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Flux Quantization on M5-branes

Grigorios Giotopoulos, Hisham Sati, Urs Schreiber

TL;DR

This work reframes flux quantization on M5-branes as a global completion problem within twisted co-Homotopy by leveraging super-spacetime Bianchi identities and the super-embedding formalism. It provides a streamlined derivation of the M5 worldvolume 3-form flux H_3 from 1/2BPS super-immersions and shows that self-duality constraints are absorbed into super-Bianchi identities, enabling a complete on-shell field content without ad hoc global data. Adopting Hypothesis H, the authors classify admissible flux laws by a twisted differential cohomology theory whose rational type matches the quaternionic Hopf fibration, yielding Skyrmion-like worldvolume solitons and, for open M5-branes, anyonic boundary states via co-Homotopy. The framework unifies local gauge potentials with global flux data through concordances, and predicts rich topological features (e.g., configurational moduli, braid-group statistics) that connect M5-brane physics to strongly coupled quantum systems and holography. Overall, the paper advances a cohesive, geometrically rigorous picture in which M5-brane flux quantization is governed by non-abelian, twisted cohomological structures rather than classical abelian cohomology alone.

Abstract

We highlight the need for global completion of the field content in the M5-brane sigma-model analogous to Dirac's charge/flux quantization, and we point out that the superspace Bianchi identities on the worldvolume and on its ambient supergravity background constrain the M5's flux-quantization law to be in a non-abelian cohomology theory rationally equivalent to a twisted form of co-Homotopy. In order to clearly bring out this subtle point we give a streamlined re-derivation of the worldvolume 3-flux via M5 "super-embeddings". Finally, assuming the flux-quantization law to actually be in co-Homotopy ("Hypothesis H") we show how this implies Skyrmion-like solitons on general M5-worldvolumes and (abelian) anyonic solitons on the boundaries of "open M5-branes" in heterotic M-theory.

Flux Quantization on M5-branes

TL;DR

This work reframes flux quantization on M5-branes as a global completion problem within twisted co-Homotopy by leveraging super-spacetime Bianchi identities and the super-embedding formalism. It provides a streamlined derivation of the M5 worldvolume 3-form flux H_3 from 1/2BPS super-immersions and shows that self-duality constraints are absorbed into super-Bianchi identities, enabling a complete on-shell field content without ad hoc global data. Adopting Hypothesis H, the authors classify admissible flux laws by a twisted differential cohomology theory whose rational type matches the quaternionic Hopf fibration, yielding Skyrmion-like worldvolume solitons and, for open M5-branes, anyonic boundary states via co-Homotopy. The framework unifies local gauge potentials with global flux data through concordances, and predicts rich topological features (e.g., configurational moduli, braid-group statistics) that connect M5-brane physics to strongly coupled quantum systems and holography. Overall, the paper advances a cohesive, geometrically rigorous picture in which M5-brane flux quantization is governed by non-abelian, twisted cohomological structures rather than classical abelian cohomology alone.

Abstract

We highlight the need for global completion of the field content in the M5-brane sigma-model analogous to Dirac's charge/flux quantization, and we point out that the superspace Bianchi identities on the worldvolume and on its ambient supergravity background constrain the M5's flux-quantization law to be in a non-abelian cohomology theory rationally equivalent to a twisted form of co-Homotopy. In order to clearly bring out this subtle point we give a streamlined re-derivation of the worldvolume 3-flux via M5 "super-embeddings". Finally, assuming the flux-quantization law to actually be in co-Homotopy ("Hypothesis H") we show how this implies Skyrmion-like solitons on general M5-worldvolumes and (abelian) anyonic solitons on the boundaries of "open M5-branes" in heterotic M-theory.
Paper Structure (12 sections, 18 theorems, 174 equations)

This paper contains 12 sections, 18 theorems, 174 equations.

Table of Contents

  1. Introduction and Overview
  2. Revisiting "super-embeddings"
  3. Darboux co-frame fields
  4. $1/2$BPS Super-immersions
  5. M5-brane super-immersions
  6. Self-dual tensors in 6d
  7. Spinors in 6d from 11d
  8. Super-flux on M5 immersions
  9. Flux quantization on M5-branes
  10. Gauge potentials on the M5-brane
  11. Skyrmions and Anyons on M5
  12. Conclusion

Key Result

Lemma 2.6

On a (pseudo-)Riemannian manifold $(X, \mathrm{d}s^2)$ there exists a (relativistic) orthonormal co-frame (Def. OrthonormalCoframes).

Theorems & Definitions (61)

  • Definition 2.2: Relativistic local co-frame field
  • Remark 2.3: Čech cocycle data from co-frames
  • Definition 2.5: Orthonormal local co-frames
  • Lemma 2.6: Existence of orthonormal co-frames
  • proof
  • Lemma 2.7: Essential uniqueness of orthonormal co-frame fields
  • proof
  • Remark 2.8: Groupoid of co-frame fields equivalent to set of metrics
  • Remark 2.9: Lifting smooth maps to covers
  • Remark 2.10: Immersions vs. embeddings
  • ...and 51 more