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arXiv:2605.10232

Bulk-Edge Correspondence via Higher Gauge Theory

Hisham Sati, Urs Schreiber

hep-thcond-mat.str-elmath-phmath.ATquant-phSubmitted May 11, 2026• Announced May 12, 2026

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.

Comments: 47 pages, 7 figures

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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.

After global completion of higher gauge fields (as appearing in higher-dimensional supergravity) by proper flux quantization in extraordinary nonabelian cohomology, the (non-perturbative, renormalized) topological quantum observables and quantum states of solitonic field histories are completely determined through a topological form of light-front quantization. We survey the logic of this construction and expand on aspects of the quantization argument. In the instructive example of 5D Maxwell-Chern-Simons theory (the gauge sector of 5D SuGra) dimensionally reduced to 3D, a suitable choice of flux quantization in Cohomotopy ("Hypothesis h") recovers this way the fine detail of the traditionally renormalized (Wilson loop) quantum observables of abelian Chern-Simons theory and makes novel predictions about anyons in fractional quantum (anomalous) Hall systems. An analogous choice ("Hypothesis H") of global completion of 11D higher Maxwell-Chern-Simons theory (the higher gauge sector of 11D SuGra) realizes various aspects of the topological sector of the conjectural "M-theory" and its M5-branes.

Topological phases of matter in (2+1) dimensions are commonly equipped with global symmetries, such as electric-magnetic duality in gauge theories and bilayer symmetry in fractional quantum Hall states. Gauging these symmetries into local dynamical ones is one way of obtaining exotic phases from conventional systems. We study this using the bulk-boundary correspondence and applying the orbifold construction to the (1+1)-dimensional edge described by a conformal field theory (CFT). Our procedure puts twisted boundary conditions into the partition function and predicts the fusion, spin, and braiding behavior of anyonic excitations after gauging. We demonstrate this for the electric-magnetic self-dual

{\mathbb{Z}}_{N}

gauge theory, the twofold symmetric

\text{SU}{(3)}_{1}

, and the

{S}_{3}

-symmetric

\text{SO}{(8)}_{1}

Wess-Zumino-Witten theories.

We develop a theory of edge excitations of fractonic systems in two dimensions, and elucidate their connections to bulk transport properties and quantum statistics of bulk excitations. The system we consider has immobile point charges, dipoles constrained to move only along lines perpendicular to their moment, and freely mobile quadrupoles and higher multipoles, realizing a bulk fractonic analog of fractional quantum Hall phases. We demonstrate that a quantized braiding phase between two bulk excitations is obtained only in two cases: when a point quadrupole braids around an immobile point charge, or when two non-orthogonal point dipoles braid with one another. The presence of a boundary edge in the system entails two types of gapless edge excitation modes, one that is fractonic with immobile charges and longitudinal dipoles, and a second non-fractonic mode consisting of transverse dipoles. We derive a novel current algebra of the fractonic edge modes. Further, investigating the effect of local edge-to-edge tunneling on these modes, we find that such a process is a relevant perturbation suggesting the possibility of edge deformation.

Collective excitations of Fractional Quantum Hall (FQH) liquids at long wavelengths are thought to be of a generally covariant geometric nature, governed by area-preserving diffeomorphisms (

\mathrm{SDiff}

). But current analyses rely solely on the corresponding perturbative

w_\infty

Lie algebra. We argue this is insufficient: We identify a non-perturbative construction of the effective Maxwell-Chern-Simons quantum field theory which carries unitary

\mathrm{SDiff}

equivariance. But this turns out to be non-differentiable, suggesting underappreciated subtleties when the usual Hilbert space truncation is removed.

We carry out numerical diagonalization for much larger systems than before by restricting the fractional quantum Hall (FQH) edge excitations to a basis that is exact for a short-range interaction and very accurate for the Coulomb interaction. This enables us to perform substantial tests of the predicted universality of the edge physics. Our results suggest the possibility that the behavior of the FQH edge is intrinsically nonuniversal, even in the absence of edge reconstruction, and therefore may not bear a sharp and unique relation to the nature of the bulk FQH state.