Abstract
We study gauge and gravitational anomalies of fermions and 2-form fields on eight-dimensional spin manifolds. Possible global gauge anomalies are classified by spin bordism groups \( {\varOmega}_9^{\mathrm{spin}}(BG) \) which we determine by spectral sequence techniques, and we also identify their explicit generator manifolds. It turns out that a fermion in the adjoint representation of any simple Lie group, and a gravitino in 8d \( \mathcal{N} \) = 1 supergravity theory, have anomalies. We discuss how a 2-form field, which also appears in supergravity, produces anomalies which cancel against these fermion anomalies in a certain class of supergravity theories. In another class of theories, the anomaly of the gravitino is not cancelled by the 2-form field, but by topological degrees of freedom. It gives a restriction on the topology of spacetime manifolds which is not visible at the level of differential-form analysis.
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D.W. Anderson, E.H. Brown and F.P. Peterson, The structure of the spin cobordism ring, Ann. Math. 86 (1967) 271.
F. Apruzzi, M. Dierigl and L. Lin, The fate of discrete 1-form symmetries in 6d, SciPost Phys. 12 (2022) 047 [arXiv:2008.09117] [INSPIRE].
L. Álvarez-Gaumé and P.H. Ginsparg, The topological meaning of non-Abelian anomalies, Nucl. Phys. B 243 (1984) 449 [INSPIRE].
L. Álvarez-Gaumé and E. Witten, Gravitational anomalies, Nucl. Phys. B 234 (1984) 269 [INSPIRE].
M.F. Atiyah, V.K. Patodi and I.M. Singer, Spectral asymmetry and Riemannian geometry 1, Math. Proc. Cambridge Phil. Soc. 77 (1975) 43 [INSPIRE].
M.F. Atiyah, V.K. Patodi and I.M. Singer, Spectral asymmetry and Riemannian geometry 2, Math. Proc. Cambridge Phil. Soc. 78 (1976) 405 [INSPIRE].
M.F. Atiyah and I.M. Singer, Dirac operators coupled to vector potentials, Proc. Nat. Acad. Sci. 81 (1984) 2597 [INSPIRE].
O. Alvarez, I.M. Singer and B. Zumino, Gravitational anomalies and the family’s index theorem, Commun. Math. Phys. 96 (1984) 409 [INSPIRE].
A. Beaudry and J.A. Campbell, A guide for computing stable homotopy groups, arXiv:1801.07530.
L. Bhardwaj, D.R. Morrison, Y. Tachikawa and A. Tomasiello, The frozen phase of F-theory, JHEP 08 (2018) 138 [arXiv:1805.09070] [INSPIRE].
A. Borel, Sur l’homologie et la cohomologie des groupes de Lie compacts connexes (in French), Amer. J. Math. 76 (1954) 273.
R. Bott and H. Samelson, Applications of the theory of Morse to symmetric spaces, Amer. J. Math. 80 (1958) 964.
M. Cvetič, M. Dierigl, L. Lin and H.Y. Zhang, String universality and non-simply-connected gauge groups in 8d, Phys. Rev. Lett. 125 (2020) 211602 [arXiv:2008.10605] [INSPIRE].
C. Córdova, D.S. Freed, H.T. Lam and N. Seiberg, Anomalies in the space of coupling constants and their dynamical applications II, SciPost Phys. 8 (2020) 002 [arXiv:1905.13361] [INSPIRE].
S. Chaudhuri, G. Hockney and J.D. Lykken, Maximally supersymmetric string theories in D < 10, Phys. Rev. Lett. 75 (1995) 2264 [hep-th/9505054] [INSPIRE].
J. Cheeger and J. Simons, Differential characters and geometric invariants, Lect. Notes Math. 1167 (1985) 50.
A. Debray, M. Dierigl, J.J. Heckman and M. Montero, The anomaly that was not meant IIB, Fortsch. Phys. 70 (2021) 2100168 [arXiv:2107.14227] [INSPIRE].
J. Davis and P. Kirk, Lecture notes in algebraic topology, American Mathematical Society, Providence, RI, U.S.A. (2001).
J. Davighi and N. Lohitsiri, Omega vs. pi, and 6d anomaly cancellation, JHEP 05 (2021) 267 [arXiv:2012.11693] [INSPIRE].
D.S. Freed and M.J. Hopkins, On Ramond-Ramond fields and k-theory, JHEP 05 (2000) 044 [hep-th/0002027] [INSPIRE].
