LIOUVILLE AND TODA W(3) GRAVITY POTENTIALS VIA MATTER-DEPENDENT ANOMALY CANCELLATION

I determine the allowed Liouville sector potentials of BRST quantized non-critical strings and W3 strings. The potentials, previously derive d from conformal-field-theory techniques, are obtained via the cancell ation of matter-dependent (potential-dependent) anomalies in the Slavn ov-Taylor-Ward identities. These anomalies arise as a consequence of a dding potentials to an existing 2D gravity or renormalized chiral W3 g ravity action. Higher order ''miracles'' and Wess-Zumino consistency a re considered as checks on the results.