Hidden sl(2,ℝ) symmetry in 2d CFTs and the wave function of 3d quantum gravity

We show that all two-dimensional conformal field theories possess a hidden sl(2,ℝ) affine symmetry. More precisely, we add appropriate ghost fields to an arbitrary CFT, and we use them to construct the currents of sl(2,ℝ). We then define a BRST operator whose cohomology defines a physical subspace where the extended theory coincides with the original CFT. We use the sl(2,ℝ) algebra to construct candidate wave functions for 3-d quantum gravity coupled to matter, and we discuss their viability.