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Finiteness of Ricci flat supersymmetric non-linear σ-models

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Abstract

Combining the constraints of Kähler differential geometry with the universality of the normal coordinate expansion in the background field method, we study the ultraviolet behavior of 2-dimensional supersymmetric non-linear σ-models with target space an arbitrary riemannian manifoldM. We show that the constraint ofN=2 supersymmetry requires that all counterterms to the metric beyond one-loop order be cohomologically trivial. It follows that such supersymmetric non-linear σ-models defined on locally symmetric spaces are super-renormalizable and thatN=4 models are on-shell ultraviolet finite to all orders of perturbation theory.

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References

  1. Alvarez-Gaumé, L., Freedman, D. Z.: Kähler geometry and the renormalization of supersymmetric σ-models. Phys.D22, 846 (1980)

    Google Scholar 

  2. Alvarez-Gaumé, L.: Three-loop finiteness in Ricci flat supersymmetric non-linear σ-models. Nucl. Phys.B184, 180 (1981)

    Google Scholar 

  3. Alvarez-Gaumé, L., Freedman, D. Z.: Geometrical structure and ultraviolet finiteness in the supersymmetric σ-model. Commun. Math. Phys.80, 443 (1981)

    Google Scholar 

  4. Green, M. B., Schwarz, J. H.: Anomaly cancellations in supersymmetricD = 10 gauge theory and superstring theory. Phys. Lett.149B, 117 (1984);

    Google Scholar 

  5. Green, M. B., Schwarz, J. H.: Infinity cancellations in SO(32) superstring theory. Phys. Lett.151B, 21 (1985)

    Google Scholar 

  6. Candelas, P., Horowitz, G., Strominger, A., Witten, E.: Vacuum Configurations for superstrings. Santa Barbara preprint NSF-ITP-84-170 (1984), to appear in Nucl. Phys. B

  7. Freedman, D. Z., Townsend, P. K.: Antisymmetric tensor gauge theories and non-linear σ-models. Nucl. Phys.B177, 282 (1981)

    Google Scholar 

  8. Zumino, B.: Supersymmetry and Kähler manifolds. Phys. Lett.87B, 203 (1979)

    Google Scholar 

  9. Goldberg, S.: Curvature and homology. New York: Dover Publications, 1982, Sect. 5.6

    Google Scholar 

  10. Calabi, E.: On Kähler manifolds with vanishing canonical class. In: Algebraic geometry and topology: a symposium in honor of S. Lefschetz. Princeton, NJ: Princeton University Press 1957, p. 78

    Google Scholar 

  11. Yau, S.-T.: Calabi's conjecture and some new results in algebraic geometry. Proc. Natl. Acad. Sci.74, 1798 (1977)

    Google Scholar 

  12. Lichnerowicz, A.: General theory of connections and the holonomy group. Amsterdam: Noordhoff 1976

    Google Scholar 

  13. Chevalley, C.: Theory of Lie groups, p. 185 Princeton, NJ: Princeton University Press 1946

    Google Scholar 

  14. Alvarez-Gaumé, L., Freedman, D. Z., Mukhi, S.: The background field method and the ultraviolet structure of the supersymmetric non-linear σ-model. Ann. Phys.134, 85 (1981)

    Google Scholar 

  15. Friedan, D.: Non-linear σ-models in 2 + ɛ dimensions. Phys. Rev. Lett.45, 1057 (1980);

    Google Scholar 

  16. Friedan, D.: Ph.D. Thesis, U. C. Berkeley (unpublished, 1980)

  17. 't Hooft, G.: Dimensional regularization and the renormalization group. Nucl. Phys.B61, 455 (1973)

    Google Scholar 

  18. Gates, J., Grisaru, M., Roček, M., Siegel, W.: Superspace. New York: Benjamin 1983

    Google Scholar 

  19. Atiyah, M., Bott, R., Patodi, V. K.: On the heat equation and the index theorem. Invent. Math.19, 279 (1973)

    Google Scholar 

  20. Bott, R.: private communication

  21. Hull, C. M.: Ultraviolet finiteness of supersymmetric non-linear σ-models. IAS preprint (1985), to appear in Nucl. Phys.B

  22. Bagger, J., Witten, E.: Matter couplings inN = 2 supergravity. Nucl. Phys.B222, 1 (1983)

    Google Scholar 

  23. Beauville, A.: Variétés Kählériennes dont la première classe de Chern est nulle. J. Differ. Geom.18, 755 (1983)

    Google Scholar 

  24. Calabi, E.: Métriques Kählériennes et fibrés holomorphes. Ann. Sci. Éc. Norm Super.12, 266 (1979)

    Google Scholar 

  25. Curtright, T. L., Freedman, D. Z.: Non-linear σ-models with extended supersymmetry in four dimensions. Phys. Lett.90B, 71 (1980); Alvarez-Gaumé L., Freedman, D. Z.: Ricci flat Kähler manifolds and supersymmetry. Phys. Lett.94B, 171 (1980)

    Google Scholar 

  26. Alvarez-Gaumé, L., Coleman, S., Ginsparg, P.: Harvard preprint HUTP-85/A037

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Authors and Affiliations

  1. Lyman Laboratory of Physics, Harvard University, 02138, Cambridge, MA, USA

    L. Alvarez-Gaumé & P. Ginsparg

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  1. L. Alvarez-Gaumé
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  2. P. Ginsparg
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Communicated by A. Jaffe

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Alvarez-Gaumé, L., Ginsparg, P. Finiteness of Ricci flat supersymmetric non-linear σ-models. Commun.Math. Phys. 102, 311–326 (1985). https://doi.org/10.1007/BF01229382

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  • DOI: https://doi.org/10.1007/BF01229382

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