D.S. Freed and M.J. Hopkins, Reflection positivity and invertible topological phases, Geom. Topol. 25 (2021) 1165 [arXiv:1604.06527] [INSPIRE].
D.S. Freed and M.J. Hopkins, Consistency of M-theory on non-orientable manifolds, Quart. J. Math. Oxford Ser. 72 (2021) 603 [arXiv:1908.09916] [INSPIRE].
D.S. Freed and G.W. Moore, Setting the quantum integrand of M-theory, Commun. Math. Phys. 263 (2006) 89 [hep-th/0409135] [INSPIRE].
D.S. Freed, G.W. Moore and G. Segal, Heisenberg groups and noncommutative fluxes, Annals Phys. 322 (2007) 236 [hep-th/0605200] [INSPIRE].
J. Francis, Integrals on spin manifolds and the K-theory of K(ℤ, 4), https://sites.math.northwestern.edu/∼jnkf/writ/bspin2011.pdf.
I. García-Etxebarria, H. Hayashi, K. Ohmori, Y. Tachikawa and K. Yonekura, 8d gauge anomalies and the topological Green-Schwarz mechanism, JHEP 11 (2017) 177 [arXiv:1710.04218] [INSPIRE].
I. García-Etxebarria and M. Montero, Dai-Freed anomalies in particle physics, JHEP 08 (2019) 003 [arXiv:1808.00009] [INSPIRE].
D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized global symmetries, JHEP 02 (2015) 172 [arXiv:1412.5148] [INSPIRE].
M.B. Green and J.H. Schwarz, Anomaly cancellation in supersymmetric D = 10 gauge theory and superstring theory, Phys. Lett. B 149 (1984) 117 [INSPIRE].
M. Guo, Some calculations of cobordism groups and their applications in physics, Ph.D. thesis, Harvard University, Cambridge, MA, U.S.A. (2018).
A. Hatcher, Spectral sequences, https://pi.math.cornell.edu/∼hatcher/AT/SSpage.html.
C.-T. Hsieh, Y. Tachikawa and K. Yonekura, Anomaly inflow and p-form gauge theories, Commun. Math. Phys. 391 (2022) 495 [arXiv:2003.11550] [INSPIRE].
Y. Hamada and C. Vafa, 8d supergravity, reconstruction of internal geometry and the swampland, JHEP 06 (2021) 178 [arXiv:2104.05724] [INSPIRE].
H. Kachi, Homotopy groups of compact Lie groups E6, E7 and E8, Nagoya Math. J. 32 (1968) 109.
A. Kapustin, Symmetry protected topological phases, anomalies, and cobordisms: beyond group cohomology, arXiv:1403.1467 [INSPIRE].
R. Kobayashi, K. Ohmori and Y. Tachikawa, On gapped boundaries for SPT phases beyond group cohomology, JHEP 11 (2019) 131 [arXiv:1905.05391] [INSPIRE].
A. Kapustin, R. Thorngren, A. Turzillo and Z. Wang, Fermionic symmetry protected topological phases and cobordisms, JHEP 12 (2015) 052 [arXiv:1406.7329] [INSPIRE].
H.-C. Kim, H.-C. Tarazi and C. Vafa, Four-dimensional N = 4 SYM theory and the swampland, Phys. Rev. D 102 (2020) 026003 [arXiv:1912.06144] [INSPIRE].
Y. Lee and Y. Tachikawa, Some comments on 6D global gauge anomalies, PTEP 2021 (2021) 08B103 [arXiv:2012.11622] [INSPIRE].
J. McCleary, A user’s guide to spectral sequences, second edition, Cambridge University Press, Cambridge, U.K. (2001)
S. Monnier and G.W. Moore, Remarks on the Green-Schwarz terms of six-dimensional supergravity theories, Commun. Math. Phys. 372 (2019) 963 [arXiv:1808.01334] [INSPIRE].
S. Monnier, Topological field theories on manifolds with Wu structures, Rev. Math. Phys. 29 (2017) 1750015 [arXiv:1607.01396] [INSPIRE].
M. Mimura and H. Toda, Topology of Lie groups. I and II, Translations of Mathematical Monographs 91, American Mathematical Society, Providence, RI, U.S.A. (1991).
