Revision 07eab1e2-00c0-4bca-83bf-eb87f64647bc

**Traditional approach.** Notice that what is considered in [[DMW00][1]; [DFM03][2]] and elsewhere to quantize the C-field flux $G_4$ is not just ordinary cohomology, but ordinary cohomology with bells and whistles added as need be:

Foremost there is the half-integral quantization of the $G_4$-flux, mentioned as (3.2) in [[DMW00][1]]. Ordinary cohomology may be modified ("shifted")  to accommodate this, as maybe first formalized in [[HS02][3]] and used in [[DFM03][2]]. 

Beyond this, [[DMW00][1]] 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.

Instead, the observation made is that the constraint $Sq^3 [F_4] = 0$ is also the first condition that appears in the Atiyah-Hirzebruch spectral sequence for lifting an ordinary cohomology class $[F_4]$ to complex topological K-theory, as demanded by the widely accepted conjecture that RR-field fluxes $F_{n}$ in string theory are quantized in topological K-theory. 

Since, moreover, $F_4$ is meant to come from $G_4$ as one lifts string theory to M-theory, the observation of [[DMW00][1]] is hence that the "integral equation of motion" in M-theory reproduces one of the constraints on one of the RR-field components given by flux quantization in K-theory. 

Suggestive as this is, this consistency check is arguably not a "derivation of K-theory" in string theory. In fact, the conjecture that string theoretic RR-flux really is quantized in K-theory remains itself being debated (for instance, it's not clear how to make it compatible with S-duality). The problem here is that little is known with certainty about non-perturbative string theory beyond a web of interlocking conjectures.


**Hypothesis H.** In view of this situation it seems worthwhile to try a strict top-down approach where a unified generalized cohomology theory in M-theory is postulated and its consequences on flux quantization rigorously derived. 

There are good hints what this M-theoretic cohomology theory ought to be: Its image in rational cohomology must see the trivialization of the cup square of the $G_4$-flux demanded by 11d supergravity -- this being the M-theoretic lift of the twisted Bianchi identities that motivate the twisted K-theory conjecture [[MSa 03, Sec. 4.2][4]; [FSS 16, Sec. 3][5]].  The condition happens to be exactly the relation that identifies the Sullivan model of the 4-sphere, thus suggesting that M-brane charge is quantized in 4-Cohomotopy theory; due to [[Sa13, Sec. 2.5][6]].

Indeed, cohomotopical charge quantization in M-theory, on the rational level, follows from a first-principles analysis of the super $p$-brane scan, and is as such the direct M-theoretic analogue of a computation that derives the twisted K-theory classification of D-brane charge ([[FSS15][7]], reviewed in [[FSS19a, Sec. 7][8]]).

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][9]") that M-brane charge is in Cohomotopy theory itself [[Sa13][6]]  (suitably twisted by the tangent bundle [[FSS19b][10]; [FSS19c][11]]).

**Implications.** Indeed, one finds that the assumption of [Hypothesis H][9], that M-brane charge is in J-twisted Cohomotopy theory, readily implies both the half-integral shifted flux quantization on $G_4$ as well as the "integral equation of motion" -- together with a list of further expected constraints [[FSS19b, Table 1][10]]. 

This way [Hypothesis H][9] 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][9] of more of the fine-print in the K-theory conjecture is in [[SS19a][12]], [[BSS18][13]], [[BMSS 18][14]].

Moreover, [Hypothesis H][9] sees the Hořava-Witten Green-Schwarz mechanism in the presence of M5-branes [[FSS19d][15]; [SS20b][16]], reveals fine-print in the M5-brane anomaly cancellation argument [[FSS19c][11]; [SS20a][17]], and seems to see a zoo of subtle brane charge effects expected in Hanany-Witten systems [[SS19b][18]]. 

Certainly none of these effects follows from flux quantization in just ordinary cohomology (nor in K-theory, for that matter).

By way of outlook, we think we see now that there is a natural chromatic character map on twisted Cohomotopy which exhibits the M5-brane partition function as charge-quantized in elliptic cohomology, matching traditional discussion of M5-brane ellitptic genera. This is work in progress.


**Conclusion.** In summary, rigorous derivation of the implications of [Hypothesis H][9] suggests that twisted Cohomotopy theory sees a fair number of subtle effects that have previously been argued informally to appear in the elusive non-perturbative completion of string theory. This may be indication that, going beyond the traditional approach of hard-coding M-theoretic folklore into a putative "C-field model", charge quantization in twisted Cohomotopy might get to the native heart of the elusive mathematical nature of "M-theory". But, of course, more analysis is necessary.


**Exposition.** More exposition of motivation and development of [Hypothesis H][9] may be found in talk notes of the recent meeting at NYU AD [*M-Theory and Mathematics*][19]: [[Sa20][20]; [Sc20][21]].

