Potts model, O(n) non-linear sigma-models and spanning forests
Seminar Room 1, Newton Institute
In modern language, Kirchhoff Matrix-Tree Theorem (of 1847) puts in relation the (multivariate) generating function for spanning trees on a graph to the partition function of the scalar fermionic free field. A trivial corollary concerns rooted spanning forests and the massive perturbation of the free field. We generalize these facts in many respects. In particular, we show that a fermionic theory with a 4-fermion interaction gives the generating function for unrooted spanning forests, which are a limit of Potts Model for q -> 0. Remarkably, this theory coincides with the perturbative theory originated from a non-linear sigma-model with OSP(1|2) symmetry, which, in Parisi-Sourlas correspondence, is expected to coincide with the analytic continuation of O(n) model to n -> -1.
The relation between spanning forests and the fermionic theory can be proven directly with combinatorial methods. However, the underlying OSP(1|2) symmetry leads to the definition of a subalgebra of Grassmann Algebra (the scalars under global rotations), with a set of surprising properties that quite simplify all the proofs. With some effort we can also generalize the whole derivation to a family of theories with OSP(1|2m) symmetry, with m a positive integer.
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