D.R. Morrison and C. Vafa, Compactifications of F-theory on Calabi-Yau threefolds. 1, Nucl. Phys. B 473 (1996) 74 [hep-th/9602114] [INSPIRE].
D.R. Morrison and C. Vafa, Compactifications of F-theory on Calabi-Yau threefolds. 2, Nucl. Phys. B 476 (1996) 437 [hep-th/9603161] [INSPIRE].
M. Montero and C. Vafa, Cobordism conjecture, anomalies, and the string lamppost principle, JHEP 01 (2021) 063 [arXiv:2008.11729] [INSPIRE].
D. Quillen, The mod2 cohomology rings of extra-special 2-groups and the spinor groups, Math. Ann. 194 (1971) 197.
J.-P. Serre, Cohomologie modulo 2 des complexes d’Eilenberg-MacLane (in French), Comment. Math. Helv. 27 (1953) 198.
R.E. Stong, Appendix: calculation of \( {\varOmega}_{11}^{\mathrm{Spin}}\left(K\left(\mathbb{Z},4\right)\right) \), in Workshop on unified string theories (1985), World Scientific, Singapore (1986), p. 430.
Y. Tachikawa, Frozen singularities in M and F-theory, JHEP 06 (2016) 128 [arXiv:1508.06679] [INSPIRE].
Y. Tachikawa, Topological modular forms and the absence of a heterotic global anomaly, PTEP 2022 (2022) 04A107 [arXiv:2103.12211] [INSPIRE].
P. Teichner, On the signature of four-manifolds with universal covering spin, Math. Ann. 295 (1993) 745 [INSPIRE].
H. Toda, Cohomology mod3 of the classifying space BF4 of the exceptional group F4, Kyoto J. Math. 13 (1973) 97.
Y. Tachikawa and M. Yamashita, Topological modular forms and the absence of all heterotic global anomalies, arXiv:2108.13542 [INSPIRE].
C. Vafa, Evidence for F-theory, Nucl. Phys. B 469 (1996) 403 [hep-th/9602022] [INSPIRE].
E. Witten, An SU(2) anomaly, Phys. Lett. B 117 (1982) 324 [INSPIRE].
E. Witten, Global gravitational anomalies, Commun. Math. Phys. 100 (1985) 197 [INSPIRE].
E. Witten, Topological tools in ten-dimensional physics, Int. J. Mod. Phys. A 1 (1986) 39 [INSPIRE].
E. Witten, On flux quantization in M-theory and the effective action, J. Geom. Phys. 22 (1997) 1 [hep-th/9609122] [INSPIRE].
E. Witten, Toroidal compactification without vector structure, JHEP 02 (1998) 006 [hep-th/9712028] [INSPIRE].
E. Witten, World sheet corrections via D instantons, JHEP 02 (2000) 030 [hep-th/9907041] [INSPIRE].
E. Witten, Supersymmetric index in four-dimensional gauge theories, Adv. Theor. Math. Phys. 5 (2002) 841 [hep-th/0006010] [INSPIRE].
E. Witten, Fermion path integrals and topological phases, Rev. Mod. Phys. 88 (2016) 035001 [arXiv:1508.04715] [INSPIRE].
E. Witten, The “parity” anomaly on an unorientable manifold, Phys. Rev. B 94 (2016) 195150 [arXiv:1605.02391] [INSPIRE].
E. Witten and K. Yonekura, Anomaly inflow and the η-invariant, in The Shoucheng Zhang memorial workshop, (2019) [arXiv:1909.08775] [INSPIRE].
K. Yonekura, On the cobordism classification of symmetry protected topological phases, Commun. Math. Phys. 368 (2019) 1121 [arXiv:1803.10796] [INSPIRE].
K. Yonekura, General anomaly matching by Goldstone bosons, JHEP 03 (2021) 057 [arXiv:2009.04692] [INSPIRE].
M. Yamashita and K. Yonekura, Differential models for the Anderson dual to bordism theories and invertible QFT’s, I, arXiv:2106.09270 [INSPIRE].
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Lee, Y., Yonekura, K. Global anomalies in 8d supergravity. J. High Energ. Phys. 2022, 125 (2022). https://doi.org/10.1007/JHEP07(2022)125
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DOI: https://doi.org/10.1007/JHEP07(2022)125