Abstract
This is a brief report on research1–5 involving the application of differential geometry to the supersymmetric nonlinear σ-model in two spacetime dimensions. We use differential geometry not from any sense of mathematical purity but because it illuminates physical aspects of the theories. In particular it is important for ultraviolet properties. The nonlinear σ-model in two dimensions is critically renormalizable. By power counting one expects ultraviolet divergences in every order of perturbation theory. The geometrical argument to be outlined here shows that the on-shell ultraviolet divergences are severely limited and actually absent to all orders in a class of supersymmetric σ-models.
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References
L. Alvarez-Gaumé and D. Z. Freedman, Phys. Rev. D22, 846 (1980).
L. Alvarez-Gaumé and D. Z. Freedman, Phys. Lett. 94B, 171 (1980).
D. Z. Freedman and P. K. Townsend, Nucl. Phys. B.
L. Alvarez-Gaume, D. Z. Freedman, and S. Mukhi, to be published in Annals of Physics.
L. Alvarez-Gaume and D. Z. Freedman, M.I.T. Mathematics Department preprint, December 1980.
D. Friedman, Phys. Rev. Lett. 1057 (1980).
G.t Hooft, Nuclear Phys. B61, 455 (1973).
J. Honerkamp, Nucl. Phys. B36, 130 (1972).
G. Ecker and J. Honerkamp, Nule. Phys. B35, 481 (1971).
B. Zumino, Phys. Lett. 87B, 203 (1979).
A. Lichnerowicz, Theorie Globale des Connections et des Groupes d’Holonomie, published by Consiglio Nazionale delle Recherche, 1955.
This result is stronger than that of Ref. 9 in that Hermiticity was assumed there and is proven in the present approach.
E. Calabi, Ann. Sc. de I’E.N.S. 12, 266 (1979).
L. Alvarez-Gaume, to be published in Nucl. Phys. B.
G. W. Gibbons and S. W. Hacking, Phys. Lett. 430 (1978).
N. Hitchin, Proc. Camb. Philos. Soc. 85. 465 (1979).
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© 1981 Springer Science+Business Media New York
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Freedman, D.Z. (1981). Differential Geometry and the Supersymmetric Nonlinear σ-Model. In: Perlmutter, A. (eds) Gauge Theories, Massive Neutrinos and Proton Decay. Studies in the Natural Sciences, vol 18. Springer, Boston, MA. https://doi.org/10.1007/978-1-4684-1107-2_25
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DOI: https://doi.org/10.1007/978-1-4684-1107-2_25
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