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Flux Quantization on M-Strings

Pinak Banerjee, Hisham Sati, Urs Schreiber

Abstract

The electric Gauss law in 11D SuGra is famously non-linear, whence its flux quantization must be in nonabelian cohomology. We have previously shown that the minimal admissible choice is 4-Cohomotopy, which in the presence of magnetized M5-probes takes its relative twistorial form. Here we discuss how this situation is further refined in the presence of M-string probes on the M5-worldvolume. Based on the superspace formulation of 11D SuGra, we find the nested Bianchi identities by iterating the superembedding construction for super p-branes. The resulting probe brane hierarchy (M1 on magnetized M5 in 11D bulk) turns out to admit flux quantization in a doubly-relative form of twisted Cohomotopy, classified by the factorization of the quaternionic Hopf fibration through the twistor fibration. The further equivariant refinement of this cohomology theory reduces on A-type singularities to a form of relative 2-Cohomotopy which geometrically engineers Chern-insulator phases on $\mathrm{M5}\cap \mathrm{A}_n$, with the M-string playing the role of gapped nodal lines.

Flux Quantization on M-Strings

Abstract

The electric Gauss law in 11D SuGra is famously non-linear, whence its flux quantization must be in nonabelian cohomology. We have previously shown that the minimal admissible choice is 4-Cohomotopy, which in the presence of magnetized M5-probes takes its relative twistorial form. Here we discuss how this situation is further refined in the presence of M-string probes on the M5-worldvolume. Based on the superspace formulation of 11D SuGra, we find the nested Bianchi identities by iterating the superembedding construction for super p-branes. The resulting probe brane hierarchy (M1 on magnetized M5 in 11D bulk) turns out to admit flux quantization in a doubly-relative form of twisted Cohomotopy, classified by the factorization of the quaternionic Hopf fibration through the twistor fibration. The further equivariant refinement of this cohomology theory reduces on A-type singularities to a form of relative 2-Cohomotopy which geometrically engineers Chern-insulator phases on , with the M-string playing the role of gapped nodal lines.
Paper Structure (14 sections, 71 equations, 2 figures)

This paper contains 14 sections, 71 equations, 2 figures.

Table of Contents

  1. Introduction
  2. M-String Superembedding into M5
  3. The Spin Geometry
  4. The Iterated Embedding
  5. The Torsion Constraint
  6. Proper Flux Quantization
  7. Flux Quantization
  8. Relative Flux Quantization
  9. Super Flux Quantization
  10. M-String Flux Quantization
  11. The Law
  12. Orbifolding
  13. A-Singularities
  14. Conclusion

Figures (2)

  • Figure 1: The quaternionic Hopf fibration$h_{\mathbb{H}}$ and its factorization through the twistor fibration$t_{\mathbb{C}}$ is given by sending $\mathbb{R}_+$-lines in quaternionic 2-space $\mathbb{H}^2$ first to the $\mathbb{C}$-lines which they span, and then further to the $\mathbb{H}$ lines which these span, using the canonical inclusions $\mathbb{R}_+ \subset \mathbb{C} \subset \mathbb{H}$ of the positive real numbers into the complex numbers and further into the quaternions (cf. FSS22-TwistorialSS26-Orb).
  • Figure 2: Overview of the system of brane embeddings, of classifying fibrations and of charge classifying maps between these, for the M-brane on magnetized M5-branes in 11D Sugra, properly flux-quantized according to \ref{['FluxQuantizingOnMStringsInMagnetizedM5TheM3Flux']}. Solid maps are given (brane embeddings and classifying fibrations) while dashed maps are dynamical data (charge sectors of fields).