# Topological Exchange Statistics in One Dimension

N.L. Harshman  American University, Washington DC, USA  A.C. Knapp  University
of Florida, Gainesville, FL, USA

###### Abstract

The standard topological approach to indistinguishable particles formulates
exchange statistics by using the fundamental group to analyze the
connectedness of the configuration space. Although successful in two and more
dimensions, this approach gives only trivial or near trivial exchange
statistics in one dimension because two-body coincidences are excluded from
configuration space. Instead, we include these path-ambiguous singular points
and consider configuration space as an orbifold. This orbifold topological
approach allows unified analysis of exchange statistics in any dimension and
predicts novel possibilities for anyons in one-dimensional systems, including
non-abelian anyons obeying alternate strand groups. These results clarify the
non-topological origin of fractional statistics in one-dimensional anyon
models.

##  I Introduction

Although all fundamental particles and most composite particles either satisfy
Bose-Einstein (BE) or Fermi-Dirac (FD) statistics, recent measurements provide
the strongest evidence yet that the quasiparticle excitations in the
fractional quantum Hall effect (FQHE) obey _fractional exchange statistics_
[1, 2]. Wilczek coined the term anyons for particles that obey fractional
exchange statistics [3] and these recent experiments demonstrate that the
excitations in the FQHE characterized by a $\nu=1/3$ filling factor exhibit a
$\theta=\pi/3$ exchange phase, satisfying a prediction made nearly forty years
ago [4, 5, 6].

The origin of fractional exchange phases in the FQHE can be traced to a
topological peculiarity: for two-dimensional systems with two-body
coincidences excluded, the configuration space is not simply-connected [7].
Not all paths in configuration space that exchange particles are equivalent,
and the equivalence classes of possible exchange paths are described by the
braid group. Abelian representations of the braid group are characterized by
an arbitrary exchange phase $\theta\in[0,2\pi)$ that determines the fractional
exchange statistics parameter $\theta$ [8, 9, 10, 11]. Because of their
topological origin, the fractional exchange statistics of these anyons can be
‘transmuted’ into a gauge interaction, i.e. the charged flux-tube model [12,
11]. Quasiparticle excitations obeying non-abelian braid group statistics may
also exist in the FQHE [13, 14, 15], but unambiguous confirmation remains
experimentally elusive. Adiabatic exchanges of non-abelian anyons could
provide an implementation for fault-tolerant quantum computing [16, 17]. This
application, combined with a fundamental interest in understanding topological
states of matter, continues to drive interest in engineering novel physical
systems that support excitations with non-standard exchange statistics [15,
18, 19].

Can the topological approach to exchange statistics in two-dimensional systems
be applied to one-dimensional systems? There is debate in the literature.
Unlike in two or more dimensions, particles in one-dimensional systems must
pass through each other to exchange positions and even short range (or zero-
range) interactions have dramatic dynamic and thermodynamic effects [20].
Given the inevitable intermingling of interactions with exchange in one
dimension, can a purely topological exchange phase be separated from the
dynamical phase accumulated along the exchange trajectory [21]? Further, for
indistinguishable particles, what does “pass through each other” even mean?
The two-body coincidence is a singular point in configuration space that
introduces ambiguity [22], so how can one distinguish trajectories in which
two particles reflect from those in which they transmit?

The standard formulation of topological exchange statistics is not sufficient
as it excludes two-body coincidences from configuration space. In one
dimension, the removal of two-body coincidences makes particle exchanges
impossible, and so the standard formulation of topological exchange statistics
allows only trivial representations [23, 15]. To overcome this technical
limitation, we extend the standard formulation of topological exchange
statistics by treating the configuration space of indistinguishable particles
as an _orbifold_ [22, 24, 25], a generalization of the idea of a manifold [26,
27]. Informally, an orbifold is locally equivalent to the linear quotient of a
Euclidean space by a finite group and ‘remembers’ that symmetry. This
extension allows trajectories that reflect and transmit at two-body
coincidences to be topologically distinguished by elements of the _orbifold
fundamental group_ , even for indistinguishable particles.

We present details about orbifolds in Sect. II, but a simple example of an
orbifold is the configuration space for two indistinguishable particles on a
line, equal to a plane modulo a reflection, ${\mathbb{R}}^{2}/S_{2}$. This
quotient identifies indistinguishable configurations $(x_{1},x_{2})$ and
$(x_{2},x_{1})$ in ${\mathbb{R}}^{2}$ and has singular locus $x_{1}=x_{2}$. In
${\mathbb{R}}^{2}$, singular points like $(x,x)$ have a Euclidean
neighborhood, but how do they look in the quotient space? One could collapse
all identified points to obtain the underlying space, the half-plane
$|{\mathbb{R}}^{2}/S_{2}|$, as a manifold with boundary [7]. Alternatively,
one could consider ${\mathbb{R}}^{2}/S_{2}$ as an orbifold, a two-dimensional
space with an internal edge of orbifold singularities on the reflection line.
To elucidate the difference, consider the perspective of an ant sitting on the
singular locus of ${\mathbb{R}}^{2}/S_{2}$. In the half-plane perspective,
there is an “edge of the universe” which no path can cross. The half-plane is
simply-connected and there are no topological exchange statistics. However, in
the orbifold perspective, the ant sees itself as sitting in the middle of an
infinite plane where the view just happens to be symmetric. Paths that cross
(transmit) and do not cross (reflect) this internal edge can be distinguished
by the local observer. The orbifold fundamental group in this case is the
symmetric group $S_{2}$, giving the possibility for both bosonic and fermionic
topological exchange statistics.

We first introduced the orbifold extension to the topological approach to
exchange statistics in [28]. This analysis was motivated by the observation
that, like two-body interactions in two dimensions, three-body interactions in
one dimension cause co-dimension two defects in configuration space.
Engineering such interactions could be feasible in ultracold atomic gases in
optical traps and optical lattices [29, 30, 31, 32]. The orbifold fundamental
group arising from hard-core three-body interactions in one dimension is
similar to the braid group: it is an infinite discrete group that descends
from the symmetric group by breaking one of the generating relations. Further,
it is realized by strand diagrams obeying certain crossing rules. We named it
the _traid_ group by analogy, but mathematicians also had discovered it in
other contexts and given it various other names [33], including the doodle
group [34, 35], the planar braid group [36], and the twin group [37, 38].

We extend these results in Sect. III and classify all possible topological
exchange statistics for distinguishable and indistinguishable particles in one
dimension. We consider the only two topological types of path-connected one-
dimensional base manifolds, the interval-type (which includes the infinite
interval of the real line) and circle, and we analyze all possible
topologically non-trivial interactions. This includes hard-core two-body
interactions, three-body interactions, and one other non-trivial form: a non-
local four-body interaction for which pairwise coincidences are allowed, but
not pairs of pairwise coincidences. This interaction leads to another
descendent of the symmetric group that we call the _fraid_ group. It is
realized by strand diagrams with non-local relations between the generators.

Our results in Sect. IIIB demonstrate that on the twisted configuration space
appropriate for indistinguishable particles on a ring, the possibility exists
for non-abelian generalized parastatistics for soft-core particles. We also
extend the traid and fraid group (and their combination) to the ring geometry
and discuss their ‘pure’ forms that apply to distinguishable particles.

Although a topological definition for exchange statistics in one dimension did
not previously exist, there is a large literature of one-dimensional models
where the particles are ‘anyons’ and/or have ‘fractional statistics’.
Continuum models include (references are representative, not exhaustive)
111Our analysis does not consider the anyon-Hubbard model [110, 80, 49, 111,
86, 87, 112] because our topological methods do not extend to configuration
spaces with discrete topology, although we note that the continuum limit of
the anyon-Hubbard model has recently been derived [54].: (1) Leinaas-Myrheim
anyons [7, 40, 41]; (2) Calogero-Sutherland anyon models [21, 23, 42]; (3)
hard-core anyon models with $\delta$-type interactions and fractional exchange
statistics [43, 44, 45]; and (4) Kundu/Lieb-Liniger anyons with non hard-core
$\delta$-type interactions and fractional exchange statistics [46, 47, 48, 49,
50, 41, 51, 52, 53, 54, 55]. A full survey of these model exceeds the scope of
this article (for a brief review see Appendix A of Ref. [41]), but a unifying
property of these models is that they contain a parameter that interpolates
between bosonic and fermionic limits. In Sect. IV, we give a preliminary
analysis of these four classes of continuum models and conclude that any
braid-like exchange phases in models (2)-(4) originate from dynamics, i.e.,
they are non-topological in origin and cannot be absorbed into a statistical
gauge interaction.

