Skip to content



Potts model, O(n) non-linear sigma-models and spanning forests

Sportiello, A (Milano)
Wednesday 09 April 2008, 14:00-15:00

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.


[pdf ]




The video for this talk should appear here if JavaScript is enabled.
If it doesn't, something may have gone wrong with our embedded player.
We'll get it fixed as soon as possible.

Back to top ∧