Abstract
We construct the Green–Schwarz terms of six-dimensional supergravity theories on spacetimes with non-trivial topology and gauge bundle. We prove the cancellation of all global gauge and gravitational anomalies for theories with gauge groups given by products of U(n), SU(n) and Sp(n) factors, as well as for E8. For other gauge groups, anomaly cancellation is equivalent to the triviality of a certain 7-dimensional spin topological field theory. We show in the case of a finite Abelian gauge group that there are residual global anomalies imposing constraints on the 6d supergravity. These constraints are compatible with the known F-theory models. Interestingly, our construction requires that the gravitational anomaly coefficient of the 6d supergravity theory is a characteristic element of the lattice of string charges, a fact true in six-dimensional F-theory compactifications but that until now was lacking a low-energy explanation. We also discover a new anomaly coefficient associated with a torsion characteristic class in theories with a disconnected gauge group.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Taylor, W.: TASI lectures on supergravity and string vacua in various dimensions. arXiv:1104.2051
Kumar V., Morrison D.R., Taylor W.: Global aspects of the space of 6D N = 1 supergravities. JHEP 11, 118 (2010) https://doi.org/10.1007/JHEP11(2010)118 arXiv:1008.1062
Monnier, S., Moore, G.W., Park, D.S.: Quantization of anomaly coefficients in 6D \({\mathcal{N}=(1,0)}\) supergravity. arXiv:1711.04777
Freed D.S.: Anomalies and invertible field theories. Proc. Symp. Pure Math. 88, 25–46 (2014) https://doi.org/10.1090/pspum/088/01462 arXiv:1404.7224
Freed D.S., Moore G.W.: Setting the quantum integrand of M-theory. Commun. Math. Phys. 263, 89–132 (2006) https://doi.org/10.1007/s00220-005-1482-7 arXiv:hep-th/0409135
Monnier S.: Hamiltonian anomalies from extended field theories. Commun. Math. Phys. 338, 1327–1361 (2015) arXiv:1410.7442
Witten E.: World sheet corrections via D instantons. JHEP 02, 030 (2000) https://doi.org/10.1088/1126-6708/2000/02/030 arXiv:hep-th/9907041
Dai X.-Z., Freed D.S.: Eta invariants and determinant lines. J. Math. Phys. 35, 5155–5194 (1994) https://doi.org/10.1063/1.530747 arXiv:hep-th/9405012
Freed, D.S., Hopkins, M.J.: Reflection positivity and invertible topological phases. arXiv:1604.06527
Freed D.S., Quinn F.: Chern–Simons theory with finite gauge group. Commun. Math. Phys. 156, 435–472 (1993) https://doi.org/10.1007/BF02096860 arXiv:hep-th/9111004
Dijkgraaf R., Witten E.: Topological gauge theories and group cohomology. Commun. Math. Phys. 129, 393 (1990) https://doi.org/10.1007/BF02096988
Jenquin, J.A.: Classical Chern–Simons on manifolds with spin structure. arXiv:math/0504524
Jenquin, J.A.: Spin Chern-Simons and spin TQFTs. arXiv:math/0605239
Belov, D., Moore, G.W.: Classification of abelian spin Chern–Simons theories. arXiv:hep-th/0505235
Monnier, S.: Topological field theories on manifolds with Wu structures. Rev. Math. Phys. 29 (2017). arXiv:1607.01396
Seiberg N., Taylor W.: Charge lattices and consistency of 6D supergravity. JHEP 06, 001 (2011) https://doi.org/10.1007/JHEP06(2011)001 arXiv:1103.0019
Henningson M.: Self-dual strings in six dimensions: Anomalies, the ADE-classification, and the world-sheet WZW-model. Commun. Math. Phys. 257, 291–302 (2005) https://doi.org/10.1007/s00220-005-1324-7 arXiv:hep-th/0405056
Berman D.S., Harvey J.A.: The self-dual string and anomalies in the M5-brane. JHEP 11, 015 (2004) https://doi.org/10.1088/1126-6708/2004/11/015 arXiv:hep-th/0408198
Kim, H.-C., Kim, S., Park, J.: 6d strings from new chiral gauge \({\mathcal{N}=\left(1,0\right)}\) theories. https://doi.org/10.1007/JHEP11(2016)165 arXiv:1608.03919
Shimizu H., Tachikawa Y.: Anomaly of strings of 6d \({ \mathcal{N}=\left(1,0\right) }\) theories. JHEP 11, 165 (2016) arXiv:1608.05894
