The arboreal gas and the supersphere sigma model

We discuss the relationship between the phase diagram of the Q = 0 state Potts model, the arboreal gas model, and the supersphere sigma model S = OSP (1/2)/OSP (0/2). We identify the Potts antiferromagnetic critical point with the critical point of the arboreal gas (at negative tree fugacity), and with a critical point of the sigma model. We show that the corresponding conformal theory on the square lattice has a non-linearly realized OSP (2/2) = SL(1/2) symmetry, and involves non-compact degrees of freedom, with a continuous spectrum of critical exponents. The role of global topological properties in the sigma model transition is discussed in terms of a generalized arboreal gas model.