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Twisted Cohomotopy Implies M-Theory Anomaly Cancellation on 8-Manifolds

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Abstract

We consider the hypothesis that the C-field 4-flux and 7-flux forms in M-theory are in the image of the non-abelian Chern character map from the non-abelian generalized cohomology theory called J-twisted Cohomotopy theory. We prove for M2-brane backgrounds in M-theory on 8-manifolds that such charge quantization of the C-field in Cohomotopy theory implies a list of expected anomaly cancellation conditions, including: shifted C-field flux quantization and C-field tadpole cancellation, but also the DMW anomaly cancellation and the C-field’s integral equation of motion.

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Notes

  1. Here and in the following, a dashed arrow indicates a map representing a cocycle that can be freely choosen, as opposed to solid arrows indicating fixed structure maps.

  2. This is in contrast with twisting vs. differential refinement where the order does not matter – see [GS19a, GS19b].

  3. All constructions here are homotopical, in particular all group actions, principal bundles, etc. are “higher structures up to coherent homotopy”, in a sense that has been made completely rigorous via the notion of \(\infty \)-groups, and their \(\infty \)-actions on \(\infty \)-principal bundles [NSS12]. But the pleasant upshot of this theory is that when homotopy coherence is systematically accounted for, then higher structures behave in all general ways as ordinary structures, for instance in that homotopy pullbacks satisfy the same structural pasting laws as ordinary pullbacks. Beware, this means in particular that all our equivalences are weak homotopy equivalences (even when we denote them as equalities), and that all our commutative diagrams are commutative up to specified homotopies (even when we do not display these).

  4. There is an evident sign typo in the statement (but not in the proof) of [FHT00, Sec. 15, Example 4] with respect to equation (43): The standard fact that the Euler class squares to the top Pontrjagin class means that there must be the relative minus sign in (43), which is exactly what the proof of [FHT00, Sec. 15, Example 4] actually concludes.

  5. Recall that this says that if

    figure bw

    is a commuting diagram, where the right square is a pullback, then the left square is a pullback precisely if the full outer rectangle is a pullback. The same holds for homotopy-commutative diagrams and homotopy-pullback squares.

  6. [Wi19] at 21:15: “I actually believe that string/M-theory is on the right track toward a deeper explanation. But at a very fundamental level it’s not well understood. And I’m not even confident that we have a good concept of what sort of thing is missing or where to find it.”

References

  1. Acharya, B.S., Gukov, S.: M theory and singularities of exceptional holonomy manifolds. Phys. Rep. 392, 121–189 (2004). arXiv:hep-th/0409191

    ADS  MathSciNet  Google Scholar 

  2. Adams, F.: On the non-existence of elements of Hopf invariant one. Ann. Math. 72, 20–104 (1960). [jstor:1970147]

  3. Alekseevskii, D.: Riemannian spaces with exceptional holonomy groups. Funct. Anal. Appl. 2, 97–105 (1968)

    Google Scholar 

  4. Andrews, P., Arkowitz, M.: Sullivan’s minimal models and higher order whitehead products. Canad. J. Math. 30, 961–982 (1978). https://doi.org/10.4153/CJM-1978-083-6

    Article  MathSciNet  MATH  Google Scholar 

  5. Atiyah, M., Witten, E.: M-Theory dynamics on a manifold of \(G_2\)-holonomy. Adv. Theor. Math. Phys. 6, 1–106 (2003). arXiv:hep-th/0107177

    MATH  Google Scholar 

  6. Awada, M.A., Duff, M.J., Pope, C.N.: \(N=8\) supergravity breaks down to \(N=1\). Phys. Rev. Lett. 50, 294–297 (1983)

    ADS  MathSciNet  MATH  Google Scholar 

  7. Babalic, E., Lazaroiu, C.: Foliated eight-manifolds for M-theory compactification. J. High Energy Phys. 01, 140 (2015). arXiv:1411.3148

    ADS  MathSciNet  MATH  Google Scholar 

  8. Babalic, E., Lazaroiu, C.: Singular foliations for M-theory compactification. J. High Energy Phys. 03, 116 (2015). arXiv:1411.3497

    MathSciNet  MATH  Google Scholar 

  9. Babalic, E., Lazaroiu, C.: The landscape of \(G\)-structures in eight-manifold compactifications of M-theory. J. High Energy Phys. 11, 007 (2015). arXiv:1505.02270

