D-brane annihilation, renormalization-group flow and non-linear $\sigma$-model for the ADHM construction

In this note $D9$- and anti-$D9$-brane annihilation in type I string theory is probed by a $D1$-brane. We consider the covariant Green-Schwarz or twistor formulation of the probe theory. We expect the theory to be $\kappa$-invariant after the annihilation is completed. Conditions of the $\kappa$-invariance of the theory impose constraints on the background tachyon field. Solutions to the constraints define tachyon values which correspond to type I $D5$-branes as remnants of the annihilation. As a byproduct we get a theory which lies in the same universality class as the non-linear $\sigma$-model for the Atiyah-Drinfeld-Hitchin-Manin construction.