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Tropical Mirror
Andrey Losevabc, Vyacheslav Lysovcd a Wu Wen-Tsun Key Lab of Mathematics, Chinese Academy of Sciences,
USTC, No. 96, JinZhai Road Baohe District, Hefei, Anhui, 230026, P.R. China
b National Research University Higher School of Economics, Laboratory of Mirror Symmetry, NRU HSE, 6 Usacheva Str., Moscow, 119048, Russia
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
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 $\mathrm{B}$-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.
Keywords:
mirror symmetry, Gromov–Witten invariants, tropical geometry, topological quantum mechanics on trees.
Received: July 30, 2023; in final form July 24, 2024; Published online August 4, 2024
Citation:
Andrey Losev, Vyacheslav Lysov, “Tropical Mirror”, SIGMA, 20 (2024), 072, 48 pp.
Linking options:
https://www.mathnet.ru/eng/sigma2074 https://www.mathnet.ru/eng/sigma/v20/p72
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