Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2001-10-26
Commun.Math.Phys. 245 (2004) 1-25
Physics
High Energy Physics
High Energy Physics - Theory
27 pages, 3 figures. New results on the connection with Yang-Mills theories
Scientific paper
10.1007/s00220-003-1005-3
We study the two-dimensional gauge theory of the symmetric group S_n describing the statistics of branched n-coverings of Riemann surfaces. We consider the theory defined on the disk and on the sphere in the large-n limit. A non trivial phase structure emerges, with various phases corresponding to different connectivity properties of the covering surface. We show that any gauge theory on a two-dimensional surface of genus zero is equivalent to a random walk on the gauge group manifold: in the case of S_n, one of the phase transitions we find can be interpreted as a cutoff phenomenon in the corresponding random walk. A connection with the theory of phase transitions in random graphs is also pointed out. Finally we discuss how our results may be related to the known phase transitions in Yang-Mills theory. We discover that a cutoff transition occurs also in two dimensional Yang-Mills theory on a sphere, in a large N limit where the coupling constant is scaled with N with an extra log N compared to the standard 't Hooft scaling.
D'Adda Alessandro
Provero Paolo
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