Schreiber Flux Quantization on M5-Branes

An article that we are finalizing at CQTS:

---

• Grigorios Giotopoulos, $\;$ Hisham Sati $\;$ and $\,$ Urs Schreiber:

  
  

  Flux Quantization on M5-Branes

  
  

  download

  • main file:

    • pdf$\;$ (v2)

    • arXiv:2406.11304

  • ancillary file: nb (Mathematica notebook with Clifford algebra checks)

---

Abstract. The M5-brane has been argued to potentially provide much needed theoretical underpinning for various non-perturbative phenomena in strongly-coupled quantum systems (such as S-duality, confinement, skyrmions and anyons), and yet the primary non-perturbative effect already in its classical on-shell formulation has received little attention: The flux and charge-quantization of the higher gauge field on its worldvolume.

This problem appears subtle because of (1a.) the notorious self-duality of the 3-form flux density in the small field limit, combined with (1b.) its highly non-linear self-duality for strong fields, and (2a.) the twisting of its Bianchi identity by the pullback of the 11d SuGra C-field flux density, which (2b.) is subject to its own subtle flux quantization law on the ambient spacetime.

These subtleties call into question the tacit assumption that the M5 brane’s 3-flux is quantized in ordinary cohomology and hence leaves open the rather fundamental question of what the M5-brane’s worldvolume higher gauge field really is, globally, controlling in particular which torsion-charged (“fractional”) solitons actually exist on M5-worldvolumes.

Here we characterize the valid quantization laws of the M5-brane’s 3-flux in (non-abelian) twisted generalized cohomology. The key step is to pass to super-spacetime and there to combine the rational cohomology of the C-field-twisted character map with the “super-embedding”-construction of the on-shell M5-brane fields, of which we give a rigorous and streamlined re-rederivation.

We show that one admissible quantization law of the 3-flux on M5-branes is by 4-Cohomotopy-twisted 3-Cohomotopy, as predicted by “Hypothesis H”. Besides quantizing the bulk fluxes $(G_4, G_7)$ and the brane’s $H_3$-flux themselves, this law also implies the (level-)quantization of the induced Page-charge/Hopf-WZ term on the M5-brane, necessary for its action functional to be globally well-defined.

Finally we demonstrate how with this flux quantization imposed, there generically appear skyrmionic solitons on M5-branes and anyonic topological quantum states on open M5-branes.

Related articles.

• Flux Quantization

• Flux Quantization on Phase Space

• Flux Quantization on 11d Superspace

• Twisted Cohomotopy implies M5 WZ term level quantization