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arxiv: 2501.17927 · v3 · submitted 2025-01-29 · ✦ hep-th · cond-mat.str-el· math-ph· math.AT· math.MP· quant-ph

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Engineering of Anyons on M5-Probes via Flux Quantization

Hisham Sati , Urs Schreiber
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classification ✦ hep-th cond-mat.str-elmath-phmath.ATmath.MPquant-ph
keywords quantumanyonsengineeringfieldformobservablestopologicalabelian
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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.

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