    ADS  MathSciNet  MATH  Google Scholar 

  10. Babalic, E., Lazaroiu, C.: Internal circle uplifts, transversality and stratified \(G\)-structures. J. High Energy Phys. 11, 174 (2015). arXiv:1505.05238

    ADS  MathSciNet  MATH  Google Scholar 

  11. Banks, T.: On the limits of effective quantum field theory: Eternal inflation, landscapes, and other mythical beasts. arXiv:1910.12817

  12. Becker, K., Becker, M.: M-theory on eight-manifolds. Nucl. Phys. B 477, 155–167 (1996). arXiv:hep-th/9605053

    ADS  MathSciNet  MATH  Google Scholar 

  13. Becker, K., Becker, M., Schwarz, J.: String Theory and M-Theory: A Modern Introduction. Cambridge University Press, Cambridge (2006)

    MATH  Google Scholar 

  14. Becker, K., Becker, M., Morrison, D., Ooguri, H., Oz, Y., Yin, Z.: Supersymmetric cycles in exceptional holonomy manifolds and Calabi-Yau 4-folds. Nucl. Phys. B 480, 225–238 (1996). arXiv:hep-th/9608116

    ADS  MathSciNet  MATH  Google Scholar 

  15. Belhaj, A., Boya, L.J., Segui, A.: Holonomy groups coming from F-theory compactification. Int. J. Theory Phys. 49, 681–692 (2010). arXiv:0911.2125

    MathSciNet  MATH  Google Scholar 

  16. Bettiol, R.G., Mendes, R.A.E.: Flag manifolds with strongly positive curvature. Math. Z. 280, 1031–1046 (2015). arXiv:1412.0039

    MathSciNet  MATH  Google Scholar 

  17. Bonetti, F., Grimm, T., Pugh, T.: Non-supersymmetric F-theory compactifications on \({\rm Spin}(7)\) manifolds. J. High Energy Phys. 01, 112 (2014). arXiv:1307.5858

    ADS  Google Scholar 

  18. Bonetti, F., Grimm, T., Palti, E., Pugh, T.: F-theory on \({\rm Spin}(7)\) manifolds: weak-coupling limit. J. High Energy Phys. 02, 076 (2014). arXiv:1309.2287

    ADS  Google Scholar 

  19. Borel, A., Serre, J.-P.: Groupes de Lie et Puissances Reduites de Steenrod. Am. J. Math. 75, 409–448 (1953). [jstor:2372495]

  20. Borsuk, K.: Sur les groupes des classes de transformations continues. CR Acad. Sci. Paris 202, 1400–1403 (1936)

    MATH  Google Scholar 

  21. Braunack-Mayer, V., Sati, H., Schreiber, U.: Gauge enhancement for super M-branes via parameterized stable homotopy theory. Comm. Math. Phys. (2019). https://doi.org/10.1007/s00220-019-03441-4. arXiv:1805.05987 [hep-th]

  22. Brown, E.H.: The cohomology of \(B {{\rm SO}}_n\) and \(B {{\rm O}}_n\) with integer coefficients. Proc. Am. Math. Soc. 85, 283–288 (1982). [jstor:2044298]

  23. Bryant, R., Harvey, R.: Submanifolds in Hyper-Kähler geometry. J. Am. Math. Soc. 2, 1–31 (1989). [jstor:1990911]

  24. Čadek, M., Vanžura, J.: On \({\rm Sp}(2)\) and \({\rm Sp}(2) \cdot {\rm Sp}(1)\)-structures in 8-dimensional vector bundles. Publ. Matemátiques 41, 383–401 (1997). [jstor:43737249]

  25. Čadek, M., Vanžura, J.: On 4-fields and 4-distributions in 8-dimensional vector bundles over 8-complexes. Colloq. Math. 76, 213–228 (1998)

    MathSciNet  MATH  Google Scholar 

  26. Čadek, M., Vanžura, J.: Almost quaternionic structures on eight-manifolds. Osaka J. Math. 35, 165–190 (1998). euclid:1200787905

  27. Chacholski, W., Pitsch, W., Scherer, J.: Homotopy pullback squares up to localization, In: Arlettaz, D., Hess, K. (eds.) An Alpine anthology of homotopy theory. arXiv:math/0501250

  28. Chern, S.-S.: A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds. Ann. Math. 45, 747–752 (1944). [jstor:1969302]

