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
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.
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).
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.
Recall that this says that if
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.
[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.”
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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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Communicated by C. Schweigert.
To Mike Duff on the occasion of his 70th birthday.
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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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DOI: https://doi.org/10.1007/s00220-020-03707-2