D1D5 System and Noncommutative Geometry

Supergravity on $AdS_3\times S^3\times {\bf T}^4$ has a dual description as a conformal sigma-model with the target space being the moduli space of instantons on the noncommutative torus. We derive the precise relation between the parameters of this noncommutative torus and the parameters of the near-horizon geometry. We show that the low energy dynamics of the system of $D1D5$ branes wrapped on the torus of finite size is described in terms of the noncommutative geometry. As a byproduct, we give a prediction on the dependence of the moduli space of instantons on the noncommutative ${\bf T}^4$ on the metric and the noncommutativity parameter. We give a compelling evidence that the moduli space of stringy instantons on ${\bf R}^4$ with the $B$ field does not receive $\alpha'$-corrections. We also study the relation between the $D1D5$ sigma-model instantons and the supergravity instantons.