  29. Chester, S.M., Perlmutter, E.: M-theory reconstruction from \((2,0)\) CFT and the chiral algebra conjecture. J. High Energy Phys. 2018, 116 (2018). arXiv:1805.00892

    MathSciNet  MATH  Google Scholar 

  30. Condeescu, C., Micu, A., Palti, E.: M-theory compactifications to three dimensions with M2-brane potentials. J. High Energy Phys. 04, 026 (2014). arXiv:1311.5901

    ADS  Google Scholar 

  31. Cremmer, E., Julia, B., Scherk, J.: Supergravity theory in eleven-dimensions. Phys. Lett. B 76, 409–412 (1978)

    ADS  Google Scholar 

  32. Danielsson, U., Dibitetto, G., Guarino, A.: KK-monopoles and \(G\)-structures in M-theory/type IIA reductions. J. High Energy Phys. 1502, 096 (2015). arXiv:1411.0575

    ADS  MathSciNet  MATH  Google Scholar 

  33. Danielsson, U., Van Riet, T.: What if string theory has no de Sitter vacua? Int. J. Mod. Phys. D 27, 1830007 (2018). arXiv:1804.01120

    ADS  MathSciNet  MATH  Google Scholar 

  34. Diaconescu, E., Freed, D.S., Moore, G.: The M-theory 3-form and \(E_8\) Gauge Theory, Elliptic Cohomology, pp. 44–88. Cambridge University Press, Cambridge (2007). arXiv:hep-th/0312069

    MATH  Google Scholar 

  35. Diaconescu, D., Moore, G., Witten, E.: \(E_8\)-Gauge theory and a derivation of K-theory from M-theory. Adv. Theor. Math. Phys 6, 1031–1134 (2003). arXiv:hep-th/0005090

    Google Scholar 

  36. Diaconescu, D., Moore, G., Witten, E.: A Derivation of K-Theory from M-Theory. arXiv:hep-th/0005091

  37. Dror, E., Dwyer, W., Kan, D.: Equivariant maps which are self homotopy equivalences. Proc. Am. Math. Soc. 80(4), 67–672 (1980). [jstor:2043448]

  38. Dubrovin, B.A., Novikov, S.P., Fomenko, A.T.: Modern Geometry–Methods and Applications–Part II: The Geometry and Topology of Manifolds. Springer, New York (1985)

    MATH  Google Scholar 

  39. Duff, M.: M-Theory (the Theory formerly known as strings). Int. J. Mod. Phys. A 11, 5623–5642 (1996). arXiv:hep-th/9608117

    ADS  MathSciNet  MATH  Google Scholar 

  40. Duff, M.: A Layman’s Guide to M-theory. In: Abdus Salam Memorial Meeting, Trieste, Italy vol. 19–22, pp. 184–213 (1997). arXiv:hep-th/9805177

  41. Duff, M. (ed.): The World in Eleven Dimensions: Supergravity, Supermembranes and M-theory. IoP, Bristol (1999)

    MATH  Google Scholar 

  42. Duff, M.J., Liu, J.T., Minasian, R.: Eleven dimensional origin of string/string duality: a one loop test. Nucl. Phys. B 452, 261–282 (1995). arXiv:hep-th/9506126

    ADS  MathSciNet  MATH  Google Scholar 

  43. Duff, M.J., Nilsson, B.E.W., Pope, C.N.: Spontaneous supersymmetry breaking by the squashed seven sphere. Phys. Rev. Lett. 50 (1983), 2043–2046, Erratum-ibid. 51(1983), 846

  44. Duff, M., Stelle, K.: Multi-membrane solutions of \(D = 11\) supergravity. Phys. Lett. B 253, 113 (1991). https://doi.org/10.1016/0370-2693(91)91371-2. spire:299386

  45. Félix, Y., Halperin, S., Thomas, J.-C.: Rational Homotopy Theory, Graduate Texts in Mathematics, vol. 205. Springer, Berlin (2000)

    Google Scholar 

  46. Fiorenza, D., Sati, H., Schreiber, U.: Super Lie \(n\)-algebra extensions, higher WZW models, and super \(p\)-branes with tensor multiplet fields. Int. J. Geom. Meth. Mod. Phys. 12, 1550018 (2015). arXiv:1308.5264

    MathSciNet  MATH  Google Scholar 

  47. Fiorenza, D., Sati, H., Schreiber, U.: The \(E_8\) moduli 3-stack of the \(C\)-field. Commun. Math. Phys. 333, 117–151 (2015). arXiv:1202.2455

