Enumeration of rational contact curves via torus actions

We prove that some Gromov-Witten numbers associated to rational contact (Legendrian) curves in any contact complex projective space with arbitrary incidence conditions are enumerative. Also, we use Bott formula on the Kontsevich space to find the exact value of those numbers. As an example, the numbers of rational contact curves of low degree in ${\mathbb{P}}^3$ and ${\mathbb{P}}^5$ are computed. The results are consistent with existing results.