Finally, in the concluding Sect. V we summarize our results and point out
several directions for future work. In particular, we highlight two sets of
open questions: the mathematical properties and physical consequences of traid
and fraid group anyons and the conceptual and mathematical shifts required to
formulate the quantum mechanics of indistinguishable particles on orbifolds.

##  II Orbifold approach to topological exchange statistics

This section presumes that the reader has some familiarity with the
topological approach to building a quantum theory on a configuration space
${\mathcal{X}}$. The key idea is that when a configuration space is not
simply-connected, then single-valued wave functions defined on the
configuration space may not exhaust the set of allowed states. The fundamental
group $\pi_{1}({\mathcal{X}})$ of the configuration space describes the
connectivity of the ${\mathcal{X}}$ by equivalence classes of loops. The
irreducible representations of $\pi_{1}({\mathcal{X}})$ provide a
classification of possible wave functions, including single-valued and multi-
valued as well as single-component (abelian) and multi-component (non-
abelian). Alternatively, one may work with only-single valued functions by
lifting the quantum system to the unique, simply-connected universal cover of
the configuration space $\widetilde{{\mathcal{X}}}$. For completeness, we
provide a brief overview of these standard results for manifolds
${\mathcal{X}}$ in Appendix A. We also provide a brief review of the main
approaches to particle statistics, including exchange statistics and exclusion
statistics, in Appendix B.

As first demonstrated by Leinaas and Myrheim [7], topological exchange
statistics beyond FD or BE are possible when the base manifold upon which the
particles move is not simply-connected or (in one and two dimensions) when
particle interactions create topological defects. They found the possibility
for novel statistics by defining the configuration space of $N$
indistinguishable particles on a base manifold ${\mathcal{M}}$ as the quotient
of the configuration space manifold of $N$ distinguishable particles
${\mathcal{X}}={\mathcal{M}}^{N}$ by the symmetric group $S_{N}$ of particle
permutations [56, 7]:

${\mathcal{Q}}={\mathcal{X}}/S_{N}.$ (1)

Sometimes called the _intrinsic_ approach to indistinguishability [22], taking
the quotient removes physically-meaningless particle labels from the
mathematical description at the start. Although the taking the quotient
precludes making passive permutations of particle identity, active particle
exchanges are realized by loops in ${\mathcal{Q}}$.

Taking the quotient also introduces a natural orbifold structure to
${\mathcal{Q}}$. Orbifolds have singular points where non-trivial local
symmetries have topological consequences. Traditionally, these singular points
have either been removed from configuration space or trivialized. Instead, we
include these points and describe the particle exchanges of indistinguishable
particles using the orbifold fundamental group $\pi_{1}^{*}({\mathcal{Q}})$.

The generalization to $\pi^{*}_{1}$ captures the topological impact of
singular points with co-dimension $\tilde{d}=1$ or $\tilde{d}=2$ on parallel
transport. Loci of singular points with $\tilde{d}=1$ are _internal mirrors_
and occur at two-body coincidences in one-dimensional systems. Loci of
singular points with $\tilde{d}=2$ are either isolated singularities, called
_cone points_ , which occur in the case of particles in two dimensions, or
_corners_ formed by intersections of $\tilde{d}=1$ intersections of internal
mirrors and occur in the case of particles in one dimension.

Using the orbifold fundamental group, we find the relation

$\pi_{1}^{*}({\mathcal{Q}})=S_{N}({\mathcal{M}}).$ (2)

holds for any path-connected base manifold ${\mathcal{M}}$. Here
$S_{N}({\mathcal{M}})$ is the generalized symmetric group [57]:

$S_{N}({\mathcal{M}})\equiv\pi_{1}({\mathcal{M}})^{N}\rtimes
S_{N}\equiv\pi_{1}({\mathcal{M}})\wr S_{N},$ (3)

where $\wr$ denotes the wreath product, a semidirect product in which $S_{N}$
acts on the normal subgroup $\pi_{1}({\mathcal{M}})^{N}$ by permutation of
factors [58, 59]. For a simply-connected base manifold, $S_{N}({\mathcal{M}})$
reduces to the symmetric group $S_{N}$ as expected.

The result (2) was previously derived for $d=\dim{\mathcal{M}}\geq 3$ in Ref.
[57] where exchange statistics given by $S_{N}({\mathcal{M}})$ are called
generalized parastatistics. Extending the result (2) to particles on base
manifolds ${\mathcal{M}}$ with dimensions $d=1$ and $d=2$ requires the
orbifold approach that we develop over the rest of this section. The
classification of topological exchange statistics in one dimension presented
in Sect. III can be understood without these details on orbifolds and orbifold
fundamental groups.

###  II.1 Distinguishable particle

To understand the orbifold structure of the configuration space for
indistinguishable particles, first consider _distinguishable_ particles moving
on a path-connected manifold ${\mathcal{M}}$ with $\dim{{\mathcal{M}}}=d$. The
configuration space is

${\mathcal{X}}={\mathcal{M}}^{N}\equiv\overbrace{{\mathcal{M}}\times\cdots\times{\mathcal{M}}}^{N\
\mathrm{times}}.$ (4)

The fundamental group of ${\mathcal{X}}$

$\pi_{1}({\mathcal{X}})=\pi_{1}({\mathcal{M}})^{N}$ (5)

is the $N$-fold direct product of the fundamental group of ${\mathcal{M}}$.
The universal cover $\widetilde{{\mathcal{X}}}$ of ${\mathcal{X}}$ factorizes
similarly $\tilde{{\mathcal{X}}}=(\widetilde{{\mathcal{M}}})^{N}$.

Removing particle coincidences may disrupt this product structure (5) for base
manifolds ${\mathcal{M}}$ with dimension $d=1$ or $d=2$. To see this, note
that every point in ${\mathcal{X}}$ can be classified by its pattern of
coinciding coordinates. For _generic_ points
$x\equiv\\{x_{1},\ldots,x_{N}\\}\in{\mathcal{X}}$, all $N$ coordinates are
different. In contrast, the set of all points that have at least one pair of
coordinates the same is called the coincidence locus
$\Delta_{2}\subset{\mathcal{X}}$, or sometimes the ‘fat diagonal’ of
${\mathcal{X}}$ [60]. Similarly, one can define $\Delta_{3}\subset\Delta_{2}$
as the locus where at least three particle coordinates are the same,
$\Delta_{2,2}\subset\Delta_{2}$ as the locus where there are at least two
pairs of coinciding particles, etc.

More generally, the space ${\mathcal{X}}$ is stratified by integer partitions
of $N$. Each integer partition $[n_{1}\ldots n_{k}]\in P_{N}$ is a collection
of positive integers which sum to $N$, typically written in non-increasing
order. For example, for $N=5$, the configuration space
${\mathcal{X}}={\mathcal{M}}^{5}$ is stratified into seven partitions,
including the partition $[221]$ corresponding to points with two $2$-particle
coincidences and the partition $[41]$ for points with one $4$-particle
coincidence.

For each partition $[\nu]=[n_{1}\ldots n_{k}]$, we define the stratum
${\mathcal{X}}_{[\nu]}\subset{\mathcal{X}}$ of points with that partition
type. The stratum ${\mathcal{X}}_{[\nu]}$ has $h_{[\nu]}=N!/(n_{1}!\cdots
n_{k}!)$ path-connected components depending on which coordinates are equal.
The closure of each stratum $\overline{{\mathcal{X}}_{[\nu]}}$ is defined by
$\sum_{j=1}^{k}(n_{j}-1)$ equalities of $d$-dimensional variables giving us
that ${\mathcal{X}}_{[\nu]}$ is co-dimension
$\tilde{d}=\sum_{j=1}^{k}(n_{j}-1)d$ within ${\mathcal{X}}$.

The fat diagonal $\Delta_{2}=\overline{{\mathcal{X}}_{[21\ldots 1]}}$ is the
union of all ${\mathcal{X}}_{[\nu]}$ except the generic points in
${\mathcal{X}}_{[1\ldots 1]}$ and its top-dimensional stratum has co-dimension
$\tilde{d}=d$. Therefore, for base manifolds ${\mathcal{M}}$ with $d=1$ or
$d=2$, removing $\Delta_{2}$ from ${\mathcal{X}}$ disrupts the connectivity
and the fundamental group is no longer given by (5). For $d=1$ base manifolds
${\mathcal{M}}$, the coincidence loci $\Delta_{3}$ (the closure of
${\mathcal{X}}_{[31\ldots 1]}$) and $\Delta_{2,2}$ (the closure of
${\mathcal{X}}_{[221\ldots 1]}$) have co-dimensions $\tilde{d}=2d=2$, as we
discuss in detail Sect. III.

