[Paper Review] The Construction of Mirror Symmetry

Rolf Schimmrigk|ArXiv.org|Sep 6, 1992

Black Holes and Theoretical Physics7 references3 citations

TL;DR

This paper constructs mirror symmetry in heterotic string theory by analyzing Calabi-Yau and Landau-Ginzburg compactifications, showing that mirror pairs emerge systematically through fractional transformations and orbifolding of order parameters. The key contribution is demonstrating that mirror symmetry is not accidental but arises from a deeper structure in the space of string vacua, extending beyond Fermat-type models and revealing that Landau-Ginzburg orbifolds may possess full 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.

Motivation & Objective

• To systematically construct mirror symmetry in heterotic string theory beyond ad hoc examples.
• To identify the underlying structure of the configuration space of string vacua that gives rise to mirror symmetry.
• To show that mirror symmetry is not accidental but emerges from a deeper geometric and algebraic framework.
• To extend mirror symmetry beyond models based on Fermat potentials, which correspond to minimal tensor models.
• To explore whether Landau-Ginzburg orbifolds exhibit full mirror symmetry, unlike the full space of LG vacua.

Proposed method

• Classify all singularities and catastrophes that lead to ground states in the heterotic string compactified on Calabi-Yau manifolds and Landau-Ginzburg models.
• Apply fractional transformations to the order parameters of Landau-Ginzburg potentials to generate new vacua.
• Use orbifolding procedures to relate different Landau-Ginzburg and Calabi-Yau models, identifying dual pairs.
• Analyze the resulting configuration space to detect mirror symmetry patterns across distinct classes of vacua.
• Compare the structure of the full space of LG vacua with that of LG orbifolds to assess mirror symmetry properties.
• Use the framework to identify mirror pairs systematically, showing they are not isolated or accidental.

Experimental results

Research questions

• RQ1 Can mirror symmetry in heterotic string theory be constructed systematically rather than discovered case by case?
• RQ2 What algebraic and geometric structures underlie the emergence of mirror pairs in Calabi-Yau and Landau-Ginzburg compactifications?
• RQ3 Is mirror symmetry restricted to models with Fermat-type potentials, or does it extend to more general Landau-Ginzburg models?
• RQ4 Does the space of Landau-Ginzburg orbifolds exhibit full mirror symmetry, unlike the broader space of LG vacua?
• RQ5 What role do fractional transformations of order parameters play in generating mirror duals?

Key findings

• Mirror symmetry in heterotic string theory is not accidental but arises systematically from the structure of singularities and catastrophes in Calabi-Yau and Landau-Ginzburg compactifications.
• Fractional transformations of order parameters and orbifolding procedures generate dual pairs of Landau-Ginzburg and Calabi-Yau vacua, revealing a systematic construction of mirror symmetry.
• The construction extends mirror symmetry beyond models with Fermat potentials, showing it applies to a broader class of Landau-Ginzburg models.
• The full space of Landau-Ginzburg vacua is not mirror symmetric, but the subspace of Landau-Ginzburg orbifolds may possess full mirror symmetry.
• The results suggest that the current coordinate system for the configuration space of string vacua is inadequate, and a better set of coordinates is needed to reveal the full structure.
• The framework provides a large subspace of the heterotic string's configuration space, probing its structure beyond solvable models.