    ADS  MATH  Google Scholar 

  48. Fiorenza, D., Sati, H., Schreiber, U.: Multiple M5-branes, String 2-connections, and 7d nonabelian Chern–Simons theory. Adv. Theor. Math. Phys. 18, 229–321 (2014). arXiv:1201.5277

    MathSciNet  MATH  Google Scholar 

  49. Fiorenza, D., Sati, H., Schreiber, U.: A higher stacky perspective on Chern–Simons theory. In: Calaque, D. (ed.) Mathematical Aspects of Quantum Field Theories. Springer, Berlin (2014). arXiv:1301.2580

    Google Scholar 

  50. Fiorenza, D., Sati, H., Schreiber, U.: The WZW term of the M5-brane and differential cohomotopy. J. Math. Phys. 56, 102301 (2015). arXiv:1506.07557

    ADS  MathSciNet  MATH  Google Scholar 

  51. Fiorenza, D., Sati, H., Schreiber, U.: Rational sphere valued supercocycles in M-theory and type IIA string theory. J. Geom. Phys. 114, 91–108 (2017). arXiv:1606.03206

    ADS  MathSciNet  MATH  Google Scholar 

  52. Fiorenza, D., Sati, H., Schreiber, U.: T-Duality from super Lie \(n\)-algebra cocycles for super p-branes. Adv. Theor. Math. Phys. 22, 1209–1270 (2018). arXiv:1611.06536

    MathSciNet  Google Scholar 

  53. Fiorenza, D., Sati, H., Schreiber, U.: T-duality in rational homotopy theory via \(L_\infty \)-algebras. Geom., Top. and Math. Phys. 1 (2018); special volume in tribute of Jim Stasheff and Dennis Sullivan. arXiv:1712.00758 [math-ph]

  54. Fiorenza, D., Sati, H., Schreiber, U.: Higher T-duality of M-branes. arXiv:1803.05634

  55. Fiorenza, D., Sati, H., Schreiber, U.: The rational higher structure of M-theory. In: Proceedings of the LMS-EPSRC Durham Symposium Higher Structures in M-Theory, Aug 2018, Fortsch. Phys., (2019). https://doi.org/10.1002/prop.201910017. arXiv:1903.02834

  56. Fiorenza, D., Sati, H., Schreiber, U.: Twisted Cohomotopy implies level quantization of the full 6d Wess-Zumino-term of the M5-brane. arXiv:1906.07417

  57. Fiorenza, D., Sati, H., Schreiber, U.: Super-exceptional geometry origin of heterotic M-theory and super-exceptional embedding construction of M5, JHEP (2020). arXiv:1908.00042

  58. Freed, D.: Dirac charge quantization and generalized differential cohomology. Surv. Diff. Geom. 7, 129–194, Int. Press, Somerville, MA, (2000). arXiv:hep-th/0011220

  59. Freed, D.: The geometry and topology of orientifolds II, talk at topology, \(C^\ast \)-algebras, and string duality. web.ma.utexas.edu/users/dafr/tcunp.pdf

  60. Fukami, T., Ishihara, S.: Almost Hermitian structure on \(S^6\). Tohoku Math J. 7, 151–156 (1955)

    MathSciNet  MATH  Google Scholar 

  61. Gaillard, J.: On \(G\)-structures in gauge/string duality (2011). [cronfa:42569]. [spire:1340775]

  62. Gauntlett, J., Martelli, D., Pakis, S., Waldram, D.: \(G\)-structures and wrapped NS5-branes. Commun. Math. Phys. 247, 421–445 (2004). arXiv:hep-th/0205050

    ADS  MathSciNet  MATH  Google Scholar 

  63. Gluck, H., Mackenzie, D., Morgan, F.: Volume-minimizing cycles in Grassmann manifolds. Duke Math. J. 79, 335–404 (1995). [euclid:1077285156]

  64. Gluck, H., Warner, F., Ziller, W.: The geometry of the Hopf fibrations. Enseign. Math. 32, 173–198 (1986)

    MathSciNet  MATH  Google Scholar 

  65. Grady, D., Sati, H.: Twisted differential generalized cohomology theories and their Atiyah–Hirzebruch spectral sequence Algebr. Geom. Topol. 19, 2899–2960 (2019). arXiv:1711.06650 [math.AT]