As a final note, when the particles are distinguishable but identical, then
particle permutations are a symmetry of the Hamiltonian. There is a
representation $O:S_{N}\to\operatorname{Diff}({\mathcal{X}})$ denoted
$s\mapsto O_{s}\in O(S_{N})$ that acts as a _passive_ coordinate
transformation, exchanging factors in the product (4). Generic points
$x\in{\mathcal{X}}_{[1\ldots 1]}$ have orbits under permutation
$\left\\{O_{s}(x)\left|s\in S_{N}\right.\right\\}$ with $N!$ members. For all
other partitions $[\nu]$, each point $x\in{\mathcal{X}}_{[\nu]}$ has a non-
trivial stabilizer subgroup $H_{x}\subset S_{N}$, i.e., the subgroup that
exchanges coinciding coordinates. Such points have orbits with only
$N!/\left|H_{x}\right|$ members. The particular embedding
$S_{[\nu]}=S_{n_{1}}\times\cdots\times S_{n_{k}}$ into $S_{N}$ depends on
which of the $h_{[\nu]}$ components of ${\mathcal{X}}_{[\nu]}$ contains $x$,
and two points of different components will will have conjugate stabilizers.

###  II.2 Intrinsic approach

In the intrinsic approach, physics happens on the indistinguishable particle
space ${\mathcal{Q}}$ rather than ${\mathcal{X}}$. The quotient
${\mathcal{Q}}={\mathcal{X}}/S_{N}$ defines a ‘forgetful map’
$p:{\mathcal{X}}\to{\mathcal{Q}}$ that erases the particles’ identities and
sends each point of ${\mathcal{X}}$ to its $S_{N}$ orbit. For generic points
in ${\mathcal{X}}_{[1\ldots 1]}$, the map $p:{\mathcal{X}}\to\mathcal{Q}$ is
$N!:1$, but points in the coincidence locus $\Delta_{2}$ are _singular_ under
this map.

Reversing this, each possible choice of coordinate labels is equivalent to a
‘lift’ of the point $q_{0}\in{\mathcal{Q}}$ to one of its representative
points $x_{0}\in{\mathcal{X}}$ in the orbit. Since particle labels are
meaningless for indistinguishable particles, any assignment of labels to
coordinates of $q_{0}$ is a choice of gauge. The diffeomorphisms $O_{s}$ of
${\mathcal{X}}$ permute the representatives of $q_{0}$ and therefore the
particle labels, so they form a discrete gauge transformation group. From
another perspective, these label-permuting symmetries $O_{s}$ form a
generalization of the deck transformations of a covering space in the orbifold
category [61].

In contrast to the passive transformations $O_{s}$ of ${\mathcal{X}}$, closed
loops in ${\mathcal{Q}}$ realize active particle exchanges along continuous
paths. However, paths that pass through the singular points
$\Delta_{2}/S_{N}\subset\mathcal{Q}$ give an ambiguity. Did the particles
exchange at the coincidence point or not? Especially in one dimension, the
ambiguity between reflection and transmission is essential to understanding
exchange statistics. However, the fundamental group $\pi_{1}(\mathcal{Q})$
does not see the path ambiguity embodied in these singular points [22].

There are two standard solutions to the presence of these singular points that
avoid using orbifolds. First, one can consider the space

${\mathcal{Q}}_{2}={\mathcal{X}}_{2}/S_{N}={\mathcal{Q}}-\Delta_{2}/S_{N},$
(6)

where ${\mathcal{X}}_{2}={\mathcal{X}}-\Delta_{2}$. The configuration space
$\mathcal{Q}_{2}$ has no multi-body coincidences and without these points, the
restriction of $p$ that maps ${\mathcal{X}}_{2}\to{\mathcal{Q}}_{2}$ forms a
covering space in the usual sense. Therefore ${\mathcal{Q}}_{2}$ is a manifold
and exchange paths are described completely by $\pi_{1}({\mathcal{Q}}_{2})$.
Famously, for ${\mathcal{M}}={\mathbb{R}}^{2}$, $\pi_{1}({\mathcal{Q}}_{2})$
is the braid group $B_{N}$ and $\pi_{1}({\mathcal{X}}_{2})$ is the pure braid
group $PB_{N}$. On general $d=2$ surfaces ${\mathcal{M}}$, the fundamental
group $\pi_{1}({\mathcal{Q}}_{2})$ gives the generalized braid groups
$B_{N}({\mathcal{M}})$ and pure braid groups $PB_{N}({\mathcal{M}})$ [62, 63,
57, 64, 65]. For ${\mathcal{M}}={\mathbb{R}}^{3}$, all particle exchanges
which do not permute the particles are homotopically trivial and so
$\pi_{1}({\mathcal{Q}}_{2})=S_{N}$, resulting in normal exchange statistics.

Alternatively, one can consider the _underlying space_ $|\mathcal{Q}|$ of the
orbifold ${\mathcal{Q}}$, which includes set of singular points
$|\Delta_{2}/S_{N}|$ but ignores their orbifold structure [61]. The space
$|{\mathcal{Q}}|$ is a manifold with certain degeneracies along
$|\Delta_{2}/S_{N}|$ such as a boundary or corners. In this case, one uses the
usual fundamental group $\pi_{1}(|{\mathcal{Q}}|)$ to describe particle
exchanges. However, trivializing the topology like this removes the
possibility for non-trivial exchange statistics. Returning to the case of
$\mathcal{M}=\mathbb{R}^{2}$, we find $\pi_{1}(|\mathcal{Q}|)=1$. This implies
the unsatisfactory result that only trivial topological exchange statistics
(i.e., only bosons, not even fermions) would be possible for particles moving
on the plane unless interactions exclude $\Delta_{2}/S_{N}$ [7, 8].

###  II.3 Configuration space orbifold

Instead of either of these approaches, we include the singular points
$\Delta_{2}/S_{N}$ in the configuration space and consider $\mathcal{Q}$ as an
orbifold [26, 61]. Orbifolds occur naturally in the context of classifying
spaces of objects with symmetries [27]. The space ${\mathcal{Q}}$ meets the
definition of a ‘good’ orbifold because it is the quotient of a manifold by a
discrete group acting upon it by diffeomorphisms [61]. Further,
${\mathcal{Q}}$ is considered ‘very good’ as it actually is the global
quotient of a manifold by a finite group.

Points in an orbifold are classified as manifold points if their local
symmetry group is trivial and orbifold points if their local symmetry group is
non-trivial. Depending on their co-dimension, orbifold points form loci that
appear as internal mirrors, cone points or corners, and other higher order
singular points in the orbifold. Manifolds form a subclass of orbifolds where
all points have trivial local symmetry groups.

For the orbifold ${\mathcal{Q}}$, the orbifold singular points are precisely
the singular points $\Delta_{2}/S_{N}$ of $p:{\mathcal{X}}\to{\mathcal{Q}}$.
In the planar example, the orbifold locus of $\mathbb{R}^{2}/S_{2}$ has co-
dimension $\tilde{d}=1$, forming an internal mirror edge at the line
$\Delta_{2}/S_{2}$ of two-body coincidences. The local symmetry group of
points on $\Delta_{2}/S_{2}=p({\mathcal{X}}_{[2]})$ is isomorphic to $S_{2}$;
all other points on the half plane $p({\mathcal{X}}_{[11]})$ are manifold
points.

More generally, the local symmetry group of an orbifold point in
${\mathcal{Q}}$ comes from its location in the stratification of
${\mathcal{X}}$ by partitions, ${\mathcal{X}}_{[\nu]}$. Each point of
${\mathcal{X}}_{[\nu]}$ is invariant under a subgroup of $S_{N}$ isomorphic to
$S_{[\nu]}=S_{n_{1}}\times\cdots\times S_{n_{k}}$. The stratification
${\mathcal{X}}_{[\nu]}$ is invariant under the $S_{N}$ action so the
decomposition descends to a similar stratification
$p({\mathcal{X}}_{[\nu]})={\mathcal{Q}}_{[\nu]}\in{\mathcal{Q}}$ where the
stabilizers $S_{[\nu]}$ of points $x\in{\mathcal{X}}_{[\nu]}$ become the local
symmetry groups of the corresponding point $q=p(x)\in{\mathcal{Q}}_{[\nu]}$.

To understand how local symmetries act, first consider the case of a manifold
point $q\in{\mathcal{Q}}_{[1\ldots 1]}$. For a sufficiently small neighborhood
$U$ of $q$ in $|{\mathcal{Q}}|$ the preimage $p^{-1}(U)$ consists of $N!$
disjoint sets in ${\mathcal{X}}$ all isomorphic to $U$ via $p$. Each connected
component $V$ of $p^{-1}(U)$ is a local model for ${\mathcal{Q}}$ at $q$ and
corresponds to an assignment of $N$ labels to the $N$ distinct points of
${\mathcal{M}}$ defining $q$, or equivalently, an unambiguous ordering of
those points in the product
${\mathcal{X}}={\mathcal{M}}\times\cdots{\mathcal{M}}$. In other words,
$p:{\mathcal{X}}_{[1\ldots 1]}\to{\mathcal{Q}}_{[1\ldots 1]}$ is a topological
covering map and the local symmetry group of the point $q\in{\mathcal{Q}}$ is
trivial.

