Gluon scattering amplitudes with instantons and minimal surfaces with topology change

We study the instanton effect on the gluon scattering amplitudes at strong coupling and large $N$ for the $\mathcal{N}=4$ supersymmetric Yang-Mills theory. According to Alday and Maldacena, the gluon scattering amplitude corresponds holographically to the area of a worldsheet minimal surface in the T-dual AdS${}_5$ geometry. The Yang-Mills instanton introduces an instanton D-brane in the geometry, with which a particular boundary condition for the minimal surface is imposed. We show that the minimal surface undergoes a topology change depending on the size and the gluon momenta, and that the instanton amplitude exhibits a characteristic dependence on gluon momenta. More specifically, we find that when the fixed instanton size modulus $\rho$ is larger than $\mathcal{O}(\sqrt{\lambda }\mathrm{/}E)$ where $\lambda$ is the 't Hooft coupling and $E$ is the typical momentum of the scattering gluon, due to the topology change of the worldsheet minimal surface, the instanton amplitude is exponentially enhanced as $\exp (\rho E)$.