Spanning forests and OSP(N|2M) -invariant σ-models

The present paper is part of our ongoing work on OSP(N|2M) supersymmetric σ-models, their relation with the Potts model at q = 0 and spanning forests, and the rigorous analytic continuation of the partition function as an entire function of N - 2M, a feature first predicted by Parisi and Sourlas in the 1970s. Here we accomplish two main steps. First, we analyze in detail the role of the Ising variables that arise when the constraint in the OSP(1|2) model is solved, and we point out two situations in which the Ising and forest variables decouple. Second, we establish the analytic continuation for the OSP(N|2M) model in some special cases: when the underlying graph is a forest, and for the Nienhuis action on a cubic graph. We also make progress in understanding the series-parallel graphs.