    MathSciNet  MATH  Google Scholar 

  66. Grady, D., Sati, H.: Twisted differential KO-theory. arXiv:1905.09085 [math.AT]

  67. Grady, D., Sati, H.: Ramond–Ramond fields and twisted differential K-theory. arXiv:1903.08843

  68. Gray, A.: A note on manifolds whose holonomy group is a subgroup of \({{\rm Sp}}(n) \varvec {\cdot } {{\rm Sp}}(1)\). Michigan Math. J. 16, 125–128 (1969)

    MathSciNet  MATH  Google Scholar 

  69. Gray, A., Green, P.S.: Sphere transitive structures and the triality automorphism. Pacific J. Math. 34, 83–96 (1970). [euclid:1102976640]

  70. Graña, M., Shahbazi, C.S., Zambon, M.: \({\rm Spin}(7)\)-manifolds in compactifications to four dimensions. J. High Energy Phys. 11, 046 (2014). arXiv:1405.3698

    ADS  MathSciNet  MATH  Google Scholar 

  71. Gubser, S.: Special holonomy in string theory and M-theory. In: Gubser, S., Lykken, J. (eds.) Strings, Branes and Extra Dimensions–TASI 2001. World Scientific, Singapore (2004). arXiv:hep-th/0201114

    MATH  Google Scholar 

  72. Harvey, R., Lawson, H.B.: Calibrated geometries. Acta Math. 148, 47–157 (1982). [euclid:1485890157]

  73. Hatsuda, M., Tomizawa, S.: Coset for hopf fibration and squashing. Class. Quant. Grav. 26, 225007 (2009). arXiv:0906.1025

    ADS  MathSciNet  MATH  Google Scholar 

  74. Heckman, J., Lawrie, C., Lin, L., Sakstein, J., Zoccarato, G.: Pixelated dark energy. arXiv:1901.10489

  75. Heckman, J., Lawrie, C., Lin, L., Zoccarato, G.: F-theory and Dark Energy, Fortsch. der Physik, (2019). https://doi.org/10.1002/prop.201900057. arXiv:1811.01959

  76. Hilgert, J., Neeb, K.-H.: Structure and Geometry of Lie Groups. Springer, Berlin (2012)

    MATH  Google Scholar 

  77. Hull, C.: Generalised geometry for M-theory. J. High Energy Phys. 0707, 079 (2007). arXiv:hep-th/0701203

    ADS  MathSciNet  Google Scholar 

  78. Hopkins, M.J., Singer, I.M.: Quadratic functions in geometry, topology, and M-theory. J. Differ. Geom. 70(3), 329–452 (2005). arXiv:math/0211216

    MathSciNet  MATH  Google Scholar 

  79. Howe, P., Lambert, N., West, P.: The self-dual string soliton. Nucl. Phys. B 515, 203–216 (1998). arXiv:hep-th/9709014

    ADS  MathSciNet  MATH  Google Scholar 

  80. Huerta, J., Sati, H., Schreiber, U.: Real ADE-equivariant (co)homotopy of super M-branes, Commun. Math. Phys. (2019) https://doi.org/10.1007/s00220-019-03442-3. arXiv:1805.05987

  81. Intriligator, K.: Anomaly matching and a Hopf–Wess–Zumino term in \(6d\), \({\cal{N}} = (2,0)\) field theories. Nucl. Phys. B 581, 257–273 (2000). arXiv:hep-th/0001205

    ADS  MATH  Google Scholar 

  82. Isham, C.J., Pope, C.N.: Nowhere-vanishing spinors and topological obstructions to the equivalence of the NSR and GS superstrings. Class. Quantum Grav. 5, 257–274 (1988). https://doi.org/10.1088/0264-9381/5/2/006. inspire:251240

  83. Isham, C.J., Pope, C.N., Warner, N.P.: Nowhere-vanishing spinors and triality rotations in 8-manifolds. Class. Quantum Grav. 5, 1297–1311 (1988). https://doi.org/10.1088/0264-9381/5/10/009. cds:185144

  84. Jaworowski, J.: Generalized cohomotopy groups as limit groups. Fund. Math. 50, 393–402 (1962)

    MathSciNet  MATH  Google Scholar 

  85. Kobin, A.: Algebraic topology (2016). ncatlab.org/nlab/files/KobinAT2016.pdf

  86. Koerber, P.: Lectures on generalized complex geometry for physicists. Fortsch. Phys. 59, 169–242 (2011). arXiv:1006.1536

