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Non-Abelian bosonization in two dimensions. (English) Zbl 0536.58012
One of the most startling aspects of mathematical physics in \(1+1\) dimensions is the existence of a non-local transformation from local Fermi fields to local Bose fields. This fact is remarkably useful for elucidating the properties of a \(1+1\) dimensional theories. Many phenomena that are difficult to understand in the Fermi language have a simple semiclassical explanation in the Bose formalism. A major limitation of the usual bosonization procedure is that in the case of Fermi theories with non-Abelian symmetries, these symmetries are not preserved by the bosonization. In this paper the author gives a non-Abelian generalization of the usual formulas for bosonization of fermions in \(1+1\) dimensions which manifestly possess all the symmetries of the Fermi theory.
Reviewer: M.Martellini

MSC:
81S10 Geometry and quantization, symplectic methods
53D50 Geometric quantization
81T60 Supersymmetric field theories in quantum mechanics
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[1] Coleman, S.: Quantum sine-Gordon equation as the massive Thirring model. Phys. Rev. D11, 2088 (1975)
[2] Mandelstam, S.: Soliton operators for the quantized sine-Gordon equation. Phys. Rev. D11, 3026 (1975)
[3] Baluni, V.: The Bose form of two-dimensional quantum chromodynamics. Phys. Lett.90 B, 407 (1980)
[4] Steinhardt, P.J.: Baryons and baryonium in QCD2. Nucl. Phys. B176, 100 (1980) · doi:10.1016/0550-3213(80)90065-6
[5] Amati, D., Rabinovici, E.: On chiral realizations of confining theories. Phys. Lett.101 B, 407 (1981)
[6] Wess, J., Zumino, B.: Consequences of anomalous word identities. Phys. Lett.37 B, 95 (1971)
[7] D’Adda, A., Davis, A.C., DiVecchia, P.: Effective actions in non-abelian theories. Phys. Lett.121 B, 335 (1983)
[8] Polyakov, A.M., Wiegmann, P.B.: Landau Institute preprint (1983)
[9] Alvarez, O.: Berkeley preprint (1983)
[10] Witten, E.: Global aspects of current algebra. Nucl. Phys. B (to appear)
[11] Novikov, S.P.: Landau Institute preprint (1982)
[12] Polyakov, A.M.: Interaction of Goldstone particles in two dimensions. Applications to ferromagnets and massive Yang-Mills fields. Phys. Lett.59 B, 79 (1975)
[13] Belavin, A.A., Polyakov, A.M.: Metastable states of two-dimensional isotropic ferromagnets. JETP Lett.22, 245 (1975)
[14] Nappi, C.R.: Some properties of an analog of the chiral model. Phys. Rev. D21, 418 (1980)
[15] Goto, T., Imamura, I.: Note on the non-perturbation-approach to quantum field theory. Prog. Theor. Phys.14, 396 (1955) · doi:10.1143/PTP.14.396
[16] Schwinger, J.: Field-theory commutators. Phys. Rev. Lett.3, 296 (1959) · Zbl 0091.22906 · doi:10.1103/PhysRevLett.3.296
[17] Jackiw, R.: In: Lectures on current algebra and its applications, Treiman S.B., et al. (eds.): Princeton, NJ: Princeton University Press 1972
[18] Coleman, S., Gross, D., Jackiw, R.: Fermion avatars of the Sugawara model. Phys. Rev.180, 1359 (1969) · doi:10.1103/PhysRev.180.1359
[19] Kac, V.G.: J. Funct. Anal. Appl.8, 68 (1974) · Zbl 0299.17005 · doi:10.1007/BF02028313
[20] Lepowsky, J., Wilson, R.L.: Construction of the affine Lie algebra.A 1(1). Commun. Math. Phys.62, 43 (1978) · Zbl 0388.17006 · doi:10.1007/BF01940329
[21] Frenkel, I.B.: Spinor representations of affine Lie algebras. Proc. Natl. Acad. Sci. USA77, 6303 (1980); J. Funct. Anal.44, 259 (1981) · Zbl 0451.17004 · doi:10.1073/pnas.77.11.6303
[22] Feingold, A.J., Frenkel, I.B.: IAS preprint (1983)
[23] Belavin, A.M., Polyakov, A.M., Schwar, A.S., Tyupkin, Yu.S.: Pseudoparticle solutions of the Yang-Mills equations. Phys. Lett.59 B, 85 (1975)
[24] ’t Hooft, G.: Symmetry breaking through Bell-Jackiw anomalies. Phys. Rev. Lett.37, 8 (1976); · doi:10.1103/PhysRevLett.37.8
[25] Computation of the quantum effects due to a four-dimensional pseudoparticle. Phys. Rev. D14, 3432 (1976)
[26] Callan, C.G., Jr., Dashen, R., Gross, D.J.: The structure of the gauge theory vacuum. Phys. Lett.63 B, 334 (1976)
[27] Jackiw, R., Rebbi, C.: Vacuum periodicity in a Yang-Mills quantum theory. Phys. Rev. Lett.37, 172 (1976) · doi:10.1103/PhysRevLett.37.172
[28] Segal, G.: Unitary representations of some infinite-dimensional groups. Commun. Math. Phys.80, 301 (1981) · Zbl 0495.22017 · doi:10.1007/BF01208274
[29] Frenkel, I., Kac, V.G.: Basic representations. Invent Math.62, 23 (1980) · Zbl 0493.17010 · doi:10.1007/BF01391662
[30] Kac, V.G., Peterson, D.H.: Spin and wedge representations of infinite-dimensional Lie algebras and groups. Proc. Natl. Acad. Sci. USA78, 3308 (1981) · Zbl 0469.22016 · doi:10.1073/pnas.78.6.3308
[31] Frenkel, I.: Private communication
[32] Frishman, Y.: Quark trapping in a model field theory. Mexico City 1973. Berlin, Heidelberg, New York: Springer 1975
[33] Deser, S., Jackiw, R., Templeton, S.: Three-dimensional massive gauge theories. Phys. Rev. Lett.48, 975 (1982); Topologically massive gauge theories. Ann. Phys. (NY)140, 372 (1982) · doi:10.1103/PhysRevLett.48.975
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