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This way Hypothesis H implies as much "derivation of K-theory"; though one should push further to an actual derivation of the implied flux quantization of RR-fields. Derivation from Hypothesis H of more of the fine-print in the K-theory conjecture is in [SS19a], [BSS18], [BMSS 18].

This way Hypothesis H implies as much "derivation of K-theory"; though one should push further to an actual derivation of the implied flux quantization of RR-fields. Derivation from Hypothesis H of more of the fine-print in the K-theory conjecture is in [SS19a; BSS18; BMSS18].

Beyond this, [DMW00] argue that the "M-theory path integral" imposes a further condition on the shifted integral cohomology class of $G_4$, namely that it be in the kernel of the Steenrod operation $Sq^3$. Whether this constraint is enforced by flux quantization in some further modification of ordinary cohomology is not discussed there.

Beyond this, [DMW00] argue that the "M-theory path integral" imposes a further condition on the shifted integral cohomology class of $G_4$, namely that it be in the kernel of the Steenrod operation $Sq^3$. Whether this "integral equation of motion" is enforced by flux quantization in some further modification of ordinary cohomology is not discussed there.

This means that any cohomology theory which quantizes M-brane charge should rationally coincide with Cohomotopy theory. Accordingly, it is quite natural to consider the hypothesis (dubbed "Hypothesis H") that M-brane charge indeed is in Cohomotopy theory itself [Sa13] (suitably twisted by the tangent bundle [FSS19b; FSS19c]).

• [DMW00] E. Diaconescu, G. Moore, E. Witten, $E_8$ Gauge Theory, and a Derivation of K-Theory from M-Theory, Adv. Theor. Math. Phys. 6:1031-1134, 2003 (arXiv:hep-th/0005090), summarised in: A Derivation of K-Theory from M-Theory (arXiv:hep-th/0005091)

• [HS02] M Hopkins, I. Singer, Quadratic Functions in Geometry, Topology,and M-Theory, J. Differential Geom. Volume 70, Number 3 (2005), 329-452 (arXiv:math.AT/0211216, euclid:1143642908)

• [DFM03] E. Diaconescu, D. Freed, G. Moore, The $M$-theory 3-form and $E_8$-gauge theory, chapter in: Elliptic Cohomology Geometry, Applications, and Higher Chromatic Analogues, Cambridge University Press 2007 (arXiv:hep-th/0312069, [doi:10.1017/CBO9780511721489](https: //doi.org/10.1017/CBO9780511721489))

• [MSa03] V. Mathai, H. Sati, Some Relations between Twisted K-theory and E8 Gauge Theory, JHEP 0403:016, 2004 (arXiv:hep-th/0312033, doi:10.1088/1126-6708/2004/03/016)

• [Sa13] H. Sati, Framed M-branes, corners, and topological invariants, J. Math. Phys. 59 (2018), 062304 (arXiv:1310.1060)

• [FSS15] D. Fiorenza, H. Sati, U. Schreiber, The WZW term of the M5-brane and differential cohomotopy, J. Math. Phys. 56, 102301 (2015) (arXiv:1506.07557, doi:10.1063/1.4932618)

• [FSS16] D. Fiorenza, H. Sati, U. Schreiber, Rational sphere valued supercocycles in M-theory and type IIA string theory, J. Geom. Phys., Vol 114 (2017) (arXiv:1606.03206, doi:10.1016/j.geomphys.2016.11.024)

• [BMSS18] V. Braunack-Mayer, H. Sati, U. Schreiber, Gauge enhancement of super M-branes via parametrized stable homotopy theory, Comm. Math. Phys. 371: 197 (2019) (arXiv:1806.01115, doi:10.1007/s00220-019-03441-4)

• [BSS18] S. Burton, H. Sati, U. Schreiber, Lift of fractional D-brane charge to equivariant Cohomotopy theory, J. Geom. Phys., 2020 (in print) (arXiv:1812.09679)

• [FSS19a] D. Fiorenza, H. Sati, U. Schreiber, The rational higher structure of M-theory, in: Proceedings of Higher Structures in M-Theory 2018, Fortsch. Phys. 2019 (arXiv:1903.02834, doi:10.1002/prop.201910017)

• [FSS19b] D. Fiorenza, H. Sati, U. Schreiber, Twisted Cohomotopy implies M-Theory anomaly cancellation on 8-manifolds, Comm. Math. Phys. 377(3), 1961-2025 (2020) (arXiv:1904.10207, doi:10.1007/s00220-020-03707-2)

• [FSS19c] D. Fiorenza, H. Sati, U. Schreiber, Twisted Cohomotopy implies level quantization of the full 6d Wess-Zumino-term of the M5-brane, Comm. Math. Phys. 2020 (in print) (arXiv:1906.07417)

• [FSS19d] D. Fiorenza, H. Sati, U. Schreiber, Twistorial Cohomotopy Implies Green-Schwarz anomaly cancellation (arXiv:2008.08544)

• [SS19a] H. Sati, U. Schreiber, Equivariant Cohomotopy implies orientifold tadpole cancellation J. Geom. Phys. Vol 156, 2020, 103775 (arXiv:1909.12277, doi:10.1016/j.geomphys.2020.103775)

• [SS19b] H. Sati, U. Schreiber, Differential Cohomotopy implies intersecting brane observables via configuration spaces and chord diagrams (arXiv:1912.10425)

