Two-dimensional gauge theories of the symmetric group S(n) and branched n-coverings of Riemann surfaces in the large-n limit

Branched n-coverings of Riemann surfaces are described by a 2d lattice gauge theory of the symmetric group S(n) defined on a cell discretization of the surface. We study the theory in the large-n limit, and we find a rich phase diagram with first and second order transition lines. The various phases are characterized by different connectivity properties of the covering surface. We point out some interesting connections with the theory of random walks on group manifolds and with random graph theory.Comment: Talk presented at the "Light-cone physics: particles and strings", Trento, Italy, September 200

Combinatorial and topological phase structure of non-perturbative n-dimensional quantum gravity

We provide a non-perturbative geometrical characterization of the partition function of $n$-dimensional quantum gravity based on a coarse classification of riemannian geometries. We show that, under natural geometrical constraints, the theory admits a continuum limit with a non-trivial phase structure parametrized by the homotopy types of the class of manifolds considered. The results obtained qualitatively coincide, when specialized to dimension two, with those of two-dimensional quantum gravity models based on random triangulations of surfaces.Comment: 13 page

Unsigned state models for the Jones polynomial

It is well a known and fundamental result that the Jones polynomial can be expressed as Potts and vertex partition functions of signed plane graphs. Here we consider constructions of the Jones polynomial as state models of unsigned graphs and show that the Jones polynomial of any link can be expressed as a vertex model of an unsigned embedded graph. In the process of deriving this result, we show that for every diagram of a link in the 3-sphere there exists a diagram of an alternating link in a thickened surface (and an alternating virtual link) with the same Kauffman bracket. We also recover two recent results in the literature relating the Jones and Bollobas-Riordan polynomials and show they arise from two different interpretations of the same embedded graph.Comment: Minor corrections. To appear in Annals of Combinatoric