Engineering of Anyons on M5-Probes via Flux Quantization

TL;DR

The work develops a non-Lagrangian, flux-quantized M5-brane framework for realizing anyonic topological order by embedding M5-branes on Seifert orbifolds and enforcing twistorial Cohomotopy flux quantization (Hypothesis $H$). Topological observables arise as Pontrjagin algebras from mapping spaces into $S^2$, recovering abelian Chern-Simons physics and braid actions while naturally predicting defect anyons and their braid statistics. The Cohomotopy charge of solitons is tied to cobordism classes of normally framed submanifolds (Pontrjagin-Thom), with the soliton moduli described by loop spaces of maps to $S^2$, leading to ground-state degeneracies $|k|^g$ on genus-$g$ surfaces and wreath-product symmetries when defects are present. These results suggest a deep algebraic-topological foundation for topological quantum gates, offering new experimental pathways and a broader conceptual bridge between algebraic topology and quantum hardware design.

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.