However, an orbifold point $q\in{\mathcal{Q}}_{[\nu]}$, $\nu\neq[1\ldots 1]$,
lifts to only $h_{[\nu]}=N!/|S_{[\nu]}|$ points in ${\mathcal{X}}$. Therefore,
if $U$ is a small neighborhood of $q$ in $|{\mathcal{Q}}|$, then $p^{-1}(U)$
will have $h_{[\nu]}$ connected components. Although each connected component
$V$ of $p^{-1}(U)$ serves as a local model for the orbifold ${\mathcal{Q}}$ at
$q$, they differ from the neighborhoods of manifold points in that they are
not isomorphic to $U$. Instead, each component $V$ of $p^{-1}(U)$ comes with
an action of $S_{[\nu]}$ so that $U=V/S_{[\nu]}$. The realization of the
$S_{[\nu]}$ symmetries by particle-label permutations (i.e. the embedding
$S_{[\nu]}\subset S_{N}$) depends non-trivially on the specific component $V$.
Each of the $V$ do not correspond to a canonical choice of particle labels,
but instead corresponds to a choice of particle labels up to an $S_{[\nu]}$
ambiguity.

For example, take $N=3$ and consider points $q\in{\mathcal{Q}}_{[21]}$. These
points $q$ with the form $\\{a,a,b\\}$ have preimages $(a,a,b)$, $(a,b,a)$,
and $(b,a,a)$ which have neighborhoods where the label-involution is realized
by the permutations $(1,2)$, $(1,3)$, and $(2,3)$, respectively. These are
exactly the generators of the stabilizer subgroups of the respective points in
${\mathcal{X}}_{[21]}$ and the resulting copies of $S_{2}\subset S_{3}$ are
all conjugate.

###  II.4 Orbifold fundamental group

The _orbifold fundamental group_ $\pi^{*}_{1}$ classifies based loops in an
orbifold up to continuous deformation, generalizing the usual fundamental
group [26, 61, 28]. Intuitively, an orbifold path $\gamma:I\to{\mathcal{Q}}$
consists of a map $|\gamma|:I\to|{\mathcal{Q}}|$ of the underlying spaces
together with compatible lifts to local models on the orbifold. For example,
if ${\mathcal{Q}}=\mathbb{R}^{2}/S_{2}$, there are two types of paths which
touch the singular boundary: one which lifts to the transmitted path in the
local model and the other which lifts to the reflected path. These paths
realize, and are distinguished by, the two elements of
$\pi_{1}^{*}(\mathbb{R}^{2}/S_{2})=S_{2}$. As with $\pi_{1}$, choice of base
point does not affect the isomorphism type of $\pi_{1}^{*}$ for path-connected
spaces.

Note that for $d\geq 3$ the orbifold singular points have large enough co-
dimension $\tilde{d}\geq 3$ that their presence or absence has no effect on
the connectivity or the (orbifold) fundamental groups of ${\mathcal{X}}$,
${\mathcal{Q}}$, and $|{\mathcal{Q}}|$. Therefore, each of these (orbifold)
fundamental groups are canonically isomorphic with $S_{N}(\mathcal{M})$.
However, for $d=1$ and $d=2$, the orbifold fundamental group ‘feels’ the
disruption to connectedness created by the local symmetry groups of orbifold
loci. Like the fundamental group, the orbifold fundamental group keeps track
of how paths intersect with $\tilde{d}=1$ orbifold loci and wind around
$\tilde{d}=2$ loci.

Using the orbifold fundamental group, the same relation
$\pi_{1}^{*}({\mathcal{Q}})=S_{N}({\mathcal{M}})$ (2) holds for any dimension.
When $\pi_{1}({\mathcal{M}})$ is trivial, then $S_{N}({\mathcal{M}})$ reduces
to $S_{N}$ and the universal cover of ${\mathcal{Q}}$ is the simply-connected
space $\widetilde{{\mathcal{Q}}}={\mathcal{X}}$ and topological exchange
statistics reproduces the results of the symmetrization postulate in any
dimension (see Appendix B). For non-trivial $\pi_{1}({\mathcal{M}})$, the
universal covers
$\widetilde{{\mathcal{Q}}}=\widetilde{{\mathcal{X}}}=\widetilde{{\mathcal{M}}}^{N}$
coincide. The topology of the base manifold allows multiple exchange paths
supporting the same particle permutation to be distinguished. Equivalently,
there are paths that do not exchange particles that are not homotopically
trivial.

These results can be summarized in a short exact sequence, a linear sequence
of groups connected by homomorphisms such that the image of one homomorphism
is the kernel of the next, and ‘short’ in the sense that there are five group
terms beginning and ending with the trivial group. The following short exact
sequence holds for any path-connected base manifold ${\mathcal{M}}$ of any
dimension and relates the fundamental group of ${\mathcal{X}}$ to the orbifold
fundamental group of ${\mathcal{Q}}$:

$1\to\pi_{1}({\mathcal{X}})\to\pi_{1}^{*}({\mathcal{Q}})\to S_{N}\to 1,$ (7a)
which specifies to $1\to\pi_{1}(\mathcal{M})^{N}\to S_{N}(\mathcal{M})\to
S_{N}\to 1.$ (7b)

The proof of (2) and (7) in Ref. [57] holds when $d\geq 3$, but fails to
generalize to the orbifold case. However, we have an alternate proof that
employs a distinct, but isomorphic definition of the orbifold fundamental
group. In this definition, like the usual fundamental group, the orbifold
fundamental group is defined to be the deck transformation group of the
universal cover [26]. Then the short exact sequence can be extracted from a
series of covers $\widetilde{{\mathcal{X}}}\to{\mathcal{X}}\to{\mathcal{Q}}$.
Here, the covering $\widetilde{{\mathcal{X}}}\to{\mathcal{X}}$ is the usual
universal covering of the manifold ${\mathcal{X}}$, which has deck
transformation group isomorphic to $\pi_{1}({\mathcal{X}})$ and the
composition $\widetilde{{\mathcal{X}}}\to{\mathcal{Q}}$ is the universal cover
of the orbifold ${\mathcal{Q}}$. Since ${\mathcal{X}}\to{\mathcal{Q}}$ comes
from a group quotient, the action of the deck transformation group is
transitive, and thus the cover is regular. This means that
$\pi_{1}({\mathcal{X}})$ includes into $\pi_{1}^{*}({\mathcal{Q}})$ as a
normal subgroup with quotient equal to the deck transformation group $S_{N}$
of the cover ${\mathcal{X}}\to{\mathcal{Q}}$, giving the short exact sequence
(7).

However, we should warn the reader that we have done something sneaky here.
When passing to the deck transformation interpretation of the the orbifold
fundamental group, we have changed the object of study! That is, for each
cover, there are two groups at work here. One is the group of passive
transformations, acting as diffeomorphisms of the covering space, forming a
discrete gauge group. In the case of ${\mathcal{X}}\to{\mathcal{Q}}$, it is
the group that permutes choices of particle labels and in the case of
$\widetilde{{\mathcal{Q}}}=\widetilde{{\mathcal{X}}}\to{\mathcal{Q}}$, it
additionally intertwines how the constituent particles have wound around
${\mathcal{M}}$. The other group is the “point-pushing” group of orbifold-
homotopy loops in the configuration space that realizes active transformations
of indistinguishable particles. These groups are isomorphic and that allows
the proof of (2) and (7), but the methods of action on
$\widetilde{{\mathcal{Q}}}$ are distinct and commute. In Sect. III.1 below, we
contrast these actions for the simplest case of indistinguishable particles on
a one-dimensional interval.

##  III Application to one dimension

The orbifold fundamental group (2) of ${\mathcal{Q}}$ describes how two
factors determine the connectedness of configuration space:
indistinguishability and the fundamental group of the base manifold. In three
dimensions and higher, those are the only two factors that contribute to
topological exchange statistics. However, hard-core or singular interactions
exclude points from configuration space, and in $d=1$ and $d=2$, excluding
points of coincidence alters $\pi_{0}$ (the set of path-connected components)
and $\pi_{1}$ of configuration space because this locus has co-dimension
$\tilde{d}\leq 2$.

