Closed string tachyon potential and $t{t}^{\ast }$ equation

Recently Dabholkar and Vafa proposed that closed string tachyon potential for non-supersymmetric orbifold $\text{C}\mathrm{/}{\mathbb{Z}}_3$ in terms of the solution of a $t{t}^{\ast }$ equation. We extend this result to ${\text{C}}^2\mathrm{/}{\mathbb{Z}}_n$ 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 ${\text{C}}^1\mathrm{/}{\mathbb{Z}}_3$ with different boundary conditions. For ${\text{C}}^2\mathrm{/}{\mathbb{Z}}_5$ case, governing equations are of generalized Toda type. The potential is monotonically decreasing function of RG flow.