**References.**

* **[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](http://arxiv.org/abs/hep-th/0005090)), summarised in: _A Derivation of K-Theory from M-Theory_ ([arXiv:hep-th/0005091](http://arxiv.org/abs/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](http://arxiv.org/abs/math.AT/0211216), [euclid:1143642908](https://projecteuclid.org/euclid.jdg/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](http://arxiv.org/abs/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](https://arxiv.org/abs/hep-th/0312033), [doi:10.1088/1126-6708/2004/03/016](https://iopscience.iop.org/article/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](http://arxiv.org/abs/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](http://arxiv.org/abs/1506.07557), [doi:10.1063/1.4932618](http://scitation.aip.org/content/aip/journal/jmp/56/10/10.1063/1.4932618))

 
* **[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](http://arxiv.org/abs/1606.03206), [doi:10.1016/j.geomphys.2016.11.024](http://dx.doi.org/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](https://arxiv.org/abs/1806.01115), [doi:10.1007/s00220-019-03441-4](https://doi.org/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](https://arxiv.org/abs/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](https://arxiv.org/abs/1903.02834), [doi:10.1002/prop.201910017](https://doi.org/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](https://arxiv.org/abs/1904.10207), [doi:10.1007/s00220-020-03707-2](https://doi.org/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](https://arxiv.org/abs/1906.07417))

* **[FSS19d]** D. Fiorenza, H. Sati, U. Schreiber, _Twistorial Cohomotopy Implies Green-Schwarz anomaly cancellation_ ([arXiv:2008.08544](https://arxiv.org/abs/2008.08544))

* **[SS19a]** H. Sati, U. Schreiber, _Equivariant Cohomotopy implies orientifold tadpole cancellation_ J. Geom. Phys. Vol 156, 2020, 103775 ([arXiv:1909.12277](https://arxiv.org/abs/1909.12277), [doi:10.1016/j.geomphys.2020.103775](https://doi.org/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](https://arxiv.org/abs/1912.10425))

* **[SS20a]** H. Sati, U. Schreiber, _Twisted Cohomotopy implies M5-brane anomaly cancellation_ ([arXiv:2002.07737](https://arxiv.org/abs/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](https://arxiv.org/abs/2011.06533))

* **[Sa20]** H. Sati, _M-theory and cohomotopy_, talk at _M-Theory and Mathematics_, NYUAD 2020 ([pdf](https://ncatlab.org/nlab/files/SatiMTheoryCohomotopy2020.pdf))

* **[Sc20]** U. Schreiber, _Microscopic brane physics from Cohomotopy theory_, talk at _M-Theory and Mathematics_, NYUAD 2020 ([pdf](https://ncatlab.org/schreiber/files/Schreiber-MTheoryMathematics2020-v200126a.pdf))


  [1]: http://arxiv.org/abs/hep-th/0005091
  [2]: https://arxiv.org/abs/hep-th/0312069
  [3]: https://arxiv.org/abs/math/0211216
  [4]: https://arxiv.org/abs/hep-th/0312033
  [5]: https://ncatlab.org/schreiber/show/Rational+sphere+valued+supercocycles+in+M-theory
  [6]: https://arxiv.org/abs/1310.1060
  [7]: https://ncatlab.org/schreiber/show/The+WZW+term+of+the+M5-brane
  [8]: https://ncatlab.org/schreiber/show/The+rational+higher+structure+of+M-theory
  [9]: https://ncatlab.org/schreiber/show/Hypothesis+H
  [10]: https://ncatlab.org/schreiber/show/Twisted+Cohomotopy+implies+M-theory+anomaly+cancellation+on+8-manifolds
  [11]: https://ncatlab.org/schreiber/show/Twisted+Cohomotopy+implies+M5+WZ+term+level+quantization
  [12]: https://ncatlab.org/schreiber/show/Equivariant+Cohomotopy+implies+orientifold+tadpole+cancellation
  [13]: https://ncatlab.org/schreiber/show/Lift+of+fractional+D-brane+charge+to+equivariant+Cohomotopy+theory
  [14]: https://ncatlab.org/schreiber/show/Gauge+enhancement+of+Super+M-Branes
  [15]: https://ncatlab.org/schreiber/show/Twistorial+Cohomotopy+implies+Green-Schwarz+anomaly+cancellation
  [16]: https://ncatlab.org/schreiber/show/The+Character+Map+in+Equivariant+Twistorial+Cohomotopy
  [17]: https://www.ncatlab.org/schreiber/show/Twisted+Cohomotopy+implies+M5-brane+anomaly+cancellation
  [18]: https://ncatlab.org/schreiber/show/Differential+Cohomotopy+implies+intersecting+brane+observables
  [19]: https://hisham-sati.github.io/M-theory-and-Mathematics/
  [20]: https://ncatlab.org/nlab/files/Sati_MTheoryCohomotopy_2020.pdf
  [21]: https://ncatlab.org/schreiber/show/Microscopic+Brane+Physics+from+Cohomotopy