A co-dimension $\tilde{d}=1$ defect, such as the set of two-body coincidences
$\Delta_{2}$ in one dimension, locally splits configuration space into
connected components. Recall that $\Delta_{2}$ is formed as the union of sets
defined by equations of the form $x_{i}=x_{j}$, so for a generic point, the
local splitting is into two pieces in the same manner as a point on a line, a
line in a plane, or a plane in a three-dimensional space. When the
intersections of all two-body coincidences $\Delta_{2}$ are removed to form
${\mathcal{X}}_{2}={\mathcal{X}}-\Delta_{2}$, particle exchanges through
coincidences become impossible and the particles can be given a consistent
order on each element of $\pi_{0}({\mathcal{X}}_{2})$.

The removal of codimension $\tilde{d}=2$ defects, which sit in configuration
space like a point in a plane or a line in a space, also disrupts the
connectivity of configuration space. Paths that wind around the defects lead
to new, non-trivial elements of $\pi_{1}$ that serve as (often non-abelian)
winding numbers. Such defects occur as the result of the following few-body
coincidences in ${\mathcal{X}}$ [28, 66]:

  1. 1.

two-body coincidences $\Delta_{2}$ defined by $x_{j}=x_{k}$ and $y_{j}=y_{k}$
in two dimensions,

  2. 2.

three-body coincidences $\Delta_{3}$ defined by $x_{i}=x_{j}=x_{k}$ in one
dimension, and

  3. 3.

a non-local, partial four body coincidences $\Delta_{2,2}$ formed by the
intersection of two two-body coincidences $x_{i}=x_{j}$ and $x_{k}=x_{l}$.

The first case famously leads to the braid group and generalization (see Sect.
II.2). We call (orbifold) fundamental groups that derive from the exclusion of
these coincidence loci _strand groups_ because, like the braid group, they
lead to configuration space described by generalizations of the symmetric
group that can be realized by strand diagrams. However, we note that for the
case of $\Delta_{2,2}$, this notion of strand group involves non-local
constraints, as we discuss below.

Therefore, for $d=1$ base manifolds $\mathcal{M}$, we define the additional
configuration spaces

$\displaystyle{\mathcal{X}}_{3}=\mathcal{M}^{N}-\Delta_{3},{\mathcal{Q}}_{3}={\mathcal{X}}_{3}/S_{N}$
(8)
$\displaystyle{\mathcal{X}}_{2,2}=\mathcal{M}^{N}-\Delta_{2,2},{\mathcal{Q}}_{2,2}={\mathcal{X}}_{2,2}/S_{N}$
$\displaystyle{\mathcal{X}}_{\\{3;2,2\\}}=\mathcal{M}^{N}-\Delta_{3}\cup\Delta_{2,2},{\mathcal{Q}}_{\\{3;2,2\\}}={\mathcal{X}}_{\\{3;2,2\\}}/S_{N}$

where the spaces ${\mathcal{X}}_{3},{\mathcal{Q}}_{3}$ excluding three-body
coincidence hold interest for $N\geq 3$ and spaces excluding double two-body
coincidences are relevant for $N\geq 4$. Because these ${\mathcal{Q}}$-spaces
include single two-body coincidence orbifold locus $\Delta_{2}/S_{N}$, we
consider them as orbifolds and use $\pi_{1}^{*}$.

There are only two homotopy types for manifolds in one dimension: the interval
type with $\pi_{1}(\mathcal{M})=1$ (a type that includes the infinite interval
$\mathbb{R}$) and the circle $S^{1}$ with $\pi_{1}(\mathcal{M})=\mathbb{Z}$.
We classify the possible topological exchange statistics for both manifold
types below for indistinguishable and distinguishable particles with and
without the relevant coincident loci removed.

###  III.1 Particles on interval-type manifolds

When ${\mathcal{M}}$ is of the simply connected interval type, the
configuration space for distinguishable particles ${\mathcal{X}}$ is also
path-connected and simply-connected and there are no non-trivial exchange
statistics. For a finite interval, ${\mathcal{X}}$ is an $N$-dimensional
hypercube, but without loss of generality we extend to the infinite interval
and consider ${\mathcal{M}}={\mathbb{R}}$ and
${\mathcal{X}}={\mathbb{R}}^{N}$. If the particles are distinguishable but
identical, then particle permutations are a symmetry of the Hamiltonian. A
permutation $s\in S_{N}$ is represented by an orthogonal transformation
$O_{s}\in O(S_{N})\subset O(N)$ on ${\mathcal{X}}={\mathbb{R}}^{N}$; see Ref.
[67, 59] for a pedagogical introduction. The symmetrization postulate uses the
representations $O(S_{N})$ to decompose the Hilbert space of wave functions on
${\mathcal{X}}$ into symmetric and antisymmetric subspaces (see Appendix B).

The intrinsic approach to indistinguishable particles, also gives the
symmetric group $\pi_{1}^{*}({\mathcal{Q}})=S_{N}$, as expected, but with a
different interpretation in terms of exchange loops. The orbifold fundamental
group is generated by $N-1$ pairwise exchanges $\sigma_{1}$ through
$\sigma_{N-1}$ satisfying the relations

$\displaystyle\sigma_{i}^{2}$ $\displaystyle=$ $\displaystyle 1$ (9a)
$\displaystyle\sigma_{i}\sigma_{i+1}\sigma_{i}$ $\displaystyle=$
$\displaystyle\sigma_{i+1}\sigma_{i}\sigma_{i+1}$ (9b)
$\displaystyle\sigma_{i}\sigma_{j}$ $\displaystyle=$
$\displaystyle\sigma_{j}\sigma_{i}\ \mbox{for}\ j>i+1.$ (9c)

Recall that ${\mathcal{Q}}$ is the space of indistinguishable particles, so
the generators $\sigma_{i}\in\pi_{1}^{*}({\mathcal{Q}})$ are not the discrete
permutations of labeled particles. Instead, they are continuous paths that
actively exchange particle _orderings_ , where each configuration of points
has a natural particle order determined by the position of the particles in
${\mathbb{R}}$. The interpretation of $\sigma_{1}$ is that it realizes a loop
in ${\mathcal{Q}}$ that exchanges the first and second particle; $\sigma_{2}$
exchanges the second and third particles, etc. These exchanges are realized as
strand diagrams in Fig. 1.

Figure 1: Depiction of the three types of generator relations (9) for
$\pi_{1}^{*}({\mathcal{Q}})$ when $N=4$ as strand diagrams, read from the
bottom: (a) the self-inverse relation $\sigma_{1}^{2}=1$ that is broken for
the braid group generators; (b) the Yang-Baxter relation
$\sigma_{1}\sigma_{2}\sigma_{1}=\sigma_{2}\sigma_{1}\sigma_{2}$ that is broken
for the traid group generators; (c) the locality relation
$\sigma_{1}\sigma_{3}=\sigma_{3}\sigma_{1}$ that is broken for the fraid group
generators.

We want to emphasize the difference between these two appearances of the
symmetric group: (1) the active, continuous particle exchanges
$\pi_{1}^{*}({\mathcal{Q}})\sim S_{N}$ represented as closed loops on
${\mathcal{Q}}$; and (2) the group of passive particle label permutations
$O(S_{N})\sim S_{N}$ represented as orthogonal transformations on
${\mathcal{X}}$. We compare the action of these groups on the points of
${\mathcal{X}}_{2}={\mathcal{X}}-\Delta_{2}$ by restricting the action of
$O(S_{N})$ from ${\mathcal{X}}$ to ${\mathcal{X}}_{2}$ and lifting the action
of $\pi_{1}^{*}({\mathcal{Q}})$ from ${\mathcal{Q}}$ to ${\mathcal{X}}_{2}$.
See Fig. 2 for a depiction of the these different actions for the case of
$N=3$.

