Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 072, 48 pages      arXiv:2204.06896      https://doi.org/10.3842/SIGMA.2024.072

Tropical Mirror

Andrey Losev abc and Vyacheslav Lysov cd
a) National Research University Higher School of Economics, Laboratory of Mirror Symmetry, NRU HSE, 6 Usacheva Str., Moscow, 119048, Russia
b) Wu Wen-Tsun Key Lab of Mathematics, Chinese Academy of Sciences, USTC, No. 96, JinZhai Road Baohe District, Hefei, Anhui, 230026, P.R. China
c) Shanghai Institute for Mathematics and Interdisciplinary Sciences, Building 3, 62 Weicheng Road, Yangpu District, Shanghai, 200433, P.R. China
d) Okinawa Institute of Science and Technology, 1919-1 Tancha, Onna-son, Okinawa 904-0495, Japan

Received July 30, 2023, in final form July 24, 2024; Published online August 04, 2024

Abstract
We describe the tropical curves in toric varieties and define the tropical Gromov-Witten invariants. We introduce amplitudes for the higher topological quantum mechanics (HTQM) on special trees and show that the amplitudes are equal to the tropical Gromov-Witten invariants. We show that the sum over the amplitudes in $A$-model HTQM equals the total amplitude in B-model HTQM, defined as a deformation of the $A$-model HTQM by the mirror superpotential. We derived the mirror superpotentials for the toric varieties and showed that they coincide with the superpotentials in the mirror Landau-Ginzburg theory. We construct the mirror dual states to the evaluation observables in the tropical Gromov-Witten theory.

Key words: mirror symmetry; Gromov-Witten invariants; tropical geometry; topological quantum mechanics on trees.

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References

  1. Cooper F., Khare A., Sukhatme U., Supersymmetry and quantum mechanics, Phys. Rep. 251 (1995), 267-385, arXiv:hep-th/9405029.
  2. Feynman R.P., Hibbs A.R., Quantum mechanics and path integrals, Dover Publications, Inc., Mineola, NY, 2010.
  3. Frenkel E., Losev A., Mirror symmetry in two steps: A-I-B, Comm. Math. Phys. 269 (2007), 39-86, arXiv:hep-th/0505131.
  4. Fulton W., Introduction to toric varieties, Ann. of Math. Stud., Vol. 131, Princeton University Press, Princeton, NJ, 1993.
  5. Gathmann A., Markwig H., Kontsevich's formula and the WDVV equations in tropical geometry, Adv. Math. 217 (2008), 537-560, arXiv:math.AG/0509628.
  6. Givental A., Kim B., Quantum cohomology of flag manifolds and Toda lattices, Comm. Math. Phys. 168 (1995), 609-641, arXiv:hep-th/9312096.
  7. Gross A., Intersection theory on tropicalizations of toroidal embeddings, Proc. Lond. Math. Soc. 116 (2018), 1365-1405, arXiv:1510.04604.
  8. Gross M., Mirror symmetry for $\mathbb P^2$ and tropical geometry, Adv. Math. 224 (2010), 169-245, arXiv:0903.1378.
  9. Gross M., Tropical geometry and mirror symmetry, CBMS Reg. Conf. Ser. Math., Vol. 114, American Mathematical Society, Providence, RI, 2011.
  10. Hori K., Katz S., Klemm A., Pandharipande R., Thomas R., Vafa C., Vakil R., Zaslow E., Mirror symmetry, Clay Math. Monogr., Vol. 1, American Mathematical Society, Providence, RI, 2003.
  11. Hori K., Vafa C., Mirror symmetry, arXiv:hep-th/0002222.
  12. Kontsevich M., Manin Yu., Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), 525-562, arXiv:hep-th/9402147.
  13. Losev A., TQFT, homological algebra and elements of K. Saito's theory of Primitive form: an attempt of mathematical text written by mathematical physicist, in Primitive Forms and Related Subjects - Kavli IPMU 2014, Adv. Stud. Pure Math., Vol. 83, Mathematical Society of Japan, Tokyo, 2019, 269-293.
  14. Losev A., Lysov V., Tropical mirror symmetry: correlation functions, arXiv:2301.01687.
  15. Losev A., Lysov V., Tropical mirror for toric surfaces, arXiv:2305.00423.
  16. Losev A., Shadrin S., From Zwiebach invariants to Getzler relation, Comm. Math. Phys. 271 (2007), 649-679, arXiv:math.QA/0506039.
  17. Lysov V., Anticommutativity equation in topological quantum mechanics, JETP Lett. 76 (2002), 724-727, arXiv:hep-th/0212005.
  18. Mandel T., Ruddat H., Descendant log Gromov-Witten invariants for toric varieties and tropical curves, Trans. Amer. Math. Soc. 373 (2020), 1109-1152, arXiv:1612.02402.
  19. Mandel T., Ruddat H., Tropical quantum field theory, mirror polyvector fields, and multiplicities of tropical curves, Int. Math. Res. Not. 2023 (2023), 3249-3304, arXiv:1902.07183.
  20. Mikhalkin G., Amoebas of algebraic varieties and tropical geometry, in Different Faces of Geometry, Int. Math. Ser. (N.Y.), Vol. 3, Kluwer, New York, 2004, 257-300, arXiv:math.AG/0403015.
  21. Mikhalkin G., Enumerative tropical algebraic geometry in $\mathbb R^2$, J. Amer. Math. Soc. 18 (2005), 313-377, arXiv:math.AG/0312530.
  22. Mikhalkin G., Introduction to tropical geometry (notes from the IMPA lectures in Summer 2007), arXiv:0709.1049.
  23. Mikhalkin G., Rau J., Tropical geometry, available at https://www.math.uni-tuebingen.de/user/jora/downloads/main.pdf.
  24. Nishinou T., Siebert B., Toric degenerations of toric varieties and tropical curves, Duke Math. J. 135 (2006), 1-51, arXiv:math.AG/0409060.
  25. Polchinski J., String theory. Vol. I. An introduction to the bosonic string, Cambridge Monogr. Math. Phys., Cambridge University Press, Cambridge, 2005.
  26. Polchinski J., String theory. Vol. II. Superstring theory and beyond, Cambridge Monogr. Math. Phys., Cambridge University Press, Cambridge, 2005.
  27. Ranganathan D., Skeletons of stable maps I: Rational curves in toric varieties, J. Lond. Math. Soc. 95 (2017), 804-832, arXiv:1506.03754.
  28. Saito K., Period mapping associated to a primitive form, Publ. Res. Inst. Math. Sci. 19 (1983), 1231-1264.
  29. Witten E., Topological sigma models, Comm. Math. Phys. 118 (1988), 411-449.

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