    ADS  MathSciNet  MATH  Google Scholar 

  87. Kollross, A.: A classification of hyperpolar and cohomogeneity one actions. Trans. Am. Math. Soc. 354, 571–612 (2002). [jstor:2693761]

  88. Kosinski, A.: Differential Manifolds. Academic Press Inc, Boston, MA (1993)

    MATH  Google Scholar 

  89. Marchiafava, S., Romani, G.: Sui fibrati con struttura quaternionale generalizzata. Ann. Mat. Pura Appl. (IV) CVII, 131–157 (1976). https://doi.org/10.1007/BF02416470

    Article  MathSciNet  MATH  Google Scholar 

  90. Mathai, V., Quillen, D.: Superconnections, Thom classes, and equivariant differential forms. Topology 25, 85–110 (1986). https://doi.org/10.1016/0040-9383(86)90007-8

    Article  MathSciNet  MATH  Google Scholar 

  91. Mathew, A.: Notes on the J-homomorphism (2012). ncatlab.org/nlab/files/MathewJHomomorphism.pdf

  92. McInnes, B.: The semispin groups in string theory. J. Math. Phys. 40, 4699–4712 (1999). arXiv:hep-th/9906059

    ADS  MathSciNet  MATH  Google Scholar 

  93. McNamara, J., Vafa, C.: Cobordism Classes and the Swampland. arXiv:1909.10355

  94. de Medeiros, P., Figueroa-O’Farrill, J.: Half-BPS M2-brane orbifolds. Adv. Theor. Math. Phys. 16(5), 1349–1408 (2012). arXiv:1007.4761

    MathSciNet  MATH  Google Scholar 

  95. de Medeiros, P., Figueroa-O’Farrill, J., Gadhia, S., Méndez-Escobar, E.: Half-BPS quotients in M-theory: ADE with a twist. J. High Energy Phys. 0910, 038 (2009). arXiv:0909.0163

    MathSciNet  Google Scholar 

  96. Mitchell, S.: Notes on principal bundles and classifying spaces. Lecture notes (2011). sites.math.washington.edu/mitchell/Notes/prin.pdf

  97. Montgomery, D., Samelson, H.: Transformation groups of spheres. Ann. Math. 44, 454–470 (1943). [jstor:1968975]

  98. Moore, G.: Anomalies, Gauss laws, and page charges in M-theory. C.R. Phys. 6, 251–259 (2005). arXiv:hep-th/0409158

    ADS  MathSciNet  Google Scholar 

  99. Moore, G.: Physical mathematics and the future, talk at strings (2014). http://www.physics.rutgers.edu/gmoore/PhysicalMathematicsAndFuture.pdf

  100. Nicolai, H., Helling, R.: Supermembranes and M(atrix) Theory. In: Duff, M., et al. (eds.) Nonperturbative Aspects of Strings, Branes and supersymmetry. World Scientific, Singapore (1999). arXiv:hep-th/9809103

    MATH  Google Scholar 

  101. Nikolaus, T., Schreiber, U., Stevenson, D.: Principal \(\infty \)-bundles–general theory. J. Homotopy Related Structr. 10, 749–801 (2015). arXiv:1207.0248

    MathSciNet  MATH  Google Scholar 

  102. Onishchik, A.L. (ed.) Lie groups and lie algebras I. A. L. Onishchik, E. B. Vinberg, Foundations of Lie Theory, II. V. V. Gorbatsevich, A. L. Onishchik, Lie Transformation Groups, Encyclopedia of Mathematical Sciences, Volume 20, Springer, Berlin (1993)

  103. Ornea, L., Parton, M., Piccinni, P., Vuletescu, V.: \({{\rm Spin}}(9)\) geometry of the octonionic Hopf fibration. Transform. Groups 18, 845–864 (2013). arXiv:1208.0899

    MathSciNet  MATH  Google Scholar 

  104. Ornea, L., Piccinni, P.: Cayley 4-frames and a quaternion-Kähler reduction related to \({{\rm Spin}}(7)\), Global differential geometry: the mathematical legacy of Alfred Gray (Bilbao, 2000), 401–405, Contemp. Math. 288, Am. Math. Soc., Providence, RI, 2001. arXiv:math/0106116

  105. Page, D.: Classical stability of round and squashed seven-spheres in eleven-dimensional supergravity. Phys. Rev. D 28, 2976 (1983). https://doi.org/10.1103/PhysRevD.28.2976. inspire:14480