Figure 2: Graphical comparison of the actions of $O(S_{N})\sim S_{N}$ and
$\pi^{*}_{1}({\mathcal{Q}})\sim S_{N}$ on ${\mathcal{X}}_{2}$ for $N=3$. (a)
The relative configuration space for three distinguishable particles on a line
with coordinates $(x_{1},x_{2},x_{3})$. The horizontal coordinate is
$z_{1}=(x_{1}-x_{2})/\sqrt{2}$ and the vertical coordinate is
$z_{2}=(x_{1}+x_{2}-x_{3})/\sqrt{6}$. The black lines are the two-body
coincidence locus $\Delta_{2}$ and they section ${\mathcal{X}}$ into six
alcoves ${\mathcal{Y}}_{\omega}\in\pi_{0}({\mathcal{X}}_{2})$ where the
particles have different position orders
$x_{\omega_{1}}<x_{\omega_{2}}<x_{\omega_{3}}$. (b) Reflections across each of
the three colored lines are orthogonal transformations $O_{s}$ of
${\mathcal{X}}$ that represent the passive permutation of particle labels. For
example, $O_{(12)}$ is a reflection across the vertical line (red) that
permutes the labels of particle 1 and particle 2 and $O_{(23)}$ is a
reflection across the diagonal line with positive slope (green) that permutes
the labels of particles 2 and 3. The double-sided arrows (red) connect points
$x_{\omega}\in p^{-1}(q)$ and indicate how the
${\mathcal{Y}}_{\omega}\in\pi_{0}({\mathcal{X}}_{2})$ are permuted by
$O_{(12)}$. (c) The relative components of the orbifold ${\mathcal{Q}}$ and a
based loop $\sigma_{1}=\pi^{*}_{1}({\mathcal{Q}})$ starting and ending at
manifold point $q$. This path realizes an exchange $\sigma_{1}$ of the first
two particles. (d) The based loop $\sigma_{1}$ in ${\mathcal{Q}}$ is lifted to
six paths in ${\mathcal{X}}_{2}$ that start at $x_{\omega}$ and end at
$x_{\omega^{\prime}}$. For visual clarity, half of the path lifts are
represented as dashed lines. These six lifts of element
$\sigma_{1}=\pi^{*}_{1}({\mathcal{Q}})$ define a map $\tilde{\sigma}_{1}$ on
${\mathcal{X}}_{2}$ that permutes the
${\mathcal{Y}}_{\omega}\in\pi_{0}({\mathcal{X}}_{2})$.

####  III.1.1 Singular two-body interactions

For particles in one dimension with hard-core two-body interactions, the two-
body coincidence locus $\Delta_{2}$ divides ${\mathcal{X}}$ into $N!$ simply-
connected alcoves ${\mathcal{Y}}_{\omega}\in\pi_{0}({\mathcal{X}}_{2})$. Each
alcove ${\mathcal{Y}}_{\omega}$ is an open subset of ${\mathcal{X}}$ such that
the quotient map $p({\mathcal{Y}}_{\omega})$ consists entirely of manifold
points ${\mathcal{Q}}_{2}\subset{\mathcal{Q}}$. Each simply-connected
component $\mathcal{Y}_{\omega}$ has the geometry of an open cone on an open
simplex on the $N-1$ sphere [68]. Conversely, each manifold point $q$ in
${\mathcal{Q}}_{2}$ lifts to $N!$ points $p^{-1}(q)=\left\\{x_{\omega}\mid
x_{\omega}\in{\mathcal{Y}}_{\omega},p(x_{\omega})=q\right\\}$. Each alcove
index $\omega=[\omega_{1}\omega_{2}\ldots\omega_{N}]$ is a permutation of the
set $\\{12\ldots N\\}$ that indicates the positional ordering of labeled
particles in ${\mathbb{R}}$; see Fig. 2. When ${\mathcal{M}}$ is of interval
type, sorting in this manner gives a one-to-one correspondence between alcove
orderings ${\mathcal{Y}}_{\omega}$ and permutations in $S_{N}$. However, as we
explore in the next section, when ${\mathcal{M}}=S^{1}$ we can only order
particles cyclically and so this correspondence develops a similar cyclic
ambiguity.

For indistinguishable particles, $\mathcal{Q}_{2}$ is isomorphic to
$\mathcal{Y}_{\omega}$ and the fundamental group
$\pi_{1}({\mathcal{Q}}_{2})=1$ is trivial. This provides a topological
perspective on the so-called fermionization of hard-core bosons [69, 70].
Because no exchanges are possible, there are no topological exchange
statistics that differentiate fermions and bosons from the intrinsic
perspective. As we discuss in Sect. IV below, non-topological exchange
statistics that interpolate between bosonic and fermionic solutions (sometimes
called Leinaas-Myrheim anyons) have been defined by imposing Robin boundary
conditions on $\Delta_{2}/S_{N}$ (equivalent to delta-interactions on
$\Delta_{2}\subset{\mathcal{X}}$) [7, 41, 71].

####  III.1.2 Singular few-body interactions

In contrast to ${\mathcal{Q}}_{2}$, removing the $\tilde{d}=2$ few-body
coincidences (8) gives non-trivial tolopogical exchange statistics. The
following strand groups are defined as:

$\displaystyle\pi^{*}_{1}(\mathcal{Q}_{3})$ $\displaystyle=$ $\displaystyle
T_{N}$ (10a) $\displaystyle\pi^{*}_{1}(\mathcal{Q}_{2,2})$ $\displaystyle=$
$\displaystyle F_{N}$ (10b)
$\displaystyle\pi^{*}_{1}(\mathcal{Q}_{\\{3;2,2\\}})$ $\displaystyle=$
$\displaystyle W_{N}$ (10c)

We call these discrete, infinite, non-abelian groups the traid group $T_{N}$
[28] (aka doodle group, planar braid group, twin group), the fraid group
$F_{N}$, and the free Coxeter group $W_{N}$ (aka universal Coxeter group
[72]).

Like the braid group $B_{N}$, the strand groups $T_{N}$, $F_{N}$ and $W_{N}$
can be understood as resulting from eliminating generator relations of the
symmetric group [28]. Relaxing relation (9a), the generators $\sigma_{i}$ give
the braid group $B_{N}$. Relaxing (9b) gives the traid group $T_{N}$ and
relaxing (9c) gives the fraid group $F_{N}$. Relaxing both (9b) and (9c) means
that only the self-inverse relations remain and the resulting group is the
free Coxeter group $W_{N}$. Each of these groups admit natural homomorphisms
to $S_{N}$ obtained by reintroducing the lost relations.

These relations have a topological basis. When $d\geq 2$, each $\sigma_{i}$
can be represented by a path that avoids the coincidence locus $\Delta_{2}$
and the loci $\Delta_{3}$ and $\Delta_{2,2}$ are codimension $2d\geq 4$ and
their removal does not affect the fundamental group as any null-homotopy of a
loop can be arranged to avoid theses sets. However for $d=2$, the relation
$\sigma_{i}^{2}=e$ comes from a null-homotopy of a path which must pass
through the codimension $2$ locus $\Delta_{2}$, so the presence of
$\Delta_{2}$ determines the presence of the relation (9a). In dimension $d=1$,
paths representing $\sigma_{i}$ cannot be represented disjointly from
$\Delta_{2}$. However, the loci $\Delta_{3}$ and $\Delta_{2,2}$ are co-
dimension $2d=2$ and the relations (9b) and (9c) are induced by null-
homotopies that must pass through these sets, respectively. For example, any
null-homotopy of the natural representative of $(\sigma_{i}\sigma_{i+1})^{3}$
must pass through have a strand diagram which has at least on one triple
point. In other words, the null-homotopy passes through $\Delta_{3}$.

The pure version of each of these groups in (10) is defined by looking at the
corresponding distinguishable particle configuration space. For example, we
can define the pure traid as $PT_{N}=\pi_{1}({\mathcal{X}}_{3})$. Following
the argument of Sect. II.4, we construct the following short exact sequences
analogous to (7):

$\displaystyle 1\to PT_{N}\to T_{N}\to S_{N}\to 1$ (11a) $\displaystyle 1\to
PF_{N}\to F_{N}\to S_{N}\to 1$ (11b) $\displaystyle 1\to PW_{N}\to W_{N}\to
S_{N}\to 1$ . (11c)

Some mathematical results for the pure traid group $PT_{N}$ can be found in
[37, 38, 73, 74].

The possible topological exchange statistics for these novel strand groups are
classified by the irreducible representations of these groups. The irreducible
representations of $T_{N}$ and $PT_{N}$ have not received much attention from
mathematicians, although the abelian representations of $T_{N}$ are classified
and several non-abelian representations found in Ref. [28]. However, there is
still not a complete classification of the non-abelian irreducible
representations of the braid group 75 years after the group was first
described [75], so one should expect classifying the non-abelian
representations of the traid and fraid groups will be a similarly complicated
and long-standing mathematical project.

###  III.2 Particles on rings

For $\mathcal{M}=S^{1}$, the distinguishable particle configuration space
${\mathcal{X}}=T^{N}$ is the $N$-torus $T^{N}=S^{1}\times\cdots\times S^{1}$
with fundamental group $\pi_{1}(T^{N})={\mathbb{Z}}^{N}$. This abelian group
of $N$-tuples of integers under addition describes equivalences classes of
paths by how many times each particle winds around the ring. It is generated
by the $N$ translations $t_{i}$ of a single particle around the circle. All
irreducible representations of ${\mathbb{Z}}^{N}$ are classified by $N$-tuples
of phases $(\phi_{1},\cdots,\phi_{N})$ with $\phi_{i}\in[0,2\pi)$. Only for
all $\phi_{i}=0$ are the wave functions on ${\mathcal{X}}$ single-valued. For
identical but distinguishable particles, all phases take the same value
$\phi_{i}=\phi$.

