Engineering of Anyons on M5-Probes via Flux Quantization

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, results from 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.

48 pages with many figures, extended lecture notes based on parts of the course "Introduction to Hypothesis H" held at 45th Srni Winter School GEOMETRY AND PHYSICS in Jan 2025; v2: added §6 "Digest for Algebraic Topologists" as presented at ICMS25; v3: published version with small corrections in the very last proof

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Cited by in corpus (2)

• Identifying Anyonic Topological Order in Fractional Quantum Anomalous Hall Systems
• Flux Quantization on M-Strings