Tropical Mirror

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.