• [SS20a] H. Sati, U. Schreiber, Twisted Cohomotopy implies M5-brane anomaly cancellation (arXiv:2002.07737)

• [SS20b] H. Sati, U. Schreiber, The character map in equivariant twistorial Cohomotopy implies the Green-Schwarz mechanism with heterotic M5-branes (arXiv:2011.06533)

• [Sa20] H. Sati, M-theory and cohomotopy, talk at M-Theory and Mathematics, NYUAD 2020 (pdf)

• [Sc20] U. Schreiber, Microscopic brane physics from Cohomotopy theory, talk at M-Theory and Mathematics, NYUAD 2020 (pdf)

This means that any cohomology theory which quantizes M-brane charge should rationally coincide with Cohomotopy theory. Accordingly, it is quite natural to consider the hypothesis (dubbed "Hypothesis H") that M-brane charge is in Cohomotopy theory itself [Sa13] (suitably twisted by the tangent bundle [FSS19b; FSS19c]).

• [DMW00] E. Diaconescu, G. Moore, E. Witten, $E_8$ Gauge Theory, and a Derivation of K-Theory from M-Theory, Adv. Theor. Math. Phys. 6:1031-1134, 2003 (arXiv:hep-th/0005090), summarised in: A Derivation of K-Theory from M-Theory (arXiv:hep-th/0005091)

• [HS02] M Hopkins, I. Singer, Quadratic Functions in Geometry, Topology,and M-Theory, J. Differential Geom. Volume 70, Number 3 (2005), 329-452 (arXiv:math.AT/0211216, euclid:1143642908)

• [DFM03] E. Diaconescu, D. Freed, G. Moore, The $M$-theory 3-form and $E_8$-gauge theory, chapter in: Elliptic Cohomology Geometry, Applications, and Higher Chromatic Analogues, Cambridge University Press 2007 (arXiv:hep-th/0312069, doi:10.1017/CBO9780511721489)

• [MSa03] V. Mathai, H. Sati, Some Relations between Twisted K-theory and E8 Gauge Theory, JHEP 0403:016, 2004 (arXiv:hep-th/0312033, doi:10.1088/1126-6708/2004/03/016)

• [Sa13] H. Sati, Framed M-branes, corners, and topological invariants, J. Math. Phys. 59 (2018), 062304 (arXiv:1310.1060)

• [FSS15] D. Fiorenza, H. Sati, U. Schreiber, The WZW term of the M5-brane and differential cohomotopy, J. Math. Phys. 56, 102301 (2015) (arXiv:1506.07557, doi:10.1063/1.4932618)

• [FSS16] D. Fiorenza, H. Sati, U. Schreiber, Rational sphere valued supercocycles in M-theory and type IIA string theory, J. Geom. Phys., Vol 114 (2017) (arXiv:1606.03206, doi:10.1016/j.geomphys.2016.11.024)

• [BMSS18] V. Braunack-Mayer, H. Sati, U. Schreiber, Gauge enhancement of super M-branes via parametrized stable homotopy theory, Comm. Math. Phys. 371: 197 (2019) (arXiv:1806.01115, doi:10.1007/s00220-019-03441-4)

• [BSS18] S. Burton, H. Sati, U. Schreiber, Lift of fractional D-brane charge to equivariant Cohomotopy theory, J. Geom. Phys., 2020 (in print) (arXiv:1812.09679)

• [FSS19a] D. Fiorenza, H. Sati, U. Schreiber, The rational higher structure of M-theory, in: Proceedings of Higher Structures in M-Theory 2018, Fortsch. Phys. 2019 (arXiv:1903.02834, doi:10.1002/prop.201910017)

• [FSS19b] D. Fiorenza, H. Sati, U. Schreiber, Twisted Cohomotopy implies M-Theory anomaly cancellation on 8-manifolds, Comm. Math. Phys. 377(3), 1961-2025 (2020) (arXiv:1904.10207, doi:10.1007/s00220-020-03707-2)

• [FSS19c] D. Fiorenza, H. Sati, U. Schreiber, Twisted Cohomotopy implies level quantization of the full 6d Wess-Zumino-term of the M5-brane, Comm. Math. Phys. 2020 (in print) (arXiv:1906.07417)

• [FSS19d] D. Fiorenza, H. Sati, U. Schreiber, Twistorial Cohomotopy Implies Green-Schwarz anomaly cancellation (arXiv:2008.08544)

• [SS19a] H. Sati, U. Schreiber, Equivariant Cohomotopy implies orientifold tadpole cancellation J. Geom. Phys. Vol 156, 2020, 103775 (arXiv:1909.12277, doi:10.1016/j.geomphys.2020.103775)

• [SS19b] H. Sati, U. Schreiber, Differential Cohomotopy implies intersecting brane observables via configuration spaces and chord diagrams (arXiv:1912.10425)

• [SS20a] H. Sati, U. Schreiber, Twisted Cohomotopy implies M5-brane anomaly cancellation (arXiv:2002.07737)

• [SS20b] H. Sati, U. Schreiber, The character map in equivariant twistorial Cohomotopy implies the Green-Schwarz mechanism with heterotic M5-branes (arXiv:2011.06533)

• [Sa20] H. Sati, M-theory and cohomotopy, talk at M-Theory and Mathematics, NYUAD 2020 (pdf)

• [Sc20] U. Schreiber, Microscopic brane physics from Cohomotopy theory, talk at M-Theory and Mathematics, NYUAD 2020 (pdf)