Statistical entropy of charged two-dimensional black holes

The statistical entropy of a five-dimensional black hole in Type II string theory was recently derived by showing that it is U-dual to the three-dimensional Bañados–Teitelboim–Zanelli black hole, and using Carlip’s method to count the microstates of the latter. This is valid even for the non-extremal case, unlike the derivation which relies on D-brane techniques. In this letter, I shall exploit the U-duality that exists between the five-dimensional black hole and the two-dimensional charged black hole of McGuigan, Nappi and Yost, to microscopically compute the entropy of the latter. It is shown that this result agrees with previous calculations using thermodynamic arguments. String theory (and its extended brane descendants) has reached a critical stage at which it is able to begin addressing fundamental issues in quantum gravity, such as the statistical origin of black hole entropy. This, in particular, has been an outstanding problem ever since Bekenstein [1] proposed in 1972 that the entropy of a black hole is proportional to its area, and Hawking [2] subsequently found the constant of proportionality to be 1/4G (in units where h̄ = 1), G being Newton’s constant. But in 1996, Strominger and Vafa [3] used D-brane techniques to derive the Bekenstein–Hawking area formula for an extremally charged five-dimensional black hole in Type II string theory. They did this by mapping it to an equivalent weakly coupled D-brane configuration, and enumerating the microstates using a certain conformal field theory. D-brane counting has also been extended to near-extremal black holes [4,5], as well as rotating [6,7] and four-dimensional ones [8,9,10]. In all these cases, precise numerical agreement with the Bekenstein–Hawking formula was found. However, this method crucially relies on supersymmetry, to ensure that BPS states do not receive quantum corrections when moving to and from the corresponding D-brane picture. Thus, it can only be applied to extreme, or at most slightly non-extremal, black holes. (For reviews of these calculations, see [11,12]. The latter reference also describes more recent progress using Matrix theory.) In a separate development, Carlip [13] succeeded in microscopically computing the entropy of the three-dimensional constant-curvature black hole of Bañados, Teitelboim and Zanelli [14]. Although gravity in three dimensions has no local dynamics, the black hole horizon induces an effective boundary to space-time which is dynamical. This boundary is described by a two-dimensional Wess–Zumino–Witten model [15], and Carlip derived the Bekenstein–Hawking formula by enumerating the states of the theory. This method, unlike the one using D-branes, does not assume supersymmetry and hence applies to the extreme and non-extremal cases alike. But it only appears to work in three dimensions. Subsequently, Hyun [16] realised that the three-dimensional BTZ black hole is connected to the above-mentioned fourand five-dimensional charged black holes [5,17,10] by U-duality. Thus, these black holes are equivalent as far as string theory is concerned, even though they look remarkably different as space-times. For instance, the BTZ black hole is asymptotically anti-de Sitter, while the higher-dimensional ones are asymptotically flat. Crucial in getting from one case to the other are the so-called shift transformations [18]. These are T-duality transformations with respect to isometries that are space-like everywhere but asymptotically null. (This type of transformation is, in fact, not new. It