Chiral fermions in 2d quantum gravity

Two theories of chiral fermions coupled to different quantum gravities in two dimensions are studied. One employs Jackiw's ansatz for classical gravity by introducing an auxiliary scalar, the other is based on the induced quantum gravity of Polyakov, which has no classical analogue. By investigating a localized theory of the effective action we show that in both cases a limited number of fermions of either chirality may couple consistently. It is stressed that the Weyl variable has to be quantized properly, which is related to recent work done on non-critical strings.