Tropical Mirror

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.

References in corpus (9)

• Supersymmetry and Quantum Mechanics
• Gromov-Witten classes, quantum cohomology, and enumerative geometry
• Quantum cohomology of flag manifolds and Toda lattices
• Mirror symmetry in two steps: A-I-B
• Tropical quantum field theory, mirror polyvector fields, and multiplicities of tropical curves
• Introduction to Tropical Geometry (notes from the IMPA lectures in Summer 2007)
• From Zwiebach invariants to Getzler relation
• Tropical Mirror Symmetry: Correlation functions
• Tropical mirror for toric surfaces