Engineering of Anyons on M5-Probes via Flux Quantization

TL;DR

This paper presents a novel, non-perturbative geometric approach to engineering anyons on M5-branes, linking advanced topology with topological quantum computing applications.

Contribution

It introduces a new non-Lagrangian framework using twisted equivariant Cohomotopy to derive anyonic topological order on M5-branes.

Findings

Constructs topological quantum observables as Pontrjagin homology algebras.

Connects algebraic topology results to Chern-Simons theory and braid group actions.

Provides a geometric foundation for topologically protected quantum gates.

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.