Abstract
We consider global anomalies for heterotic string theory formulated on orbifolds. The vanishing of certain characteristic classes in group cohomology provides sufficient conditions for the absence of global anomalies. For abelian orbifolds level matching implies these cohomology conditions, so suffices for the absence of anomalies. For nonabelian orbifolds level matching does not suffice, and there are additional constraints. We give some examples to illustrate these new constraints.
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References
Vafa, C.: Modular invariance and discrete torsion on orbifolds. Nucl. Phys. B273, 592 (1986)
Freed, D.S.: Determinants, torsion, and strings. Commun. Math. Phys.107, 483–513 (1986)
Gross, D.J., Harvey, J.A., Martinec, E., Rohm, R.: Heterotic string theory (I). The free heterotic string. Nucl. Phys. B256, 253 (1985)
Dixon, L., Harvey, J., Vafa, C., Witten, E.: Strings on orbifolds (II). Nucl. Phys. B274, 285 (1986)
Narain, K.S., Sarmadi, M.H., Vafa, C.: Asymmetric orbifolds. Harvard University preprint, HUTP-86/A089
Kawai, H., Lewellen, D.C., Tye, S.-H.H.: Phys. Rev. Lett.57, 1832 (1986); Cornell preprint CLNS 86/751
Meuller, M., Witten, E.: Princeton preprint (1986)
Antoniadis, I., Bachas, C., Kounnas, C.: To appear
Witten, E.: Global anomalies in string theory. In: Anomalies, geometry, and topology. White, A. (ed.). World Scientific 1985, pp. 61–99
Wen, X.-G., Witten, E.: Electric and magnetic charges in superstring models. Princeton preprint
Witten, E.: Global gravitational anomalies. Commun. Math. Phys.100, 197–229 (1985)
Witten, E.: Topological tools in ten dimensional physics. Lectures at the Workshop on Unified String Theories, Santa Barbara
Atiyah, M.F.: The logarithm of the Dedekind η-function. Preprint
Kubert, D.S., Lang, S.: Modular units: Berlin, Heidelberg, New York: Springer 1981
Freed, D.S.: On determinant line bundles. In: Mathematical aspects of string theory. Yau, S.T. (ed.). New York: World Scientific (to appear)
Frenkel, I., Lepowsky, J., Meurmen, A.: In: Proceedings of a conference on vertex operators in mathematics and physics. Lepowsky, J. (ed.). Berlin, Heidelberg, New York: Springer 1984
Landweber, P.S., Ravenel, D., Strong, R.: Periodic cohomology theories defined by elliptic curves. Preprint
Witten, E.: Elliptic genera and quantum field theory. Preprint
Alvarez-Gaumé, L., Witten, E.: Gravitational anomalies. Nucl. Phys. B234, 269 (1983)
Lang, S.: Elliptic functions.Berlin, Heidelberg, New York: Springer 1973
Atiyah, M.F.: Riemann surfaces and spin structures. Ann. Scient. Éc. Norm. Sup. 4e série, t. 4, fasc. 1, 47–62 (1971)
Alvarez-Gaumé, L., Moore, G., Vafa, C.: Theta functions, modular invariance, and strings. Commun. Math. Phys.106, 1 (1986)
Mumford, D.: Tata lectures on theta, Vols. I and II. Boston: Birkhäuser 1983
Igusa, J.: Theta functions. Berlin, Heidelberg, New York. Springer 1972
Bott, R.: Lectures on Morse theory, old and new. Bull. AMS7, 331–358 (1982)
Atiyah, M.F., Bott, R.: The moment map and equivariant cohomology. Topology23, 1–28 (1984)
Atiyah, M.F., Singer, I.M.: Dirac operators coupled to vector potentials. Proc. Nat. Acad. Sci.81, 259 (1984)
Quillen, D.: Determinants of Cauchy-Riemann operators over a Riemann surface. Funk. Anal. iprilozen19, 3 (1985)
Bismut, J.M., Freed, D.S.: The analysis of elliptic families: Metrics and connections on determinant bundles. Commun. Math. Phys.106, 159–176 (1986)
Bismut, J.M., Freed, D.S.: The analysis of elliptic families: Dirac operators, eta invariants, and the holonomy theorem of Witten. Commun. Math. Phys.107, 103–160 (1986)
Narain, K.S., Sarmadi, M.H., Witten, E.: Nucl. Phys. B279, 369 (1987)
Ginsparg, P., Vafa, C.: Toroidal compactification of non-supersymmetric heterotic strings. Harvard University preprint HUTP-86/A064
Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry. I. Math. Proc. Cambridge Philos. Soc.77, 43–69 (1975); II, Math. Proc. Cambridge Philos. Soc.78, 405–432 (1975); III, Math. Proc. Cambridge Philos. Soc.79, 71–99 (1976)
Alvarez-Gaumé, L., Ginsparg, P.: The structure of gauge and gravitational anomalies. Ann. Phys.161, 423 (1985)
Elitzur, S., Nair, V.: Non-perturbative anomalies in higher dimensions. Nucl. Phys. B243, 205 (1984)
Pontrjagin, L.: In: Topological groups. London, New York, Paris: Gordon and Breach 1966, Chap. 1
Lee, R., Miller, E.Y., Weintraub, S.H.: Spin structures: their relationship to quadratic pairings, Arf invariants, Rohlin invariants, and the holonomy of determinant line bundles, preprint
Lewis, G.: The integral cohomology rings of groups of orderp 3. Trans. AMS132, 501–529 (1968)
Dixon, L., Harvey, J.: String theories in ten dimensions without spacetime supersymmetry. Nucl. Phys. B274, 93 (1986)
Evens, L.: On the Chern classes of representations of finite groups. Trans. AMS108, 180–193 (1965)
Cartan, H., Eilenberg, S.: Homological algebra. Princeton: Princeton University Press 1956
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Communicated by L. Alvarez-Gaumé
The first author is partially supported by an NSF Postdoctoral Research Fellowship. The second author is supported in part by the NSF contract no. PHY 82-15249, and in part by a fellowship from the Harvard Society of Fellows
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Freed, D.S., Vafa, C. Global anomalies on orbifolds. Commun.Math. Phys. 110, 349–389 (1987). https://doi.org/10.1007/BF01212418
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DOI: https://doi.org/10.1007/BF01212418