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Exact duality transformations for sigma models and gauge theories. (English) Zbl 1062.82009
Summary: We present an exact duality transformation in the framework of statistical mechanics for various lattice models with non-Abelian global or local symmetries. The transformation applies to sigma models with variables in a compact Lie group \(G\) with global \(G\times G\)-symmetry (the chiral model) and with variables in coset spaces \(G/H\) and a global \(G\)-symmetry [for example, the nonlinear \(\text{O}(N)\) or \(\mathbb {RP}^N\) models] in any dimension \(d \geq 1\). It is also available for lattice gauge theories with local gauge symmetry in dimensions \(d\geq 2\) and for the models obtained from minimally coupling a sigma model of the type mentioned above to a gauge theory. The duality transformation maps the strong coupling regime of the original model to the weak coupling regime of the dual model. Transformations are available for the partition function, for expectation values of fundamental variables (correlators and generalized Wilson loops) and for expectation values in the dual model which correspond in the original formulation to certain ratios of partition functions (free energies of dislocations, vortices or monopoles). Whereas the original models are formulated in terms of compact Lie groups \(G\) and \(H\), coset spaces \(G/H\) and integrals over them, the configurations of the dual model are given in terms of representations and intertwiners of \(G\) and \(H\). They are spin networks and spin foams. The partition function of the dual model describes the group -theoretic aspects of the strong coupling expansion in a closed form.
MSC:
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
81T25 Quantum field theory on lattices
81T10 Model quantum field theories
81T13 Yang-Mills and other gauge theories in quantum field theory
82B10 Quantum equilibrium statistical mechanics (general)
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References:
[1] Kramers, Phys. Rev. 60 pp 252– (1941)
[2] Wegner, J. Math. Phys. 12 pp 2259– (1971)
[3] Savit, Phys. Rev. Lett. 39 pp 55– (1977)
[4] Peskin, Ann. Phys. 113 pp 122– (1978)
[5] Savit, Rev. Mod. Phys. 52 pp 453– (1980)
[6] Einhorn, Phys. Rev. D 17 pp 2583– (1978)
[7] Drouffe, Phys. Rep. 102 pp 1– (1983)
[8] Anishetty, Phys. Lett. B 314 pp 387– (1993)
[9] Mathur
[10] Ooguri, Mod. Phys. Lett. A 7 pp 2799– (1992)
[11] J. C. Baez, ”An introduction to spin foam models of BF theory and quantum gravity,” inGeometry and Quantum Physics, Lecture Notes in Physics 543 (Springer, Berlin, 2000), pp. 25–93; · Zbl 0978.81043
[12] Ooguri
[13] Oriti, Rep. Prog. Phys. 64 pp 1703– (2001)
[14] Oriti
[15] Baez, Class. Quantum Grav. 15 pp 1827– (1998)
[16] Baez
[17] Reisenberger
[18] Reisenberger, Phys. Rev. D 56 pp 3490– (1997)
[19] Rovelli
[20] Oeckl, Nucl. Phys. B 598 pp 400– (2001)
[21] Pfeiffer
[22] Pfeiffer, Nucl. Phys. B, Proc. Suppl. 106 pp 1010– (2002)
[23] Oeckl
[24] Pfeiffer, J. Math. Phys. 42 pp 5272– (2001)
[25] Pfeiffer
[26] Oeckl, J. Geom. Phys. 46 pp 308– (2003)
[27] Grosse, Int. J. Theor. Phys. 40 pp 459– (2001)
[28] S. Majid,Foundations of Quantum Group Theory(Cambridge University Press, Cambridge, 1995). · Zbl 0857.17009
[29] R. Carter, G. Segal, and I. Macdonald,Lectures on Lie Groups and Lie Algebras, London Mathematical Society Student Texts 32 (Cambridge University Press, Cambridge, 1995). · Zbl 0832.22001
[30] N. J. Vilenkin and A. U. Klimyk,Representations of Lie Groups and Special Functions-Class I Representations, Special Functions and Integral Transforms, Volume 2(Kluwer Academic, Doordrecht, 1993). · Zbl 0809.22001
[31] Pfeiffer, Class. Quantum Grav. 19 pp 1109– (2002)
[32] Pfeiffer
[33] Cucchieri, J. Stat. Phys. 86 pp 581– (1997)
[34] Sokal
[35] R. Penrose, ”Angular momentum: An approach to combinatorial space-time,” inQuantum Theory and Beyond: Essays and Discussions Arising from a Colloquium, edited by T. Bastin (Cambridge University Press, Cambridge, 1971), pp. 151–180.
[36] Rovelli, Phys. Rev. D 52 pp 5743– (1995)
[37] Smolin
[38] Fröhlich, Commun. Math. Phys. 112 pp 343– (1987)
[39] Barrett · Zbl 1405.94065
[40] H. J. Rothe,Lattice Gauge Theories–An Introduction(World Scientific, Singapore, 1992). · Zbl 0875.81030
[41] I. Montvay and G. Munster,Quantum Fields on a Lattice(Cambridge University Press, Cambridge, 1994).
[42] Jersák, Phys. Rev. D 60 pp 054502– (1999)
[43] Pfeiffer
[44] Hari Dass, Nucl. Phys. B, Proc. Suppl. 94 pp 670– (2001)
[45] Shin · Zbl 1185.94055
[46] Baez, Class. Quantum Grav. 19 pp 4627– (2002)
[47] Tsang
[48] Pfeiffer
[49] Jaimungal, Nucl. Phys. B 542 pp 441– (1999)
[50] Jaimungal
[51] Barrett, Trans. Am. Math. Soc. 348 pp 3997– (1996)
[52] Westbury
[53] Crane, J. Knot Theory Ramif. 6 pp 177– (1997)
[54] Yetter
[55] Barrett, Class. Quantum Grav. 17 pp 3101– (2000)
[56] Crane · Zbl 1349.90074
[57] José, Phys. Rev. B 16 pp 1217– (1977)
[58] Banks, Nucl. Phys. B 129 pp 493– (1977)
[59] Campostrini, Phys. Rev. D 52 pp 358– (1995)
[60] Vicari
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