tt* Geometry and Closed String Tachyon Potential

TL;DR

This paper develops a closed string tachyon action for non-supersymmetric orbifolds using tt* equations, linking geometry of vacua to solutions of integrable equations like Painleve III.

Contribution

It introduces a novel tachyon action framework based on tt* equations and explicitly connects it to integrable systems for specific orbifolds.

Findings

Tachyon action derived from tt* equations for ${f C}/Z_n$

Solutions involve affine Toda and Painleve III equations

Detailed analysis of ${f C}/Z_3$ case

Abstract

We propose a closed string tachyon action including kinetic and potential terms for non-supersymmetric orbifolds. The action is given in terms of solutions to $tt^*$ equations which captures the geometry of vacua of the corresponding N=2 worldsheet theory. In certain cases the solutions are well studied. In case of tachyons of ${\bf C}/Z_n$, solutions to affine toda equations determine the action. We study the particular case of ${\bf C}/Z_3\to {\bf C}$ in detail and find that the Tachyon action is determined in terms of a solution to Painleve III equation.