The Construction of Mirror Symmetry

TL;DR

This paper reviews the construction of mirror symmetry in heterotic string theory, highlighting its systematic identification, its non-accidental nature, and the broader configuration space it reveals beyond solvable models.

Contribution

It introduces a framework for constructing and understanding mirror symmetry in heterotic strings, extending beyond Fermat potentials and revealing the symmetry in Landau-Ginzburg orbifolds.

Findings

Systematic identification of mirror pairs in heterotic string vacua.

Mirror symmetry is not coincidental but inherent in the configuration space.

Landau-Ginzburg orbifolds exhibit mirror symmetry.

Abstract

The construction of mirror symmetry in the heterotic string is reviewed in the context of Calabi-Yau and Landau-Ginzburg compactifications. This framework has the virtue of providing a large subspace of the configuration space of the heterotic string, probing its structure far beyond the present reaches of solvable models. The construction proceeds in two stages: First all singularities/catastrophes which lead to ground states of the heterotic string are found. It is then shown that not all ground states described in this way are independent but that certain classes of these LG/CY string vacua can be related to other, simpler, theories via a process involving fractional transformations of the order parameters as well as orbifolding. This construction has far reaching consequences. Firstly it allows for a systematic identification of mirror pairs that appear abundantly in this class of string vacua, thereby showing that the emerging mirror symmetry is not accidental. This is important because models with mirror flipped spectra are a priori independent theories, described by distinct CY/LG models. It also shows that mirror symmetry is not restricted to the space of string vacua described by theories based on Fermat potentials (corresponding to minimal tensor models). Furthermore it shows the need for a better set of coordinates of the configuration space or else the structure of this space will remain obscure. While the space of LG vacua is {\it not} completely mirror symmetric, results described in the last part suggest that the space of Landau--Ginburg {\it orbifolds} possesses this symmetry.