The wrong kind of gravity

The KPZ formula [V.G. Knizhnik, A.M. Polyakov, and A.B. Zamolodchikov, Mod. Phys. Lett. A 3 (1988) 819] shows that coupling central charge c less than or equal to 1 spin models to 2D quantum gravity dresses the conformal weig hts to get new critical exponents, where the relation between the original and dressed weights depends only on c. At the discrete level the coupling t o 2D gravity is effected by putting the spin models on annealed ensembles o f Phi(3) planar random graphs or their dual triangulations, where the conne ctivity fluctuates on the same time-scale as the spins. Since the sole determining factor in the dressing is the central charge, on e could contemplate putting a spin model on a quenched ensemble of 2D gravi ty graphs with the "wrong" c value. We might then expect to see the critica l exponents appropriate to the c value used in generating the graphs. In su ch cases the KPZ formula could be interpreted as giving a continuous line o f critical exponents which depend on this central charge. We note that rati onal exponents other than the KPZ values can be generated using this proced ure for the Ising, tricritical Ising and 3-state Potts models. (C) 1999 Pub lished by Elsevier Science B.V. All rights reserved.