Higher-Dimensional Anyons via Higher Cohomotopy

TL;DR

The paper proves that higher-dimensional analogues of abelian FQH anyons arise from flux quantization in Cohomotopy, formalized via the fundamental groups of mapping spaces. Using H-group theory and Samelson/Whitehead products, it shows that for $k\in\{1,2,4\}$ the non-torsion part of $\\pi_1 Map^*_0((S^{2k-1})^2,S^{2k})$ is the integer Heisenberg group at level 2, with explicit torsion factors in higher degrees. In particular, $\\pi_1 Map_n((S^1)^2,S^2) \cong \mathrm{Heis}_3(\\mathbb{Z},2n)$ and $\\pi_1 Map_n((S^3)^2,S^4) \cong \mathrm{Heis}_3(\\mathbb{Z},0) \times\\mathbb{Z}_{/12}$, $\\pi_1 Map_n((S^7)^2,S^8) \cong \mathrm{Heis}_3(\\mathbb{Z},0) \times\\mathbb{Z}_{/120}$. These results extend to general CW complexes and propose a unifying framework (Hypothesis H) linking 2- and 4-cohomotopy flux quantization to observable anyon braiding, with potential implications for FQH/FQAH physics and flux quantization in 11D supergravity. The work suggests a rigorous algebro-topological route to understanding higher-dimensional anyons and their realizations in quantum materials and M-theory backgrounds.

Abstract

We highlight that integer Heisenberg groups at level 2 underlie topological quantum phenomena: their group algebras coincide with the algebras of quantum observables of abelian anyons in fractional quantum Hall (FQH) systems on closed surfaces. Decades ago, these groups were shown to arise as the fundamental groups of the space of maps from the surface to the 2-sphere -- which has recently been understood as reflecting an effective FQH flux quantization in 2-Cohomotopy. Here we streamline and generalize this theorem using the homotopy theory of H-groups, showing that for $k \in \{1,2,4\}$, the non-torsion part of $π_1 \mathrm{Map}\big({(S^{2k-1})^2, S^{2k}}\big)$ is an integer Heisenberg group of level 2, where we identify this level with 2 divided by the Hopf invariant of the generator of $π_{4k-1}(S^{2k})$. This result implies the existence of higher-dimensional analogs of FQH anyons in the cohomotopical completion of 11D supergravity ("Hypothesis H").

Figures (2)

• Figure 1: The anyons of fractional quantum Hall systems are vortices in the 2D electon gas induced by surplus magnetic flux quanta on top of an exact rational filling fraction of $K$ flux quanta per electron. Under each braiding of their worldlines the quantum state $\vert \psi \rangle$ transforms by multiplication with a braiding phase$\zeta = \exp(\pi \mathrm{i}/K)$.
• Figure 2: The adiabatic tuning of classical parameters $p$ along paths $\gamma$ in parameter space induces unitary transformations $U_\gamma$ between corresponding Hilbert spaces $\mathcal{H}$ of gapped ground states. For topological states these transformations depend only on the homotopy class of $\gamma$, exhibiting a local system or flat bundle of Hilbert spaces over the parameter space. These are equivalently linear representations of the fundamental groups of parameter loops $\ell$ at each base point, reflecting the topological order of the system in any topological phase.

Theorems & Definitions (42)

• Definition 2.1: Ordinary Heisenberg group
• Definition 2.2: Integer Heisenberg group at level $\ell=2$
• Remark 2.3: Other levels
• Definition 2.4: cf. Whitehead1978
• Definition 2.5: cf. FHT2000
• Proposition 2.6: cf. Whitehead1978
• Lemma 2.7: cf. Whitehead1978
• Proposition 2.8
• proof
• Remark 2.9