Classical {\it vs.}\ Landau-Ginzburg Geometry of Compactification

TL;DR

This paper compares classical Calabi-Yau compactifications with Landau-Ginzburg models, revealing how twisted sectors and spectral sequences provide a unified understanding of string theory geometries and spectra.

Contribution

It demonstrates the correspondence between Landau-Ginzburg chiral rings and Calabi-Yau cohomology, clarifying the role of twisted sectors and spectral flow in string compactifications.

Findings

Landau-Ginzburg models account for complex structure obstructions.

Spectral flow arguments are mathematically justified via Koszul complexes.

Stringy features are recovered through point-field analysis.

Abstract

We consider superstring compactifications where both the classical description, in terms of a Calabi-Yau manifold, and also the quantum theory is known in terms of a Landau-Ginzburg orbifold model. In particular, we study (smooth) Calabi-Yau examples in which there are obstructions to parametrizing all of the complex structure cohomology by polynomial deformations thus requiring the analysis based on exact and spectral sequences. General arguments ensure that the Landau-Ginzburg chiral ring copes with such a situation by having a nontrivial contribution from twisted sectors. Beyond the expected final agreement between the mathematical and physical approaches, we find a direct correspondence between the analysis of each, thus giving a more complete mathematical understanding of twisted sectors. Furthermore, this approach shows that physical reasoning based upon spectral flow arguments for determining the spectrum of Landau-Ginzburg orbifold models finds direct mathematical justification in Koszul complex calculations and also that careful point- field analysis continues to recover suprisingly much of the stringy features.