Freed, D.S.: Dirac charge quantization and generalized differential cohomology. arXiv:hep-th/0011220
Freed D.S., Moore G.W., Segal G.: Heisenberg groups and noncommutative fluxes. Ann. Phys. 322, 236–285 (2007) https://doi.org/10.1016/j.aop.2006.07.014 arXiv:hep-th/0605200
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
Distler, J., Freed, D.S., Moore, G.W.: Orientifold precis. arXiv:0906.0795
Monnier, S., Moore, G.W.: A brief summary of global anomaly cancellation in six-dimensional supergravity. arXiv:1808.01335
Avramis, S.D.: Anomaly-free supergravities in six dimensions. arXiv:hep-th/0611133
Polchinski J.: Monopoles, duality, and string theory. Int. J. Mod. Phys. A 19S1, 145–156 (2004) https://doi.org/10.1142/S0217751X0401866X arXiv:hep-th/0304042
Banks T., Seiberg N.: Symmetries and strings in field theory and gravity. Phys. Rev. D 83, 084019 (2011) https://doi.org/10.1103/PhysRevD.83.084019 arXiv:1011.5120
Green M.B., Schwarz J.H., West P.C.: Anomaly free chiral theories in six-dimensions. Nucl. Phys. B 254, 327–348 (1985) https://doi.org/10.1016/0550-3213(85)90222-6
Sagnotti A.: A note on the Green–Schwarz mechanism in open string theories. Phys. Lett. B 294, 196–203 (1992) https://doi.org/10.1016/0370-2693(92)90682-T arXiv:hep-th/9210127
Sadov V.: Generalized Green–Schwarz mechanism in F theory. Phys. Lett. B 388, 45–50 (1996) https://doi.org/10.1016/0370-2693(96)01134-3 arXiv:hep-th/9606008
Riccioni F.: All couplings of minimal six-dimensional supergravity. Nucl. Phys. B 605, 245–265 (2001) https://doi.org/10.1016/S0550-3213(01)00199-7 arXiv:hep-th/0101074
Green M.B., Schwarz J.H.: Anomaly cancellation in supersymmetric D = 10 gauge theory and superstring theory. Phys. Lett. B 149, 117–122 (1984) https://doi.org/10.1016/0370-2693(84)91565-X
Witten E.: Global gravitational anomalies. Commun. Math. Phys. 100, 197–229 (1985) https://doi.org/10.1007/BF01212448
Monnier S.: The global anomalies of (2,0) superconformal field theories in six dimensions. JHEP 09 (2014). https://doi.org/10.1007/JHEP09(2014)088 arXiv:1406.4540
Bismut J.-M., Freed D.S.: The analysis of elliptic families. II. Dirac operators, eta invariants, and the holonomy theorem. Commun. Math. Phys. 107, 103–163 (1986) https://doi.org/10.1016/S0393-0440(97)80160-X
Fiorenza, D., Valentino, A.: Boundary conditions for topological quantum field theories, anomalies and projective modular functors. Commun. Math. Phys. 338 (2015). arXiv:1409.5723
Atiyah M.F., Patodi V.K., Singer I.M.: Spectral asymmetry and Riemannian geometry. I. Math. Proc. Cambr. Philos. Soc. 77, 43–69 (1975)
Witten E.: Five-brane effective action in M-theory. J. Geom. Phys. 22, 103–133 (1997) arXiv:hep-th/9610234
Maldacena J.M., Moore G.W., Seiberg N.: D-brane charges in five-brane backgrounds. JHEP 10, 005 (2001) arXiv:hep-th/0108152
Belov, D., Moore, G.W.: Holographic action for the self-dual field. arXiv:hep-th/0605038
Monnier S.: The global gravitational anomaly of the self-dual field theory. Commun. Math. Phys. 325, 73–104 (2014) https://doi.org/10.1007/s00220-013-1845-4 arXiv:1110.4639
Taylor, L.R.: Gauss sums in algebra and topology. http://www3.nd.edu/~taylor/papers/Gauss_sums.pdf. Accessed 3 Feb 2019
Monnier, S.: The anomaly field theories of six-dimensional (2,0) superconformal theories. arXiv:1706.01903
Bershadsky, M., Vafa, C.: Global anomalies and geometric engineering of critical theories in six-dimensions. arXiv:hep-th/9703167
Suzuki R., Tachikawa Y.: More anomaly-free models of six-dimensional gauged supergravity. J. Math. Phys. 47, 062302 (2006) https://doi.org/10.1063/1.2209767 arXiv:hep-th/0512019
Cheeger, J., Simons, J.: Differential characters and geometric invariants. In: Geometry and Topology, vol. 1167 of Lecture Notes in Mathematics, pp. 50–80. Springer Berlin (1985). https://doi.org/10.1007/BFb0075216
