Flux Quantization on Phase Space

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. In this short note, we observe that all higher Maxwell-type equations 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 valued in a characteristic \(L_\infty \)-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 rational Whitehead \(L_\infty \)-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 (Sati and Schreiber in Quantum observables on quantized fluxes. arXiv:2312.13037). Moreover, for the case of NS/RR-fields in type II supergravity/string theory, the traditional “Hypothesis K” of flux quantization in topological K-theory is naturally implied, without the need, on phase space, of the notorious further duality constraint. Finally, as a genuinely non-abelian example we consider flux quantization of the C-field in 11d supergravity/M-theory given by unstable differential 4-Cohomotopy (“Hypothesis H”) and emphasize again that, implemented on Cauchy data, this qualifies as the full phase space without the need for a further duality constraint.