Free-field Representations and Geometry of some Gepner models

TL;DR

This paper explores the geometric structure of certain Gepner models using free-field representations, revealing their relation to Landau-Ginzburg orbifolds and deformations leading to Calabi-Yau sigma models.

Contribution

It provides a direct free-field approach to understanding Gepner models' geometry and their deformations to Calabi-Yau sigma models.

Findings

Gepner models correspond to Landau-Ginzburg orbifolds.

Deformations relate models to Calabi-Yau manifolds as double covers.

Holomorphic sectors exhibit chiral de Rham complex structures.

Abstract

The geometry of $k^{K}$ Gepner model, where $k+2=2K$ is investigated by free-field representation known as "$bc\bet\gm$"-system. Using this representation it is shown directly that internal sector of the model is given by Landau-Ginzburg $\mathbb{C}^{K}/\mathbb{Z}_{2K}$-orbifold. Then we consider the deformation of the orbifold by marginal anti-chiral-chiral operator. Analyzing the holomorphic sector of the deformed space of states we show that it has chiral de Rham complex structure of some toric manifold, where toric dates are given by certain fermionic screening currents. It allows to relate the Gepner model deformed by the marginal operator to the $\sigma$-model on CY manifold realized as double cover of $\mathbb{P}^{K-1}$ with ramification along certain submanifold.