Quantum-corrected Cardy entropy for generic (1+1)-dimensional gravity

Various studies have explored the possibility of explaining the Bekenstein-Hawking (black hole) entropy by way of some suitable state-counting procedure. Notably, many of these treatments have used the well-known Cardy formula as an intermediate step. Our current interest is a recent calculation in which Carlip has deduced the leading-order quantum correction to the (otherwise) classical Cardy formula. In this paper, we apply Carlip's formulation to the case of a generic model of two-dimensional gravity with coupling to a dilaton field. We find that the corrected Cardy entropy includes the anticipated logarithmic 'area' term. Such a term is also evident when the entropic correction is derived independently by thermodynamic means. However, there is an apparent discrepancy between the two calculations with regard to the factor in front of the logarithm. In fact, the two values of this prefactor can only agree for very specific two-dimensional models, such as that describing Jackiw-Teitelboim theory.