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We are finalizing a short note on the subject of higher Gauss laws and flux quantization:
Abstract: While it has become widely appreciated that (higher) gauge theories need, besides their variational phase space data, to be equipped with “flux quantization laws” in generalized differential cohomology, there used to be no general prescription for how to define and construct the resulting flux-quantized phase space stacks.
We observe here that all higher Maxwell-type equations — in vacuum but on curved gravitational backgrounds and possibly with non-linear self-couplings among the higher fluxes — have solution spaces given by flux densities on a Cauchy surface subject to a higher Gauß law and no further constraint: The metric duality-constraint is all absorbed into the evolution equation away from the Cauchy surface.
Moreover, we observe that the higher Gauß law characterizes the Cauchy data as flat differential forms on the Cauchy surface valued in a characteristic -algebra. Using the recent construction of the non-abelian Chern-Dold character map, this implies that compatible flux quantization laws on phase space have classifying spaces whose Whitehead -algebra is this characteristic one. The flux-quantized higher phase space stack of the theory is then simply the corresponding (generally non-abelian) differential cohomology moduli stack on the Cauchy surface.
We show how this systematic prescription reproduces existing proposals for flux quantized phase spaces of vacuum Maxwell theory and of the chiral boson and its higher siblings, but reveals that there are other choices of (non-abelian) flux quantization laws even in these basic cases, further discussed in a companion article.
For the case of NS/RR-fields in type II supergravity, the traditional “Hypothesis K” of flux quantization in topological K-theory follows on phase space, without need of the notorious further duality constraint.
Finally, as a genuinely non-abelian example we consider flux-quantization of the C-field in 11d supergravity given by unstable differential 4-Cohomotopy (“Hypothesis H”) and emphasize again that, implemented on Cauchy data, this gives the full phase space without need of a further duality constraint.
Our latest pdf version is kept behind the above link. Comments are welcome.
small typo in Eq (4), should be D instead of 11
Thanks! Fixed now (here).
added pointer to the eponymous original:
Carl F. Gauß, Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum, methodo nova tractata, Commentationes Societatis Regiae Scientiarum Göttingensis Recentiores. Comm. Class. Math. 2 (1813) 1–24 [PPN35283028X_0002_2NS]
reprinted in: Carl Friedrich Gauss – Werke 5, Springer (1877) 2-22 [doi:10.1007/978-3-642-49319-5_1]
added references on (non-)locality of charged field operators due to the Gauss law:
Giovanni Morchio, Franco Strocchi, Localization and symmetries, J. Phys. A 40 (2007) 3173-3188 [arXiv:math-ph/0607015, doi:10.1088/1751-8113/40/12/S17]
Detlev Buchholz, Fabio Ciolli, Giuseppe Ruzzi, Ezio Vasselli, Gauss’s law, the manifestations of gauge fields, and their impact on local observables [arXiv:2212.11009]
Detlev Buchholz, Gauss’ law and the infraparticle problem, Phys. Lett. B 174 (1986) 331-334 [doi:10.1016/0370-2693(86)91110-X]
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