Liouville Lost, Liouville Regained: Central Charge in a Dynamical Background

Several recent approaches to black hole entropy obtain the density of states from the central charge of a Liouville theory. If Liouville theory is coupled to a dynamical spacetime background, however, the classical central charge vanishes. I show that the central charge can be restored by introducing appropriate constraints, which may be interpreted as fall-off conditions at a boundary such as a black hole horizon. ∗email: [email protected] In an effort to understand the microscopic degrees of freedom responsible for black hole entropy, a number of researchers have recently begun to look at Liouville theories, either at spatial infinity [1, 2, 3, 4, 5, 6, 7] or in a neighborhood of the horizon [8]. Such theories naturally arise at the asymptotic boundary of the (2+1)-dimensional BTZ black hole [1], and are relevant to many higher-dimensional black holes whose near-horizon behavior resembles that of the BTZ black hole [9]; they may also be obtained near the horizon of an arbitrary black hole by dimensionally reducing to the r–t plane [8]. Since Liouville theory is a conformal field theory, its density of states can be inferred from its central charge by means of the Cardy formula [10]. There is some debate as to whether the Liouville states represent the genuine gravitational degrees of freedom or merely give an effective “thermodynamic” description [7,11,12,13], but in either case, the result offers a potential explanation for the universality of the Bekenstein-Hawking entropy: the density of states may be determined by conformal symmetry, independent of the details of quantum gravity [14]. The computation of black hole entropy from Liouville theory depends sensitively on the central charge. In particular, it is the appearance of a classical central charge—that is, a central term that is already present in the Poisson brackets—that leads to an order 1/h̄ contribution to the entropy. Unfortunately, as I shall demonstrate below, when Liouville theory is coupled to a dynamical two-dimensional metric, the classical central charge vanishes. This problem is evaded in some approaches to Liouville theory at spatial infinity, in which the boundary conditions freeze the metric, but it is present in other treatments of spatial infinity [3, 4, 5, 6] and in the near-horizon derivation [8]. Of course, one can recover the central charge, and thus the Bekenstein-Hawking entropy, by freezing the dynamics of the metric. But this seems too strong a condition. The main goal of this paper is to demonstrate that it is sufficient to impose asymptotic fall-off conditions on the metric, either at infinity or at the horizon. To show this, I will introduce a new method for treating such fall-off conditions, via second class constraints and Dirac brackets, which I hope may be more generally useful. 1. Liouville central charge in a dynamic background We begin with a generalized Liouville action, IL[φ, g] = 1 4π ∫ dx √−g { 1 2 g∂aφ∂bφ+ 1 γ φR + V [φ] } . (1.1) For standard Liouville theory, the potential is V [φ] = μ 2γ e, (1.2) but more general forms are possible [15]. After the usual ADM decomposition of the metric, ds = Ndt − σ (dx+ βdt) , (1.3) a tedious but straightforward computation brings the action to the canonical form IL = ∫