Comments on the Fate of unstable orbifolds

TL;DR

This paper analyzes the decay processes of unstable orbifolds $C^r/Z_n$ through localized tachyon condensation, providing a unified method applicable to all cases and establishing equivalence with toric geometry for $C^2/Z_n$.

Contribution

It introduces a simple, uniform method for determining decay modes of unstable orbifolds under tachyon condensation, extending previous work and proving equivalence with toric geometry in specific cases.

Findings

Decay modes of $C^r/Z_n$ orbifolds are fully determined.

The new method is applicable to all $C^r/Z_n$ cases.

For $C^2/Z_n$, the method is shown to be equivalent to toric geometry.

Abstract

We study the localized tachyon condensation in their mirror Landau-Ginzburg picture. We completely determine the decay mode of an unstable orbifold $C^r/Z_n$, $r=1,2,3$ under the condensation of a tachyon with definite R-charge and mass by extending the Vafa's work hep-th/0111105. Here, we give a simple method that works uniformly for all $C^r/Z_n$. For $C^2/Z_n$, where method of toric geometry works, we give a proof of equivalence of our method with toric one. For $C^r/Z_n$ cases, the orbifolds decay into sum of $r$ far separated orbifolds.