×
Alvarez-Gaumé, L.; Ginsparg, P.

Finiteness of Ricci flat supersymmetric nonlinear \(\sigma\)-models. (English) Zbl 0597.53070

Commun. Math. Phys. 102, 311-326 (1985).
The paper deals with the problem of super-normalisation of nonlinear \(\sigma\)-models. Using the dimension independence of the background field method and the geometrical properties of Kähler manifolds, the authors show the on-shell finiteness of \(N=4\) models with target space an arbitrary Ricci flat manifold to all orders of perturbation theory. Some comments are made on finiteness of \(N=2\) and \(N=1\) \(\sigma\)-models.
Reviewer: Z.F.Seidov

Cited in 30 Documents

MSC:

53C80 Applications of global differential geometry to the sciences
81T17 Renormalization group methods applied to problems in quantum field theory
81T60 Supersymmetric field theories in quantum mechanics

Keywords:

quantum field theories; super-normalisation; \(\sigma \) -models; perturbation theory
Cite Review PDF
Full Text: DOI

References:

[1] Alvarez-Gaumé, L., Freedman, D. Z.: Kähler geometry and the renormalization of supersymmetric ?-models. Phys.D22, 846 (1980)
[2] Alvarez-Gaumé, L.: Three-loop finiteness in Ricci flat supersymmetric non-linear ?-models. Nucl. Phys.B184, 180 (1981) · doi:10.1016/0550-3213(81)90214-5
[3] Alvarez-Gaumé, L., Freedman, D. Z.: Geometrical structure and ultraviolet finiteness in the supersymmetric ?-model. Commun. Math. Phys.80, 443 (1981) · doi:10.1007/BF01208280
[4] Green, M. B., Schwarz, J. H.: Anomaly cancellations in supersymmetricD = 10 gauge theory and superstring theory. Phys. Lett.149B, 117 (1984);
[5] Green, M. B., Schwarz, J. H.: Infinity cancellations in SO(32) superstring theory. Phys. Lett.151B, 21 (1985)
[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 ?-models. Nucl. Phys.B177, 282 (1981) · Zbl 0983.81513 · doi:10.1016/0550-3213(81)90392-8
[8] Zumino, B.: Supersymmetry and Kähler manifolds. Phys. Lett.87B, 203 (1979)
[9] Goldberg, S.: Curvature and homology. New York: Dover Publications, 1982, Sect. 5.6 · Zbl 0479.53014
[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 · Zbl 0080.15002
[11] Yau, S.-T.: Calabi’s conjecture and some new results in algebraic geometry. Proc. Natl. Acad. Sci.74, 1798 (1977) · Zbl 0355.32028 · doi:10.1073/pnas.74.5.1798
[12] Lichnerowicz, A.: General theory of connections and the holonomy group. Amsterdam: Noordhoff 1976 · Zbl 0337.53031
[13] Chevalley, C.: Theory of Lie groups, p. 185 Princeton, NJ: Princeton University Press 1946 · Zbl 0063.00842
[14] Alvarez-Gaumé, L., Freedman, D. Z., Mukhi, S.: The background field method and the ultraviolet structure of the supersymmetric non-linear ?-model. Ann. Phys.134, 85 (1981) · doi:10.1016/0003-4916(81)90006-3
[15] Friedan, D.: Non-linear ?-models in 2 + ? dimensions. Phys. Rev. Lett.45, 1057 (1980); · doi:10.1103/PhysRevLett.45.1057
[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) · doi:10.1016/0550-3213(73)90376-3
[18] Gates, J., Grisaru, M., Ro?ek, M., Siegel, W.: Superspace. New York: Benjamin 1983
[19] Atiyah, M., Bott, R., Patodi, V. K.: On the heat equation and the index theorem. Invent. Math.19, 279 (1973) · Zbl 0257.58008 · doi:10.1007/BF01425417
[20] Bott, R.: private communication
[21] Hull, C. M.: Ultraviolet finiteness of supersymmetric non-linear ?-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) · doi:10.1016/0550-3213(83)90605-3
[23] Beauville, A.: Variétés Kählériennes dont la première classe de Chern est nulle. J. Differ. Geom.18, 755 (1983) · Zbl 0537.53056
[24] Calabi, E.: Métriques Kählériennes et fibrés holomorphes. Ann. Sci. Éc. Norm Super.12, 266 (1979)
[25] Curtright, T. L., Freedman, D. Z.: Non-linear ?-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)
[26] Alvarez-Gaumé, L., Coleman, S., Ginsparg, P.: Harvard preprint HUTP-85/A037
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.