Toda systems in closed string tachyon condensation

TL;DR

This paper explores Toda systems arising in localized tachyon condensation in string theory, calculating tachyon potentials and analyzing their relation to singularity deformations, revealing both specific and complex cases.

Contribution

It identifies and analyzes Toda systems in tachyon condensation, linking them to singularity deformations and providing explicit solutions for certain cases.

Findings

Toda systems describe tachyon condensation in specific geometries

Tachyon potentials are obtained as solutions to Toda equations

Complex cases pose challenges in analysis

Abstract

We consider $tt^*$ equations appearing in the study of localized tachyon condensations. They are described by various Toda system when we consider the condensation by the lowest tachyon corresponding to the monomial $xy$. The tachyon potential is calculated as a solution to these equations. The Toda system appearing in the deformation of $\C^2/\Z_n$ by $xy$ is identical to that of $D_n$ singularity deformed by $x$. For $\C^3/\Z_n$ with $xyz$ deformation, we find only generic non-simple form, similar to the case appearing in $\C/\Z_5\to \C/\Z_3$ and we discuss the difficulties in these cases.