Schreiber Engineering of Anyons on M5-Probes

An article that we are finalizing at CQTS:

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

  
  

  Engineering of Anyons on M5-Probes

  via Flux-Quantization

  
  

  lecture notes

  part II of: Introduction to Hypothesis H

  
  

  download:

  • article-like formatting:

    • pdf (v2)

    • arXiv:2501.17927

  • slide-like formatting:

    • pdf

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Abstract: These extended lecture notes survey a novel derivation of anyonic topological order (as seen in fractional quantum Hall systems) on single magnetized M5-branes probing Seifert orbi-singularities (“geometric engineering” of anyons), which we motivate from fundamental open problems in the field of quantum computing.

The rigorous construction is non-Lagrangian and non-perturbative, based on previously neglected global completion of the M5-brane‘s tensor field by flux-quantization consistent with its non-linear self-duality and its twisting by the bulk C-field: This exists only in little-studied non-abelian generalized cohomology theories, notably in a twisted equivariant (and “twistorial”) form of unstable Cohomotopy (“Hypothesis H”).

As a result, topological quantum observables form Pontrjagin homology algebras of mapping spaces from the orbi-fixed worldvolume into a classifying 2-sphere. Remarkably, theorems of algebraic topology imply from this the quantum observables and modular functor of abelian Chern-Simons theory, as well as braid group actions on defect anyons of the kind envisioned as hardware for topologically protected quantum gates.

Readers are expected to be acquainted with the topic of part I of the lecture notes:

• Flux Quantization

Presented at:

• 45th Srní Winter School on Geometry & Physics

  18-25 Jan 2025, Srní, Czechia

Based on:

• Flux Quantization on M5-Branes

• Abelian Anyons on Flux-Quantized M5-Branes

• Anyons on M5-Probes of Seifert 3-Orbifolds

Part of:

• Introduction to Hypothesis H

Related presentations:

• Homotopy Theory for Topological Quantum Computing