In this paper, a new concept has been introduced related to the construction of inequivalent quantizations for a system, which is called “history-like path” for particles. This is a collection of non-causal homotopical paths. Each homotopy class of paths will be labelled with a “word” constructed from the generators of fundamental group 1(QN(i)) of a configuration space of the system in a space-like i which is a deformation retract of Mi region that is a sub-space-time of M in which there is no singular slice (a slice Mc in M that contains a singular point). The labels determine the generators and their relations in constructing the fundamental group 1(QN) of the system.
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∈{1,2,4}, the non-torsion part of
π1Map((S2k−1)2,S2k) is an integer Heisenberg group of level 2, where we identify this level with 2 divided by the Hopf invariant of the generator of
π4k−1(S2k). This result implies the existence of higher-dimensional analogs of FQH anyons in the cohomotopical completion of 11D supergravity ("Hypothesis H").