Duality Symmetry and the Cardy Limit

TL;DR

This work studies extremal black holes in Type IIA string theory on $K3\times T^2$ with duality group $O(6,22,\mathbb{Z})\times SL(2,\mathbb{Z})$ and asks whether generic charge configurations can be mapped into the Cardy limit where the near-horizon geometry yields a BTZ black hole and the entropy follows the Cardy formula. It develops a detailed analysis of the D0-D4 and D0-D6 systems, showing that generic charges cannot be duality-transformed into the Cardy regime, but that in the non-supersymmetric case one can reach a Cardy-limit configuration after a small fractional shift in charges (scaling as $\Delta Q/Q\sim 1/\sqrt{Q}$) by appropriate SL$(2,\mathbb{Z})$ and $O(6,22,\mathbb{Z})$ transformations; in SUSY this non-genericity is more restrictive. The D0-D6 system cannot reach Cardy-limit either, though nearby charge configurations can be arranged to lie in Cardy-limit subsets with additional charges. A key result is that AdS$_3$-lift and Cardy-limit descriptions hinge on the absence of D6 charge, linking discrete duality invariants to the microscopic entropy and suggesting a moduli-insensitive entropy for charges admitting a Cardy-limit description. The study motivates broader exploration of discrete invariants and potential microscopic entropy determinations for extremal non-supersymmetric black holes, including rotating cases.

Abstract

We study supersymmetric and non-supersymmetric extremal black holes obtained in Type IIA string theory compactified on K3 X T^2, with duality group O(6,22,Z) X SL(2,Z). In the Cardy limit an internal circle combines with the AdS_2 component in the near horizon geometry to give a BTZ black hole whose entropy is given by the Cardy formula. We study black holes carrying D0-D4 and D0-D6 brane charges. We find, both in the supersymmetric and non-supersymmetric cases, that a generic set of charges cannot be brought to the Cardy limit using the duality symmetries. In the non-supersymmetric case, unlike the supersymmetric one, we find that when the charges are large, a small fractional change in them always allows the charges to be taken to the Cardy limit. These results could lead to a microscopic determination of the entropy for extremal non-supersymmetric black holes, including rotating cases like the extreme Kerr black hole in four dimensions.