Closed string tachyon potential and $tt^*$ equation

TL;DR

This paper extends the understanding of closed string tachyon potentials in non-supersymmetric orbifolds, linking them to solutions of integrable equations like Painleve III and Toda, and analyzing their RG flow behavior.

Contribution

It generalizes the connection between tachyon potentials and integrable equations from $ ext{C}/ ext{Z}_3$ to higher-dimensional orbifolds $ ext{C}^2/ ext{Z}_n$, revealing new mathematical structures.

Findings

Tachyon potentials for $n=3,4$ relate to Painleve III equations.

For $n=5$, potentials are governed by generalized Toda equations.

Potential decreases monotonically along RG flow.

Abstract

Recently Dabholkar and Vafa proposed that closed string tachyon potential for non-supersymmetric orbifold $\C/\Z_3$ in terms of the solution of a $tt^*$ equation. We extend this result to $\C^2/\Z_n$ for $n=3,4,5$. Interestingly, the tachyon potentials for $n=3$ and 4 are still given in terms of the solutions of Painleve III type equation that appeared in the study of $\C^1/\Z_3$ with different boundary conditions. For $\C^2/\Z_5$ case, governing equations are of generalized Toda type. The potential is monotonically decreasing function of RG flow.