  106. Pittie, H.: The integral homology and cohomology rings of \({{\rm SO}}(n)\) and \({{\rm Spin}}(n)\). J. Pure Appl. Algebra 73, 105–153 (1991). https://doi.org/10.1016/0022-4049(91)90108-E

    Article  MathSciNet  MATH  Google Scholar 

  107. Pope, C.N., Warner, N.P.: An \({{\rm SU}}(4)\) invariant compactification of \(d=11\) supergravity on a stretched seven-sphere. Phys. Lett. B 150, 352–356 (1985)

    ADS  MathSciNet  Google Scholar 

  108. Porteous, I.: Clifford Algebras and the Classical Groups. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  109. Prins, D., Tsimpis, D.: IIA supergravity and M-theory on manifolds with \({{\rm SU}}(4)\) structure. Phys. Rev. D 89, 064030 (2014). arXiv:1312.1692

    ADS  Google Scholar 

  110. Sati, H., Schreiber, U.: Equivariant cohomotopy implies orientifold tadpole cancellation. arXiv:1909.12277

  111. Sati, H., Schreiber, U.: Lift of fractional D-brane charge to equivariant Cohomotopy theory. arXiv:1812.09679

  112. Sati, H., Schreiber, U.: Differential cohomotopy implies intersecting brane observables via configuration spaces and chord diagrams. arXiv:1912.10425

  113. Rudolph, G., Schmidt, M.: Differential geometry and mathematical physics: Part II. Fibre Bundles, Topology and Gauge Fields, Springer, Berlin (2017). https://doi.org/10.1007/978-94-024-0959-8

  114. Salamon, S.: Quaternionic Kähler manifolds. Invent. Math. 67, 143–171 (1982). https://doi.org/10.1007/BF01393378

    Article  ADS  MathSciNet  MATH  Google Scholar 

  115. Sati, H.: M-theory and characteristic classes. J. High Energy Phys. 0508, 020 (2005). arXiv:hep-th/0501245

    ADS  MathSciNet  Google Scholar 

  116. Sati, H.: Flux quantization and the M-theoretic characters. Nucl. Phys. B 727, 461 (2005). arXiv:hep-th/0507106

    ADS  MathSciNet  MATH  Google Scholar 

  117. Sati, H.: Duality symmetry and the form-fields in M-theory. J. High Energy Phys. 0606, 062 (2006). arXiv:hep-th/0509046

    ADS  MathSciNet  Google Scholar 

  118. Sati, H.: Geometric and topological structures related to M-branes. Proc. Symp. Pure Math. 81, 181–236 (2010). arXiv:1001.5020 [math.DG]

    MathSciNet  MATH  Google Scholar 

  119. Sati, H.: Framed M-branes, corners, and topological invariants. J. Math. Phys. 59, 062304 (2018). arXiv:1310.1060 [hep-th]

    ADS  MathSciNet  MATH  Google Scholar 

  120. Sati, H., Schreiber, U.: Higher T-duality of M-branes via local supersymmetry. Phys. Lett. B 781, 694–698 (2018). arXiv:1805.00233 [hep-th]

    ADS  MATH  Google Scholar 

  121. Sati, H., Schreiber, U., Stasheff, J.: Twisted differential string and fivebrane structures. Commun. Math. Phys. 315, 169–213 (2012). arXiv:0910.4001 [math-AT]

    ADS  MathSciNet  MATH  Google Scholar 

  122. Sati, H., Shim, H.-B.: String structures associated to indefinite Lie groups. J. Geom. Phys. 140, 246–264 (2019). arXiv:1504.02088 [math-ph]

    ADS  MathSciNet  MATH  Google Scholar 

  123. Sethi, S., Vafa, C., Witten, E.: Constraints on low-dimensional string compactifications. Nucl. Phys. B 480, 213–224 (1996). arXiv:hep-th/9606122

    ADS  MathSciNet  MATH  Google Scholar 

  124. Shahbazi, C.S.: M-theory on non-Kähler manifolds. J. High Energy Phys. 09, 178 (2015). arXiv:1503.00733

    ADS  MATH  Google Scholar 

  125. Souères, B., Tsimpis, D.: The action principle and the supersymmetrisation of Chern–Simons terms in eleven-dimensional supergravity. Phys. Rev. D 95, 026013 (2017). arXiv:1612.02021