For indistinguishable particles, the orbifold fundamental group allows
generalized parastatistics given by

$\pi^{*}_{1}(\mathcal{Q})=S_{N}(S^{1})={\mathbb{Z}}\wr
S_{N}=S_{N}\ltimes{\mathbb{Z}}^{N}.$ (12)

This group is non-abelian even for $N=2$ and in addition to multi-valued,
scalar representations [10], there are multi-valued, multi-component wave
functions on the orbifold $\mathcal{Q}$ that realize states with generalized
parastatistics. In principle, a complete classification of the irreducible
representations of (12) can be found using the method of induced
representations of the normal subgroup ${\mathbb{Z}}^{N}$ [76].

Figure 3: Depictions of six different elements of
$S_{4}(S^{1})={\mathbb{Z}}\wr S_{4}={\mathbb{Z}}^{4}\rtimes S_{4}$ mentioned
in the text. In these strand diagrams, the ring has been cut flattened into a
line and the gray dashed lines on either side are the cut. The elements
depicted include: $\sigma_{4}$, the pairwise exchange of the first and fourth
particle (relative to the cut) constructed from $S_{N}$ generators as in first
line of (13); $t_{1}$, one of the $4$ generators of ${\mathbb{Z}}^{4}$
subgroup of $S_{4}(S^{1})$; its inverse $t_{1}^{-1}$; the ‘around-the-back’
operator $\sigma_{0}$ defined in (13b) and simplified using
$\sigma_{i}^{2}=1$; the $\zeta$ operators defined in (13c) corresponding to a
shift in the center-of-mass by $\pi/2$ around the ring; and $\zeta^{-1}$ its
inverse.

####  III.2.1 Twisted subgroup

The group $\pi^{*}_{1}(\mathcal{Q})=S_{N}(S^{1})$ contains the affine
symmetric group $\tilde{S}_{N}$ [72] (also called the twisted symmetric group
[77]) as a normal subgroup. To show this, denote the $N-1$ generators of
$S_{N}$ by $\sigma_{i}$ as in (9) and the $N$ generators of ${\mathbb{Z}}^{N}$
by $t_{i}$. The semidirect action of $S_{N}$ on ${\mathbb{Z}}^{N}$ in the
wreath product (12) implies the generator relations
$t_{i}\sigma_{i}=\sigma_{i}t_{i+1}$ and $t_{i}\sigma_{j}=\sigma_{j}t_{i}$ for
$j\neq i$ and $j\neq i+1$. In terms of these generators, define the following
three elements:

$\displaystyle\sigma_{N}$ $\displaystyle=$
$\displaystyle\sigma_{1}\sigma_{2}\ldots\sigma_{N-2}\sigma_{N-1}\sigma_{N-2}\ldots\sigma_{2}\sigma_{1}$
$\displaystyle=$
$\displaystyle\sigma_{N-1}\sigma_{N-2}\ldots\sigma_{2}\sigma_{1}\sigma_{2}\ldots\sigma_{N-2}\sigma_{N-1}$
$\displaystyle\sigma_{0}$ $\displaystyle=$ $\displaystyle
t_{1}\sigma_{N}t_{1}^{-1}=t_{1}t_{N}^{-1}\sigma_{N}$ (13b)
$\displaystyle\zeta$ $\displaystyle=$ $\displaystyle
t_{1}\sigma_{1}\sigma_{2}\ldots\sigma_{N-1}=\sigma_{1}\ldots\sigma_{N-1}t_{N}.$
(13c)

These elements are depicted as strand diagrams in Fig. 3. The element
$\sigma_{N}$ is the pairwise exchange of the ‘first’ and ‘last’ particles
(with respect to some starting angle on the ring) in which they pass through
all the particles between them. The element $\sigma_{0}$ is the pairwise
exchange of the first and last particle ‘around the back’, i.e., without
crossing the other particles. With the addition of $\sigma_{0}$, the set of
elements $\\{\sigma_{0},\sigma_{1},\ldots,\sigma_{N-1}\\}$ satisfies the
defining relations for generators of the affine symmetric group
$\tilde{S}_{N}$ [77]. This establishes that $\tilde{S}_{N}\subset
S_{N}(S^{1})$.

To show $\tilde{S}_{N}$ is a normal subgroup, consider the element $\zeta$. It
shifts all particles one place in the order and generates a subgroup
${\mathbb{Z}}_{\zeta}\subset S_{N}(S^{1})$ isomorphic to the integers. It also
satisfies the relation

$\zeta\sigma_{i}=\sigma_{i+1}\zeta$ (14)

and in particular $\zeta\sigma_{N-1}=\sigma_{0}\zeta$. The element
$\zeta\notin\tilde{S}_{N}$ acts as an outer automorphism on the generators of
$\tilde{S}_{N}$ (and therefore on all of $\tilde{S}_{N}$) establishing that
$\tilde{S}_{N}$ is a normal subgroup of $S_{N}(S^{1})$. Therefore, the
orbifold fundamental group can be equivalently expressed as

$S_{N}(S^{1})={\mathbb{Z}}_{\zeta}\ltimes\tilde{S}_{N}$ (15)

where the semidirect product is specified by (14). Further note that
$\zeta^{N}=t_{1}t_{2}\cdots t_{N}$ realizes a displacement of all particles
one trip around the ring, i.e. a full displacement of the center-of-mass. One
can show that $\zeta^{N}$ commutes with all elements of $\tilde{S}_{N}$.

####  III.2.2 Singular two-body interactions

Now consider ${\mathcal{X}}_{2}$. The removal of $\Delta_{2}$ divides the
$N$-torus $T^{N}$ into only $(N-1)!$ sectors
$\mathcal{Y}_{\omega}\in\pi_{0}({\mathcal{X}}_{2})$ where $\omega$ labels a
cyclic order of the particles. There is one cyclic order for every coset
$S_{N}/C_{N}$ of the symmetric group by the cyclic group
$C_{N}\cong{\mathbb{Z}}/N\equiv{\mathbb{Z}}_{N}$. For $N>2$, hard-core two-
body interactions lock the particles into a particular cyclic order $\omega$
and ${\mathcal{X}}_{2}$ is not path-connected. Spaces with multiple path
components may, a priori, have non-isomorphic fundamental groups for each
component. However, within each ordering sector $\mathcal{Y}_{\omega}$, the
fundamental group $\pi_{1}(\mathcal{Y}_{\omega})$ is naturally isomorphic to
the integers. We denote this group as $N{\mathbb{Z}}_{\zeta}$ as it is
generated by a full cycle around the ring by particles $\zeta^{N}$. This “full
rotation” interpretation also gives a natural isomorphism from
$\pi_{1}(\mathcal{Y}_{\omega})$ to the fundamental group of the base space,
$\pi_{1}(S^{1})$.

As in the interval case, $p:\mathcal{X}_{2}\to\mathcal{Q}_{2}$ is not a
connected cover, but unlike the interval case, $p$ is not simply an
isomorphism from each $\mathcal{Y}_{\omega}$ to $\mathcal{Q}_{2}$. Rather, the
restriction of $p$ to $\mathcal{Y}_{\omega}$ is a connected $N:1$ cover. Each
point in $\mathcal{Q}_{2}$ corresponds to $N$ distinct points in
$\mathcal{Y}_{\omega}$ that differ by a cyclic rotation $c\in C_{N}$ acting as
a deck/gauge transformation of distinguishable particles. The fundamental
group $\pi_{1}(\mathcal{Q}_{2})={\mathbb{Z}}_{\zeta}$ generated by $\zeta$
naturally contains $\pi_{1}(\mathcal{Y}_{\omega})\sim\pi_{1}(S^{1})$ as its
$N{\mathbb{Z}}_{\zeta}$ subgroup generated by $\zeta^{N}$.

These results establish that the corresponding short exact sequence of abelian
groups for a connected component
${\mathcal{Y}}_{\omega}\subset{\mathcal{X}}_{2}$ and ${\mathcal{Q}}_{2}$ is

$\displaystyle
1\to\pi_{1}({\mathcal{Y}}_{\omega})\to\pi_{1}({\mathcal{Q}}_{2})\to C_{N}\to
1$ $\displaystyle 1\to
N{\mathbb{Z}}_{\zeta}\to{\mathbb{Z}}_{\zeta}\to{\mathbb{Z}}/N\to 1.$ (16)

Note that the symmetric group $S_{N}$ no longer appears in this sequence and
therefore there are no FD statistics or parastatistics possible. Instead the
abelian group $\pi_{1}({\mathcal{Q}}_{2})={\mathbb{Z}}_{\zeta}$ has one-
dimensional representations characterized by a single phase $\phi$.

