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We show that all two-dimensional conformal field theories possess a hiddensl(2,R) affine symmetry. More precisely, we add appropriate ghost fields to anarbitrary CFT, and we use them to construct the currents of sl(2,R). We thendefine a BRST operator whose cohomology defines a physical subspace where theextended theory coincides with the original CFT. We use the sl(2,R) algebra toconstruct candidate wave functions for 3-d quantum gravity coupled to matter,and we discuss their viability.Comment: Minor misprints corrected.Eight references added. To appear in  JHEP.34 pages, LaTe

HIDDEN sl(2,R) SYMMETRY IN 2D CFTs AND THE WAVE FUNCTION OF 3D QUANTUM GRAVITY