    ADS  MathSciNet  Google Scholar 

  126. Spanier, E.: Borsuk’s cohomotopy groups. Ann. Math. 50, 203–245 (1949). [jstor:1969362]

  127. Tsimpis, D.: 11D supergravity at \({\cal{O}}(\ell ^3)\). J. High Energy Phys. 0410, 046 (2004). arXiv:hep-th/0407271

    ADS  MathSciNet  Google Scholar 

  128. Tsimpis, D.: M-theory on eight-manifolds revisited: \(N=1\) supersymmetry and generalized \({\rm Spin}(7)\) structures. J. High Energy Phys. 0604, 027 (2006). arXiv:hep-th/0511047

    ADS  MathSciNet  Google Scholar 

  129. Vafa, C.: Evidence for F-theory. Nucl. Phys. B 469, 403–418 (1996). arXiv:hep-th/9602022

    ADS  MathSciNet  MATH  Google Scholar 

  130. Vafa, C., Witten, E.: A one-loop test of string duality. Nucl. Phys. B 447, 261–270 (1995). arXiv:hep-th/9505053

    ADS  MathSciNet  MATH  Google Scholar 

  131. Varadarajan, V.: \({{\rm Spin}}(7)\)-subgroups of \({\rm SO}(8)\) and \({\rm Spin}(8)\). Expos. Math. 19, 163–177 (2001). https://doi.org/10.1016/S0723-0869(01)80027-X

    Article  MATH  Google Scholar 

  132. Walschap, G.: Metric Structures in Differential Geometry. Graduate Texts in Mathematics. Springer, Berlin (2004)

    MATH  Google Scholar 

  133. Whitehead, G.: On the homotopy groups of spheres and rotation groups. Ann. Math. 43, 634–640 (1942) [jstor:1968956]

  134. Witten, E.: Strong coupling and the cosmological constant. Mod. Phys. Lett. A 10, 2153–2156 (1995). arXiv:hep-th/9506101

    ADS  MathSciNet  MATH  Google Scholar 

  135. Witten, E.: Some comments on string dynamics, talk at Strings95. arXiv:hep-th/9507121

  136. Witten, E.: On flux quantization in M-theory and the effective action. J. Geom. Phys. 22, 1–13 (1997). arXiv:hep-th/9609122

    ADS  MathSciNet  MATH  Google Scholar 

  137. Witten, E.: Five-brane effective action In M-theory. J. Geom. Phys. 22, 103–133 (1997). arXiv:hep-th/9610234

    ADS  MathSciNet  MATH  Google Scholar 

  138. Witten, E.: The cosmological constant from the viewpoint of string theory, Lecture at DM2000. In: Kline, D. (ed.) Sources and Detection of Dark Matter and Dark Energy in the Universe 2000, 27–36, Springer, 2001. https://doi.org/10.1007/978-3-662-04587-9. arXiv:hep-ph/0002297

  139. Witten, E.: In: G. Farmelo, The Universe Speaks in numbers, interview 5 (2019) grahamfarmelo.com/the-universe-speaks-in-numbers-interview-5

  140. Wu, S.: Mathai-Quillen Formalism, pp. 390-399, Encyclopedia of Mathematical Physics (2006). arXiv:hep-th/0505003

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Acknowledgements

D. F. would like to thank NYU Abu Dhabi for hospitality during the writing of this paper. We thank Paolo Piccinni for useful discussion and Martin Čadek for useful communication. We also thank Mike Duff for comments on an earlier version.

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Authors and Affiliations

  1. Dipartimento di Matematica, La Sapienza Università di Roma, Piazzale Aldo Moro 2, 00185, Rome, Italy

    Domenico Fiorenza

  2. Mathematics, Division of Science, New York University Abu Dhabi, Abu Dhabi, UAE

    Hisham Sati & Urs Schreiber

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  1. Domenico Fiorenza
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  2. Hisham Sati
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Correspondence to Urs Schreiber.

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Communicated by C. Schweigert.

To Mike Duff on the occasion of his 70th birthday.

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Urs Schreiber: On leave from Czech Academy of Science, Prague.

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Fiorenza, D., Sati, H. & Schreiber, U. Twisted Cohomotopy Implies M-Theory Anomaly Cancellation on 8-Manifolds. Commun. Math. Phys. 377, 1961–2025 (2020). https://doi.org/10.1007/s00220-020-03707-2

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