A Note on D1-D5-J System and 5D Small Black Ring
Norihiro Iizuka, Masaki Shigemori
TL;DR
This work investigates whether a 5D small black ring formed by the D1-D5 system with added angular momentum develops a horizon when higher-derivative stringy corrections are taken into account. By exploiting a duality chain and the 4D-5D connection, the authors map the 5D D1-D5-J system to a 4D small non-rotating black hole whose horizon arises from $R^2$ corrections, and they show that the 4D macroscopic entropy matches the 5D microscopic entropy, $S_{ m micro}=4\pi\sqrt{-N_FN_P-n_fn_p}=4\pi\sqrt{N_1N_5+n_pn_{kk}}$. This entropy matching provides indirect evidence that a nonvanishing horizon forms for the 5D small black ring under stringy corrections, supporting the robustness of the 4D-5D connection even for small black objects. The results reinforce the role of higher-derivative corrections in horizon formation and offer a bridge between microscopic D-brane counting and macroscopic geometries across dimensions.
Abstract
The ``small'' black ring in 5D obtained by giving angular momentum to the D1-D5 system compactified on S^1 x K3 is a very interesting object in the sense that it does not have an event horizon in the supergravity limit whereas it microscopically has a finite entropy. The microscopic origin of this small black ring can be analyzed in detail since it is constructed by adding angular momentum to the well-studied D1-D5 system. On the other hand, its macroscopic, geometrical picture is difficult to study directly. In this note, by duality transformations and the 4D-5D connection, we relate this 5D small black ring to a 4D small non-rotating black hole, where the latter is known to develop a non-vanishing horizon due to stringy R^2 corrections to the supergravity action. This gives an indirect evidence that a non-vanishing horizon is formed for the 5D small black ring. We also show that the entropy of the 4D small black hole agrees with the microscopic entropy of the 5D small black ring, which supports that the 4D-5D connection is indeed valid even for small black objects.