Closed string tachyon potential and $tt^*$ equation

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.Comment: 15 pages, reference added. to appear in PL