Description of Black Hole Microstates by Means of a Free Affine-Scalar   Field
  In this article we will investigate the origin of central extensions in the Poisson algebra of charges, which arise in the dimensionally reduced theories describing black holes. We will see that the equations of motion and constraints arising from the dimensionally reduced action involve two fields i.e. the dilaton and the conformal factor. This fields can be integrated by means of a free field. The transformation properties of this field are studied. It will be shown that in the near horizon approximation this field must transform like an affine scalar. The stress tensor that generates such affine transformations is the improved stress tensor. The second derivative term in the tensor is responsible for the central extension in the Poisson algebra. It is therefore the affine transformation property that is responsible for the arising of central charges The central charge can be used to compute the black hole entropy by means of the Cardy formula. 
Alex Giacomini