vafa, C., Mirror symmetry, D-branes and counting holomorphic discs, Preprint (2000), arXiv:hep-th/0012041.Google Scholar
Alexeev, V., Complete moduli in the presence of semiabelian group action, Ann. of Math. (2) 155 (2002), 611–708; MR 1923963.CrossRefGoogle Scholar
Allegretti, D. G. L. and Kyu Kim, H., A duality map for quantum cluster varieties from surfaces, Adv. Math. 306 (2017), 1164–1208; MR 3581328.CrossRefGoogle Scholar
Auroux, D., Mirror symmetry and T-duality in the complement of an anticanonical divisor, J. Gökova Geom. Topol. GGT 1 (2007), 51–91; MR 2386535.Google Scholar
Auroux, D., Katzarkov, L. and Orlov, D., Mirror symmetry for del Pezzo surfaces: vanishing cycles and coherent sheaves, Invent. Math. 166 (2006), 537–582; MR 2257391 (2007g:14045).CrossRefGoogle Scholar
Bezrukavnikov, R. and Kaledin, D., Fedosov quantization in algebraic context, Mosc. Math. J. 4 (2004), 559–592; 782; MR 2119140.CrossRefGoogle Scholar
Boalch, P., Hyperkähler manifolds and nonabelian Hodge theory of (irregular) curves, Preprint (2012), arXiv:1203.6607.Google Scholar
Bousseau, P., The quantum tropical vertex, Preprint (2018), arXiv:1806.11495.Google Scholar
Bousseau, P., Tropical refined curve counting from higher genera and lambda classes, Invent. Math. 215 (2019), 1–79; MR 3904449.CrossRefGoogle ScholarPubMed
Bryan, J. and Pandharipande, R., Curves in Calabi–Yau threefolds and topological quantum field theory, Duke Math. J. 126 (2005), 369–396; MR 2115262.CrossRefGoogle Scholar
Carl, M., Pumperla, M. and Siebert, B., A tropical view of Landau-Ginzburg models, Preprint (2010), http://www.math.uni-hamburg.de/home/siebert/Preprints/LGtrop.pdf.Google Scholar
Cecotti, S. and Vafa, C., BPS wall crossing and topological strings, Preprint (2009), arXiv:0910.2615.Google Scholar
Chen, Q., Stable logarithmic maps to Deligne-Faltings pairs I, Ann. of Math. (2) 180 (2014), 455–521; MR 3224717.CrossRefGoogle Scholar
Etingof, P. and Ginzburg, V., Noncommutative del Pezzo surfaces and Calabi–Yau algebras, J. Eur. Math. Soc. (JEMS) 12 (2010), 1371–1416; MR 2734346.CrossRefGoogle Scholar
Etingof, P., Oblomkov, A. and Rains, E., Generalized double affine Hecke algebras of rank 1 and quantized del Pezzo surfaces, Adv. Math. 212 (2007), 749–796; MR 2329319.CrossRefGoogle Scholar
Filippini, S. A. and Stoppa, J., Block-Göttsche invariants from wall-crossing, Compos. Math. 151 (2015), 1543–1567; MR 3383167.CrossRefGoogle Scholar
Fock, V. V. and Goncharov, A. B., Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 865–930; MR 2567745.CrossRefGoogle Scholar
Fock, V. V. and Goncharov, A. B., Cluster ensembles, quantization and the dilogarithm. II. The intertwiner, in Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I, Progress in Mathematics, vol. 269 (Birkhäuser, Boston, 2009), 655–673; MR 2641183.CrossRefGoogle Scholar
Friedman, R., On the geometry of anticanonical pairs, Preprint (2015), arXiv:1502.02560.Google Scholar
Fukaya, K., Multivalued Morse theory, asymptotic analysis and mirror symmetry, in Graphs and patterns in mathematics and theoretical physics, Proceedings of Symposia in Pure Mathematics, vol. 73 (American Mathematical Society, Providence, RI, 2005), 205–278; MR 2131017.CrossRefGoogle Scholar
Ginzburg, V., Lectures on D-modules, with collaboration of V. Baranovsky and S. Evens, online lecture notes (1998), http://www.math.harvard.edu/∼gaitsgde/grad_2009/Ginzburg.pdf.Google Scholar
Gross, M., Tropical geometry and mirror symmetry, CBMS Regional Conference Series in Mathematics, vol. 114 (American Mathematical Society, Providence, RI, 2011), MR 2722115.CrossRefGoogle Scholar
Gross, M., Hacking, P. and Keel, S., Mirror symmetry for log Calabi–Yau surfaces I, Publ. Math. Inst. Hautes Études Sci. 122 (2015), 65–168; MR 3415066.CrossRefGoogle Scholar
