Dual Polyhedra, Mirror Symmetry and Landau-Ginzburg Orbifolds

TL;DR

This paper explores the geometric and mirror symmetry properties of Landau-Ginzburg orbifolds, establishing a correspondence between certain states and dual polyhedra, and connecting these findings to toric geometry and mirror maps for Calabi-Yau manifolds.

Contribution

It introduces a new geometrical perspective on Landau-Ginzburg orbifolds, linking states to dual polyhedra and confirming the equivalence of mirror maps with those for Calabi-Yau manifolds.

Findings

Correspondence between $(a,c)$ states and integral points on dual polyhedra

Identification of states with $(1,1)$ forms from blowing-up processes

Equivalence of the monomial-divisor mirror map for Landau-Ginzburg orbifolds and Calabi-Yau mirror maps

Abstract

New geometrical features of the Landau-Ginzburg orbifolds are presented, for models with a typical type of superpotential. We show the one-to-one correspondence between some of the $(a,c)$ states with $U(1)$ charges $(-1,1)$ and the integral points on the dual polyhedra, which are useful tools for the construction of mirror manifolds. Relying on toric geometry, these states are shown to correspond to the $(1,1)$ forms coming from blowing-up processes. In terms of the above identification, it can be checked that the monomial-divisor mirror map for Landau-Ginzburg orbifolds, proposed by the author, is equivalent to that mirror map for Calabi-Yau manifolds obtained by the mathematicians.