Abstract
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 disc 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.
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References
Kostov, I.K., Staudacher, M.: Phys. Lett. B394, 75 (1997), [hep-th/9611011]
Kostov, I.K., Staudacher, M., Wynter, T.: Commun. Math. Phys. 191, 283 (1998), [hep-th/9703189]
Billo, M., D’Adda, A., Provero, P.: hep-th/0103242
Gross, D.J., Taylor, W.I.: Nucl. Phys. B400, 181 (1993), [hep-th/9301068]
Gross, D.J., Taylor, W.I.: Nucl. Phys. B403, 395 (1993), [hep-th/9303046]
Baez, J., Taylor, W.: Nucl. Phys. B 426, 53 (1994), [arXiv:hep-th/9401041]
Billo, M., D’Adda, A., Provero, P.: Unpublished
Douglas, M.R., Kazakov, V.A.: Phys. Lett. B 319, 219 (1993), [hep-th/9305047]
Caselle, M., D’Adda, A., Magnea, L., Panzeri, S.: arXiv:hep-th/9309107
Gross, D.J., Matytsin, A.: Nucl. Phys. B 429, 50 (1994), [arXiv:hep-th/9404004]
Gross, D.J., Matytsin, A.: Nucl. Phys. B 437, 541 (1995), [arXiv:hep-th/9410054]
Douglas, M.R., Li, K., Staudacher, M.: Nucl. Phys. B 420, 118 (1994), [arXiv:hep-th/9401062]
Ganor, O., Sonnenschein, J., Yankielowicz, S.: Nucl. Phys. B 434, 139 (1995), [arXiv:hep-th/9407114]
Crescimanno, M.J., Taylor, W.: Nucl. Phys. B 437, 3 (1995), [arXiv:hep-th/9408115]
Diaconis, P., Shahshahani, M.: Z. Wahrsch. Verw. Gebiete 57, 159 (1981)
Migdal, A.A.: Sov. Phys. JETP 42, 413 (1975)
Rusakov, B.E. Mod. Phys. Lett. A5, 693 (1990)
Pak, I., Vu, V. H.: Disc. Appl. Math. 110, 251 (2001)
Gross, D.J., Witten, E.: Phys. Rev. D 21, 446 (1980)
Erdös, P., Rényi, A.: Publ. Math. Debrecen 6, 290 (1959)
Erdös, P., Rényi, A.: The Art of Counting, Cambridge: MIT, 1973
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Communicated by R.H. Dijkgraaf
Received: 19, November 2001
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D’Adda, A., Provero, P. Two-Dimensional Gauge Theories of the Symmetric Group S n in the Large-n Limit. Commun. Math. Phys. 245, 1–25 (2004). https://doi.org/10.1007/s00220-003-1005-3
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DOI: https://doi.org/10.1007/s00220-003-1005-3