Gross, M., Hacking, P. and Keel, S., Moduli of surfaces with an anti-canonical cycle, Compos. Math. 151 (2015), 265–291; MR 3314827.CrossRefGoogle Scholar
Gross, M., Hacking, P. and Keel, S., The mirror of the cubic surface, Preprint (2019), arXiv:1910.08427.Google Scholar
Gross, M., Hacking, P., Keel, S. and Siebert, B., Theta functions on varieties with effective anti-canonical class, Preprint (2016), arXiv:1601.07081.Google Scholar
Gross, M., Pandharipande, R. and Siebert, B., The tropical vertex, Duke Math. J. 153 (2010), 297–362; MR 2667135 (2011f:14093).CrossRefGoogle Scholar
Gross, M. and Siebert, B., From real affine geometry to complex geometry, Ann. of Math. (2) 174 (2011), 1301–1428; MR 2846484.CrossRefGoogle Scholar
Gross, M. and Siebert, B., Logarithmic Gromov–Witten invariants, J. Amer. Math. Soc. 26 (2013), 451–510; MR 3011419.CrossRefGoogle Scholar
Gross, M. and Siebert, B., Intrinsic mirror symmetry and punctured Gromov–Witten invariants, Preprint (2016), arXiv:1609.00624.Google Scholar
Kapranov, M., Noncommutative geometry based on commutator expansions, J. Reine Angew. Math. 505 (1998), 73–118; MR 1662244.CrossRefGoogle Scholar
Kapustin, A. and Witten, E., Electric-magnetic duality and the geometric Langlands program, Commun. Number Theory Phys. 1 (2007), 1–236; MR 2306566.CrossRefGoogle Scholar
Kontsevich, M., Deformation quantization of algebraic varieties, Lett. Math. Phys. 56 (2001), 271–294; MR 1855264.CrossRefGoogle Scholar
Kontsevich, M. and Soibelman, Y., The unity of mathematics, Progress in Mathematics, vol. 244 (Birkhäuser, Boston, 2006), 321–385; MR 2181810.Google Scholar
Kontsevich, M. and Soibelman, Y., Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants, Commun. Number Theory Phys. 5 (2011), 231–352; MR 2851153.CrossRefGoogle Scholar
Looijenga, E., Rational surfaces with an anticanonical cycle, Ann. of Math. (2) 114 (1981), 267–322; MR 632841.CrossRefGoogle Scholar
Mandel, T., Scattering diagrams, theta functions, and refined tropical curve counts, Preprint (2015), arXiv:1503.06183.Google Scholar
Maulik, D., Pandharipande, R. and Thomas, R. P., Curves on K3 surfaces and modular forms, J. Topol. 3 (2010), 937–996, with an appendix by A. Pixton; MR 2746343.CrossRefGoogle Scholar
Mikhalkin, G., Enumerative tropical algebraic geometry in ℝ2, J. Amer. Math. Soc. 18 (2005), 313–377; MR 2137980.CrossRefGoogle Scholar
Mumford, D., Towards an enumerative geometry of the moduli space of curves, in Arithmetic and geometry, Vol. II, Progress in Mathematics, vol. 36 (Birkhäuser, Boston, 1983), 271–328; MR 717614.CrossRefGoogle Scholar
Nishinou, T. and Siebert, B., Toric degenerations of toric varieties and tropical curves, Duke Math. J. 135 (2006), 1–51; MR 2259922.CrossRefGoogle Scholar
Oblomkov, A., Double affine Hecke algebras of rank 1 and affine cubic surfaces, Int. Math. Res. Not. (2004), 877–912; MR 2037756.CrossRefGoogle Scholar
Soibelman, Y., On non-commutative analytic spaces over non-Archimedean fields, in Homological mirror symmetry, Lecture Notes in Physics, vol. 757 (Springer, Berlin, 2009), 221–247; MR 2596639.Google Scholar
Strominger, A., Yau, S.-T. and Zaslow, E., Mirror symmetry is T-duality, Nuclear Phys. B 479 (1996), 243–259; MR 1429831.CrossRefGoogle Scholar
Symington, M., Four dimensions from two in symplectic topology, in Topology and geometry of manifolds (Athens, GA, 2001), Proceedings of Symposia in Pure Mathematics, vol. 71 (American Mathematical Society, Providence, RI, 2003), 153–208; MR 2024634.CrossRefGoogle Scholar
Witten, E., Chern-Simons gauge theory as a string theory, in The Floer memorial volume, Progress in Mathematics, vol. 133 (Birkhäuser, Basel, 1995), 637–678; MR 1362846.CrossRefGoogle Scholar
Yekutieli, A., Deformation quantization in algebraic geometry, Adv. Math. 198 (2005), 383–432; MR 2183259.CrossRefGoogle Scholar