  In this paper, we demonstrate that locally, the $\alpha^{\prime}$ expansion
of a string propagating in AdS can be summed into a closed expression, where
the $\alpha'$ dependence is manifested. The T-dual of this sum exactly matches
the expression controlling all genus expansion in the Goparkumar-Vafa formula,
which in turn also matches the loop expansion of the Chern-Simons gauge theory.
We therefore find an exact correspondence between the $\alpha^{\prime}$
expansion for a string moving in AdS and the genus expansion of a string
propagating in four dimensional flat spacetime. We are then able to give a
closed form of the $\alpha'$ expansion for all values of
$\sqrt{\alpha'}/R_{AdS}$. Moreover, the correspondence makes it possible to
conjecture the exact $g_s$ dependence of the strongly coupled theories.
