TL;DR
This paper develops a tropical geometric framework for Gromov-Witten invariants in toric varieties, establishing a mirror symmetry correspondence through amplitudes in topological quantum mechanics.
Contribution
It introduces tropical Gromov-Witten invariants and demonstrates their equivalence to amplitudes in higher topological quantum mechanics, linking A-model and B-model in mirror symmetry.
Findings
Tropical Gromov-Witten invariants are defined and computed.
Amplitudes in HTQM match tropical Gromov-Witten invariants.
Mirror superpotentials are derived and shown to coincide with Landau-Ginzburg superpotentials.
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 -model HTQM equals the total amplitude in B-model HTQM, defined as a deformation of the -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.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Quantum Mechanics and Non-Hermitian Physics · Nonlinear Waves and Solitons