Logarithmic correction to the Bekenstein-Hawking entropy of BTZ black hole

We derive an exact expression for the partition function of the Euclidean BTZ black hole. Using this, we show that for a black hole with large horizon area, the correction to the Bekenstein-Hawking entropy is −3/2 log(area), in agreement with that for the Schwarzschild black hole obtained in the four dimensional canonical gravity formalism and also in a Lorentzian computation of BTZ black hole entropy. We find that the right expression for the logarithmic correction in the context of the BTZ black hole comes from the modular invariance associated with the toral boundary of the black hole. PACS No.: 04.60.-m, 04.60.Kz, 04.70.Dy e-mail: trg, kaul, [email protected] 1 Within the quantum geometry formulation of gravity, it has been argued that the quantum degrees of freedom of a (3+1)-dimensional black hole can be described in terms of a ChernSimons theory on the horizon[1], [2]. For a (3+1)-dimensional Schwarzschild black hole, this allows an exact calculation of the entropy [2] which for a large horizon area yields, besides the usual Bekenstein-Hawking entropy proportional to the area, a next order log(area) correction with a definite numerical coefficient, −3/2 [3]. SU(2) Wess-Zumino conformal field theory on the boundary plays an important role in this calculation. There are other methods which can been employed to evaluate black hole entropy. Exploiting the nature of corrections to the Cardy formula for density of states in a two-dimensional conformal field theory, Carlip has evaluated this logarithmic correction in several black hole models including those from certain string theories[5]. Contrary to the expectations that these corrections may lead to distinguishing various formulations of quantum gravity, as emphasized by Carlip, the coefficient of logarithmic correction to the Bekenstein-Hawking entropy for large horizon area may have a universal character. The same value of the coefficient appears for a variety of black holes independent of the dimensions. In particular, the same correction was obtained for the entropy of the (2+1)-dimensional Lorentzian BTZ black hole [4] in [5] by studying the growth of states in the asymptotic conformal field theory at the boundary of the black hole spacetime. The semi-classical entropy of the BTZ black hole has been earlier obtained in different Lorentzian [6], [7] and Euclidean [8], [9] formulations of gravity. However, the correction term to semiclassical entropy seen in [5] has not been reproduced in the Euclidean path integral calculations for BTZ black hole, which is surprising. In this paper, we derive an exact expression for the partition function of the BTZ black hole in the Euclidean path integral approach. In this framework, three-dimensional gravity with a negative cosmological constant is described in terms of two SU(2) Chern-Simons theories [10], [11]. Then, SU(2) Wess-Zumino conformal field theories are naturally induced on the boundary [12]. The quantum degrees of freedom corresponding to the entropy of the black hole are described by these conformal field theories. From the exact expression of the partition function, we show that there is indeed a correction to the semi-classical entropy that is proportional to the logarithm of the area (horizon length in this case) with a coefficient −3/2 again in agreement with the result for a four dimensional black hole obtained in ref. [3]. The gravity action Igrav written in a first-order formalism (using triads e and spin connection ω) is the difference of two Chern-Simons actions. Igrav = ICS[A] − ICS[Ā], (1) where A = (