Noncommutative $D_3$-brane, black holes, and attractor mechanism

We revisit the $4D$ generalized black hole geometries, obtained by us [14], with a renewed interest, to unfold some aspects of effective gravity in a noncommutative $D_3$-brane formalism. In particular, we argue for the existence of extra dimensions in the gravity decoupling limit in the theory. We show that the theory is rather described by an ordinary geometry and is governed by an effective string theory in $5D$. The extremal black hole geometry ${\mathrm{AdS}}_5$ obtained in effective string theory is shown to be in precise agreement with the gravity dual proposed for $D_3$-brane in a constant magnetic field. Kaluza-Klein compactification is performed to obtain the corresponding charged black hole geometries in $4D$. Interestingly, they are shown to be governed by the extremal black hole geometries known in string theory. The attractor mechanism is exploited in effective string theory underlying a noncommutative $D_3$-brane and the macroscopic entropy of a charged black hole is computed. We show that the generalized black hole geometries in a noncommutative $D_3$-brane theory are precisely identical to the extremal black holes known in $4D$ effective string theory.