Abstract
We prove a formula for the global gravitational anomaly of the self-dual field theory in the presence of background gauge fields, assuming the results of arXiv:1110.4639. Along the way, we also clarify various points about the self-dual field theory. In particular, we give a general definition of the theta characteristic entering its partition function and settle the issue of its possible metric dependence. We treat the cohomological version of type IIB supergravity as an example of the formalism. We show the apparent existence of a mixed gauge-gravitational global anomaly, occurring when the B-field and Ramond–Ramond two-form gauge fields have non-trivial Wilson lines, and suggest a way in which it could cancel.
Article PDF
Similar content being viewed by others
References
Witten E.: Global gravitational anomalies. Commun. Math. Phys. 100, 197–229 (1985)
Monnier, S.: Global anomalies and chiral p-forms. arXiv:1211.2167
Monnier, S.: The anomaly line bundle of the self-dual field theory. Commun. Math. Phys. 325, 41–72 (2014). arXiv:1109.2904
Monnier, S.: The global gravitational anomaly of the self-dual field theory. Commun. Math. Phys. 325, 73–104 (2014). arXiv:1110.4639
Monnier, S.: Global gravitational anomaly cancellation for five-branes. arXiv:1310.2250
Witten, E.: Five-brane effective action in M-theory. J. Geom. Phys. 22, 103–133 (1997). arXiv:hep-th/9610234
Moore, G.W.: Anomalies, Gauss laws, and Page charges in M-theory. Comptes Rendus Physique 6, 251–259 (2005). arXiv:hep-th/0409158
Moore, G.W.: Quantization of Page charges in M-theory (2004) (Unpublished manuscript)
Witten, E.: Duality relations among topological effects in string theory. JHEP 05, 031 (2000). arXiv:hep-th/9912086
Belov, D., Moore, G.W.: Holographic action for the self-dual field. arXiv:hep-th/0605038
Hopkins, M.J., Singer, I.M.: Quadratic functions in geometry, topology, and M-theory. J. Differ. Geom. 70, 329 (2005). arXiv:math/0211216
Freed, D.S., Moore, G.W., Segal, G.: Heisenberg groups and noncommutative fluxes. Ann. Phys. 322, 236–285 (2007). arXiv:hep-th/0605200
Diaconescu, E., Moore, G.W., Freed, D.S.: The M-theory 3-form and E(8) gauge theory. arXiv:hep-th/0312069
Cheeger, J., Simons, J.: Differential characters and geometric invariants. In: Geometry and Topology, Lecture Notes in Mathematics, vol. 1167, pp. 50–80. Springer, Berlin (1985). doi:10.1007/BFb0075216
Witten, E.: On flux quantization in M-theory and the effective action. J. Geom. Phys. 22, 1–13 (1997). arXiv:hep-th/9609122
Alvarez-Gaume L., Witten E.: Gravitational anomalies. Nucl. Phys. B 234, 269 (1984)
Freed, D.S., Witten, E.: Anomalies in string theory with D-branes. arXiv:hep-th/9907189
Strominger, A.: Open p-branes. Phys. Lett. B 383, 44–47 (1996). arXiv:hep-th/9512059
Branson, T.: Q-curvature and spectral invariants. In: Slovák, J., M. Čadek, M. (eds.) Proceedings of the 24th Winter School “Geometry and Physics”, pp. 11–55. Circolo Matematico di Palermo, Palermo (2005)
Monnier, S.: Geometric quantization and the metric dependence of the self-dual field theory. Commun. Math. Phys. 314, 305–328 (2012). arXiv:1011.5890
Milnor, J., Stasheff, J.D.: Characteristic classes. In: Annals of mathematics studies. University of Tokyo Press, Princeton (1974)
Bismut J.-M., Freed D.S.: The analysis of elliptic families. II. Dirac operators, eta invariants, and the holonomy theorem. Commun. Math. Phys. 107(1), 103–163 (1986)
Monnier, S.: Canonical quadratic refinements of cohomological pairings from functorial lifts of the Wu class. arXiv:1208.1540
Kriz, I., Sati, H.: Type IIB string theory, S-duality, and generalized cohomology. Nucl. Phys. B 715, 639–664 (2005). arXiv:hep-th/0410293
Freed, D. S.: Classical Chern–Simons theory. Part 1. Adv. Math. 113, 237–303 (1995). arXiv:hep-th/9206021
Brumfiel G.W., Morgan J.W.: Quadratic functions, the index modulo 8 and a Z/4-Hirzebruch formula. Topology 12, 105–122 (1973)
Atiyah M.F., Patodi V.K., Singer I.M.: Spectral asymmetry and Riemannian geometry. Bull. London Math. Soc. 5, 229–234 (1973)
Taylor, L.R.: Gauss sums in algebra and topology. http://www3.nd.edu/taylor/papers/Gauss_sums
Birkenhake, C., Lange, H.: Complex abelian varieties. In: Grundlehren der Mathematischen Wissenschaften, vol. 302, 2nd edn. Springer, Berlin (2004)
Minasian, R., Moore, G.W.: K-theory and Ramond–Ramond charge. JHEP 11, 002 (1997). arXiv:hep-th/9710230
Witten, E.: D-branes and K-theory. JHEP 12, 019 (1998). arXiv:hep-th/9810188
Moore, G.W., Witten, E.: Self-duality, Ramond–Ramond fields, and K-theory. JHEP 05, 032 (2000). arXiv:hep-th/9912279
Freed, D.S.: Dirac charge quantization and generalized differential cohomology. arXiv:hep-th/0011220
Belov, D.M., Moore, G.W.: Type II actions from 11-dimensional Chern–Simons theories. arXiv:hep-th/0611020
Stong R.: Calculation of \({\Omega_{11}^{\rm spin}(K(Z,4))}\) . In: Green, M., Gross, D. (eds.) Unified String Theories, pp. 430–437. World Scientific, Singapore (1986)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Marcos Marino.
Rights and permissions
About this article
Cite this article
Monnier, S. The Global Anomaly of the Self-Dual Field in General Backgrounds. Ann. Henri Poincaré 17, 1003–1036 (2016). https://doi.org/10.1007/s00023-015-0423-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-015-0423-z