Hopkins M.J., Singer I.M.: Quadratic functions in geometry, topology, and M-theory. J. Diff. Geom. 70, 329 (2005) arXiv:math/0211216
Freed, D.S., Moore G.W., Segal G.: The uncertainty of fluxes. Commun.Math.Phys. 271, 247–274 (2007) https://doi.org/10.1007/s00220-006-0181-3 arXiv:hep-th/0605198
Witten E.: Topological tools in ten-dimensional physics. Int. J. Mod. Phys. A 1, 39 (1986) https://doi.org/10.1142/S0217751X86000034
Witten E.: Duality relations among topological effects in string theory. JHEP 05, 031 (2000) arXiv:hep-th/9912086
Diaconescu, E., Moore, G.W., Freed, D.S.: The M-theory 3-form and E(8) gauge theory. arXiv:hep-th/0312069
Monnier S.: The global anomaly of the self-dual field in general backgrounds. Ann. Henri Poincaré 17, 1003–1036 (2016) arXiv:1309.6642
Brumfiel G.W., Morgan J.W.: Quadratic functions, the index modulo 8 and a Z/4-Hirzebruch formula. Topology 12, 105–122 (1973)
Steenrod N.E.: Products of cocycles and extensions of mappings. Ann. Math. 48, 290–320 (1947)
Gilkey, P., Leahy, J., Park, J.: Spectral Geometry, Riemannian Submersions, and the Gromov–Lawson Conjecture. Studies in Advanced Mathematics. Taylor & Francis (1999)
Taylor W., Turner A.P.: An infinite swampland of U(1) charge spectra in 6d supergravity theories. JHEP 06, 010 (2018) https://doi.org/10.1007/JHEP06(2018)010 arXiv:1803.04447
Stong R.: Notes on Cobordism Theory. Princeton University Press, Princeton (1968)
Witten, E.: Anomalies revisited, talk at strings (2015). https://member.ipmu.jp/yuji.tachikawa/stringsmirrors/2015/6-2.00-2.30-Edward-Witten.pdf. Accessed 3 Feb 2019
Yuan, Q.: Answer to “Spin manifold and the second Stiefel–Whitney class” (2014). https://math.stackexchange.com/questions/808263. Accessed 3 Feb 2019
Freed, D.S.: Classical Chern–Simons theory, Part 2. Houston J. Math. 28, 293–310 (2005)
Freed, D.S.: Pions and generalized cohomology. arXiv:hep-th/0607134
García-Etxebarria, I., Montero, M.: Dai–Freed anomalies in particle physics. arXiv:1808.00009
Zhubr, A.V.: Spin bordism of oriented manifolds and the hauptvermuting for 6-manifolds. In: Turaev, V., Vershik, A., Rokhlin, V. (eds.) Topology, Ergodic Theory, Real Algebraic Geometry. Rokhlin’s memorial, pp. 263–286. American Mathematical Society (2001)
Brown E.: The cohomology of BSO n and BO n with integer coefficients. Proc. Am. Math. Soc. 85, 283–288 (1982)
Mimura, M., Toda, H.: Topology of Lie Groups I & II, vol. 91 of Translations of Mathematical Monographs. American Mathematical Society (1991)
Breen L., Mikhailov R., Touzé A.: Derived functors of the divided power functors. Geom. Topol. 20, 257–352 (2016)
Smirnov V.A.: Secondary Steenrod operations in cohomology of infinite-dimensional projective spaces. Math. Notes 79, 440–445 (2006) https://doi.org/10.1007/s11006-006-0050-6
Kapustin A., Thorngren R., Turzillo A., Wang Z.: Fermionic symmetry protected topological phases and cobordisms. JHEP 12, 052 (2015) https://doi.org/10.1007/JHEP12(2015)052 arXiv:1406.7329
Acknowledgements
We would like to thank Daniel Park for discussions that led to this project.We also thank Dan Freed,Mike Hopkins, Graeme Segal,Wati Taylor, Andrew Turner, Nathan Seiberg and Edward Witten for useful discussions. G.M. is supported by the DOE under grant DOE-SC0010008 to Rutgers University. S.M. is supported in part by the grant MODFLAT of the European Research Council, SNSF grants No. 152812, 165666, and by NCCR SwissMAP, funded by the Swiss National Science Foundation.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by N. Nekrasov
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Monnier, S., Moore, G.W. Remarks on the Green–Schwarz Terms of Six-Dimensional Supergravity Theories. Commun. Math. Phys. 372, 963–1025 (2019). https://doi.org/10.1007/s00220-019-03341-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-019-03341-7