In classical field theory, many fields and related objects are described as differential
forms. For example, in electromagnetism, the field $F := B - \mathrm dt\wedge E$ is a 2-form, and Maxwell's
equations ask that it be closed. Quantization imposes integrality constraints on $F$, which may concisely be
pacakged as asking for $F$ to be a (cocycle representative of a) differential cohomology class $\check F\in \check
H^2(M; \mathbb Z)$ (here $M$ is spacetime).


In some theories, it turns out that it makes more sense to use differential *generalized* cohomology to perform
this quantization. For example, in superstring theory, the Ramond-Ramond field is quantized in differential
$K$-theory. Determining which generalized cohomology theory $X$ to use is a bit of an art: it's part of the data of
formulating the theory, making it harder to conclusively prove that one choice is right or wrong. Instead, what one
can do is prove that quantizing a given field in differential $X$-theory recovers as a theorem something whose
physics implication is something that physicists already believe about the theory, and that other choices of
generalized cohomology theories do not yield that conclusion.


My specific question is about M-theory (or more precisely, its low-energy limit given by supergravity). There is a
field called the $C$-field, which classically is a 3-form. There are two proposals that I know of for quantizing the
$C$-field in differential generalized cohomology:


* Traditionally, one uses $X = H\mathbb Z$, i.e. ordinary cohomology, twisted by the orientation local system. This recovers some facts about M-theory that physicists expect, [including the appearance of $K$-theory in superstring theory](https://arxiv.org/abs/hep-th/0005091).
* Alternatively, [Fiorenza-Sati-Schreiber](https://arxiv.org/abs/1904.10207) posit "hypothesis H," that the
$C$-field is quantized in $J$-twisted stable cohomotopy. That is, $X = \mathbb S$, twisted on a spacetime
manifold $M$ by the composite of maps $M\to B\mathrm O$ given by the stable normal bundle, then $B\mathrm
 O\to B\mathrm{GL}\_1(\mathbb S)$ given by the $J$-homomorphism. In several papers, Fiorenza-Sati-Schreiber
prove that hypothesis H implies several facts about M-theory that physicists expected, providing evidence in
favor of hypothesis H.


To what extent does the ordinary cohomology proposal fail to satisfy these hypotheses? That is, **what is an example
of a claim which physicists believe to be true about M-theory which is true under hypothesis H, but not under the
ordinary cohomology hypothesis?**



In M-theory, what can hypothesis H tell us that quantization in ordinary cohomology cannot?
**Traditional approach.** Notice that what is considered in [[DMW00](http://arxiv.org/abs/hep-th/0005091); [DFM03](https://arxiv.org/abs/hep-th/0312069)] 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](http://arxiv.org/abs/hep-th/0005091)]. Ordinary cohomology may be modified ("shifted") to accommodate this, as maybe first formalized in [[HS02](https://arxiv.org/abs/math/0211216)] and used in [[DFM03](https://arxiv.org/abs/hep-th/0312069)].


Beyond this, [[DMW00](http://arxiv.org/abs/hep-th/0005091)] 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.


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](http://arxiv.org/abs/hep-th/0005091)] 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](https://arxiv.org/abs/hep-th/0312033); [FSS 16, Sec. 3](https://ncatlab.org/schreiber/show/Rational+sphere+valued+supercocycles+in+M-theory)]. 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](https://arxiv.org/abs/1310.1060)].


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](https://ncatlab.org/schreiber/show/The+WZW+term+of+the+M5-brane)], reviewed in [[FSS19a, Sec. 7](https://ncatlab.org/schreiber/show/The+rational+higher+structure+of+M-theory)]).


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](https://ncatlab.org/schreiber/show/Hypothesis+H)") that M-brane charge is in Cohomotopy theory itself [[Sa13](https://arxiv.org/abs/1310.1060)] (suitably twisted by the tangent bundle [[FSS19b](https://ncatlab.org/schreiber/show/Twisted+Cohomotopy+implies+M-theory+anomaly+cancellation+on+8-manifolds); [FSS19c](https://ncatlab.org/schreiber/show/Twisted+Cohomotopy+implies+M5+WZ+term+level+quantization)]).


**Implications.** Indeed, one finds that the assumption of [Hypothesis H](https://ncatlab.org/schreiber/show/Hypothesis+H), 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](https://ncatlab.org/schreiber/show/Twisted+Cohomotopy+implies+M-theory+anomaly+cancellation+on+8-manifolds)].


This way [Hypothesis H](https://ncatlab.org/schreiber/show/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](https://ncatlab.org/schreiber/show/Hypothesis+H) of more of the fine-print in the K-theory conjecture is in [[SS19a](https://ncatlab.org/schreiber/show/Equivariant+Cohomotopy+implies+orientifold+tadpole+cancellation); [BSS18](https://ncatlab.org/schreiber/show/Lift+of+fractional+D-brane+charge+to+equivariant+Cohomotopy+theory); [BMSS18](https://ncatlab.org/schreiber/show/Gauge+enhancement+of+Super+M-Branes)].


Moreover, [Hypothesis H](https://ncatlab.org/schreiber/show/Hypothesis+H) sees the Hořava-Witten Green-Schwarz mechanism in the presence of M5-branes [[FSS19d](https://ncatlab.org/schreiber/show/Twistorial+Cohomotopy+implies+Green-Schwarz+anomaly+cancellation); [SS20b](https://ncatlab.org/schreiber/show/The+Character+Map+in+Equivariant+Twistorial+Cohomotopy)], reveals fine-print in the M5-brane anomaly cancellation argument [[FSS19c](https://ncatlab.org/schreiber/show/Twisted+Cohomotopy+implies+M5+WZ+term+level+quantization); [SS20a](https://www.ncatlab.org/schreiber/show/Twisted+Cohomotopy+implies+M5-brane+anomaly+cancellation)], and seems to see a zoo of subtle brane charge effects expected in Hanany-Witten systems [[SS19b](https://ncatlab.org/schreiber/show/Differential+Cohomotopy+implies+intersecting+brane+observables)].


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](https://ncatlab.org/schreiber/show/Hypothesis+H) 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](https://ncatlab.org/schreiber/show/Hypothesis+H) may be found in talk notes of the recent meeting at NYU AD [*M-Theory and Mathematics*](https://hisham-sati.github.io/M-theory-and-Mathematics/): [[Sa20](https://ncatlab.org/nlab/files/Sati_MTheoryCohomotopy_2020.pdf); [Sc20](https://ncatlab.org/schreiber/show/Microscopic+Brane+Physics+from+Cohomotopy)].


**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))


