Notes on a singular Landau-Ginzburg family

TL;DR

This paper analyzes a specific singular Landau-Ginzburg model with a multi-variable superpotential, linking it to the topological aspects of the $c=1$ string at multiple self-dual radii and exploring its geometric singularities.

Contribution

It introduces a particular Landau-Ginzburg family and connects its IR limit to the topological degrees of freedom of the $c=1$ string at scaled radii, also discussing potential Calabi-Yau realizations.

Findings

The Landau-Ginzburg model describes the $c=1$ string at scaled self-dual radius.

The superpotential characterizes a family with singularities related to string compactification.

Possible geometric interpretations as Calabi-Yau singularities are considered.

Abstract

We study some properties of a singular Landau-Ginzburg family characterized by the multi-variable superpotential $W=-X^{-1}(Y_1Y_2)^{n-1} + {1\over n} (Y_1Y_2)^n - Y_3Y_4$. We will argue that (the infra-red limit of) this theory describes the topological degrees of freedom of the $c=1$ string compactified at $n$ times the self-dual radius. We also briefly comment on the possible realization of these line singularities as singularities of Calabi-Yau manifolds.