Correspondence between genus expansion and ${\alpha }^{\prime }$ expansion in string theory

Abstract

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 ${\alpha }^{\prime }$ 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 ${\alpha }^{\prime }$ expansion for all values of $\sqrt{{\alpha }^{\prime }}/R_{AdS}$. Moreover, the correspondence makes it possible to conjecture the exact $g_s$ dependence of the strongly coupled theories.

arXiv comment11 pages

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