Loop quantum gravity without the Hamiltonian constraint

We construct a version of loop quantum gravity on a fixed constant mean curvature Cauchy slice for general relativity conformally coupled to a scalar field. The key input is to gauge fix the Hamiltonian constraint classically with the generator of local conformal transformations. This generator coincides with the constant mean curvature condition in the presence of a conformally coupled scalar field. The resulting quantum theory is a reduced phase space quantization with respect to the Hamiltonian constraint. As an application, it allows to calculate the entropy of a certain class of black holes based on counting physical states. In addition, the interpretation of the geometric operators of loop quantum gravity changes in this framework, and an effective Planck scale emerges with a dependence on the scalar field.