Bulk-Edge Correspondence via Higher Gauge Theory

TL;DR

This paper recasts the bulk-edge correspondence in fractional quantum Hall systems using higher gauge theory, revealing geometric and topological structures related to M-branes and flux quantization.

Contribution

It introduces a higher gauge theory framework for FQH systems, linking bulk-edge effects to complex fibrations and M-brane geometric engineering.

Findings

Identifies the complex Hopf fibration as classifying bulk/boundary topological effects.

Reconstructs chiral edge currents via non-Lagrangian higher gauge theory.

Connects FQH physics to M-branes probing A-type orbi-singularities in 11D supergravity.

Abstract

More profound than bulk topological order of quantum materials is only its unwinding via gapless excitations along boundaries of the sample. We recast this bulk-edge correspondence -- for the experimentally relevant case of fractional quantum Hall (FQH) systems -- in terms of effective relative higher gauge theory, controlled by choices of classifying fibrations. For FQH systems, we identify the complex Hopf fibration as classifying the bulk/boundary topological effects, and find that it yields a non-Lagrangian reconstruction of Floreanini-Jackiw/Wess-Zumino-Witten chiral edge currents. Remarkably, the resulting effective FQH higher gauge theory turns out to be "geometrically engineered" on M2/M5-branes probing A-type orbi-singularities in 11D supergravity, globally completed by flux-quantization in twisted equivariant differential (TED) Cohomotopy: Here the M-string ends of M2-branes on M5-branes engineer the FQH liquid's boundary. This geometric engineering on M-branes might naturally elucidate the curious combination of $W_\infty$-symmetry and of super-symmetry that is known to govern the collective excitations of FQH liquids at long wavelengths.