Higher-Dimensional Anyons via Higher Cohomotopy

TL;DR

This paper connects higher-dimensional topological quantum phenomena to advanced homotopy theory, revealing new structures called higher-dimensional anyons through the study of cohomotopy and homotopy groups.

Contribution

It generalizes the understanding of integer Heisenberg groups in topological quantum systems using homotopy theory, establishing higher-dimensional analogs of anyons.

Findings

Non-torsion part of certain homotopy groups forms an integer Heisenberg group

Identifies the level of these groups with Hopf invariants

Suggests existence of higher-dimensional FQH anyons in 11D supergravity

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 $\pi_1 \mathrm{Map}\big({(S^{2k-1})^2, S^{2k}}\big)$ is an integer Heisenberg group of level 2, where we identify this level with 2 divided by the Hopf invariant of the generator of $\pi_{4k-1}(S^{2k})$. This result implies the existence of higher-dimensional analogs of FQH anyons in the cohomotopical completion of 11D supergravity ("Hypothesis H").