Closed string tachyon potential and tt∗ equation

Recently Dabholkar and Vafa proposed that closed string tachyon potential for non-supersymmetric orbifold C/Z3 in terms of the solution of a tt∗ equation. We extend this result to C2/Zn for n=3,4,5. Interestingly, the tachyon potentials for n=3 and 4 are still given in terms of the solutions of Painlevé III type equation that appeared in the study of C1/Z3 with different boundary conditions. For C2/Z5 case, governing equations are of generalized Toda type. The potential is monotonically decreasing function of RG flow.