The geometry of ${\mathcal{Q}}_{2}$ provides insights into these results for
$\pi_{1}({\mathcal{Q}})$ and $\pi_{1}({\mathcal{Q}}_{2})$. The manifold
${\mathcal{Q}}_{2}$ is a Möbius band for the case $N=2$ [7]; see Fig. 4. For
$N=3$, the space ${\mathcal{Q}}_{2}$ is a kind of Penrose triangle, a torus
with an equilateral triangle cross section that makes a $2\pi/3$ twist every
rotation; see Fig. 5. For higher dimensions, there is a generalized simplex
hyperprism with a twist, e.g. for $N=4$ the cross-section of the hyperprism is
a rhombic dispheniod. The ordering sector ${\mathcal{Y}}_{\omega}$ is an
$N$-fold cover of $\mathcal{Q}_{2}$ that is a also a simplex hyperprism, but
no longer twisted. This universal cover of ${\mathcal{Q}}_{2}$ is the product
of ${\mathbb{R}}$ and an $N$-simplex (non-regular for $N\geq 4$) and is useful
for solving the Schrödinger equation in certain polytopes [78, 79]. For
${\mathcal{Q}}$ the corresponding geometries are the same, but the orbifold
singularities at the boundaries are included.

Figure 4: Depiction of the universal cover
$\widetilde{{\mathcal{Q}}}=\widetilde{{\mathcal{X}}}$ and several useful
domains for $N=2$ and ${\mathcal{M}}=S^{1}$. The black dots are all lifts of
the same point in ${\mathcal{Q}}$ and ${\mathcal{X}}$, the black lines are the
lifts of the coincidence locus $\Delta_{2}$ to $\widetilde{{\mathcal{Q}}}$.
The red square (a) is one choice for the fundamental domain isomorphic to the
torus ${\mathcal{X}}=T^{2}$. Each congruent square would then be labeled by a
pair of integers in $\pi_{1}({\mathcal{X}})=Z^{2}$ describing the path to
fundamental domain. The green triangle (b) and purple square (c) are
equivalent alternate choices for the fundamental domain isomorphic to the
Möbius band ${\mathcal{Q}}$ or ${\mathcal{Q}}_{2}$. The blue rectangle (d) is
the double cover of ${\mathcal{Q}}$ or ${\mathcal{Q}}_{2}$. It is the smallest
torus $\overline{T}^{2}=S^{1}_{\mathrm{rel}}\times S^{1}_{\mathrm{com}}$ for
which there exists separable and single-valued center-of-mass and relative
coordinates for indistinguishable particles. Although it has the same area in
configuration space, it is not equivalent to the torus ${\mathcal{X}}=T^{2}$.
The yellow square (e) is the double cover of ${\mathcal{X}}$ that is the
smallest torus for which exist single-valued center-of-mass and relative
coordinates for distinguishable particles.

Figure 5: Two views of a depiction of the universal cover
$\widetilde{{\mathcal{Q}}}=\widetilde{{\mathcal{X}}}$ and several useful
domains for $N=3$ and ${\mathcal{M}}=S^{1}$. The black dots are all lifts of
the same point in ${\mathcal{Q}}$ and ${\mathcal{X}}$, the black planes in the
lower figure are the lifts of the coincidence locus $\Delta_{2}$ to
$\widetilde{{\mathcal{Q}}}$. The five highlighted domains are equivalent to
the five domains depicted in Fig. 4: the red cube is a fundamental domain
congruent to ${\mathcal{X}}=T^{3}$, the green 3-orthoscheme tetrahedron and
purple triangular prism are equivalent alternate choices for the fundamental
domain isomorphic to the Penrose triangle ${\mathcal{Q}}$ or
${\mathcal{Q}}_{2}$. The blue rhombic prism is the minimal separable torus for
indistinguishable particles $\overline{T}^{3}=T^{2}_{\mathrm{rel}}\times
S^{1}_{\mathrm{com}}$ and the yellow rhombic prism is the minimal separable
torus for distinguishable particles.

####  III.2.3 Singular few-body interactions

For co-dimension $\tilde{d}=2$ interactions, we define the following strand
groups

$\displaystyle\pi^{*}_{1}(\mathcal{Q}_{3})$ $\displaystyle=$ $\displaystyle
T_{N}(S^{1})$ (17a) $\displaystyle\pi^{*}_{1}(\mathcal{Q}_{2,2})$
$\displaystyle=$ $\displaystyle F_{N}(S^{1})$ (17b)
$\displaystyle\pi^{*}_{1}(\mathcal{Q}_{\\{3;2,2\\}})$ $\displaystyle=$
$\displaystyle W_{N}(S^{1})$ (17c)

Like the interval versions (10), the equivalent groups (17) result from
breaking the analogous generator relations and each of these groups admit maps
to $S_{N}(S^{1})$ from relation reintroduction.

The factorizations (12) and (15) of $S_{N}(S^{1})$ provides two alternate ways
to view the groups in (17). For the case of $\pi^{*}_{1}(\mathcal{Q}_{3})$,
the two factorizations are

$\displaystyle T_{N}(S^{1})$ $\displaystyle=$ $\displaystyle
T_{N}\ltimes{\mathbb{Z}}^{N}$ (18a) $\displaystyle=$
$\displaystyle{\mathbb{Z}}_{\zeta}\ltimes\tilde{T}_{N}.$ (18b)

In the first factorization, the normal subgroup ${\mathbb{Z}}^{N}$ of particle
translations around the ring is the same as (12), but $T_{N}$ replaces
$S_{N}$. The semidirect product in (18a) obeys the same relations between the
generators $\sigma_{i}$ and $t_{j}$ as (12). Similarly, in the second
factorization the ‘twisted’ traid group $\tilde{T}_{N}$ is defined from
$\tilde{S}_{N}$ by breaking the relation (9b) extended to the larger set of
generators $\\{\sigma_{0},\ldots,\sigma_{N-1}\\}$. The semidirect product is
inferred from the same generator relations as (15).

Equivalent constructions define the twisted fraid group $\tilde{F}_{N}$ and
the twisted universal Coxeter group $\tilde{W}_{N}$. In principle, these forms
of the factorization provide a way to construct and classify irreducible
representations of these novel strand groups on rings from the irreducible
representations of strand groups on intervals.

Finally, pure versions of (17) can be defined for the fundamental groups of
the manifolds ${\mathcal{X}}_{3}$, ${\mathcal{X}}_{2,2}$ and
$\mathcal{X}_{\\{3;2,2\\}}$ that satisfy short exact sequences like (11).

##  IV Topological analysis of models with one-dimensional anyons

For some physicists, an abelian anyon model must (by definition) have
fractional exchange statistics, i.e., a pairwise exchange of particles
transforms the wave function by a phase $\exp(i\theta)$ that interpolates
between bosons $\theta=0$ and fermions $\theta=\pi$. Therefore, much previous
work on anyons in one dimension either imposes fractional exchange statistics
on wave functions (or field operators) at the start [21, 43, 44] or derives
wave functions or field operators with fractional exchange phases from a
Hamiltonian [46, 80].

However, in the previous section, we calculated the orbifold fundamental group
(including the fundamental group in the case of ${\mathcal{Q}}_{2}$) for the
configuration space of indistinguishable particle for every possible
topologically disruptive interaction on both intervals and rings. For all
scenarios, pairwise exchanges were either absent (as in ${\mathcal{Q}}_{2}$)
or square-trivial (for ${\mathcal{Q}}$, ${\mathcal{Q}}_{3}$, and the rest). In
no one-dimensional case is the (orbifold) fundamental group the braid group or
any other group with abelian representations that furnish ‘traditional’
fractional exchange statistics.

The necessary conclusion is that if fractional exchange statistics occur in
models of particle systems in one dimension, they have a dynamical origin
(i.e., they derive from interactions) and not a topological origin. In other
words, unlike in two dimensions, fractional exchange statistics do not derive
from the configuration space of indistinguishable particles on one-dimensional
manifolds even accounting for excluded few-body coincidences. This in no way
diminishes their physical or mathematical interest, but it may alter their
interpretation and application.

To make the contrast clear, first consider the simplest case where
‘traditional’ fractional exchange statistics given by a phase $\theta$ occurs:
two particles in a plane with two-body coincidences excluded. The
configuration space ${\mathcal{Q}}_{2}=({\mathbb{R}}^{4}-\Delta_{2})/S_{2}$
can be factored into the product of a plane and cone with the point at the tip
excluded [7, 22]. This space is not simply-connected and its fundamental group
is $\pi_{1}({\mathcal{Q}}_{2})=B_{2}\cong{\mathbb{Z}}$. The group $B_{2}$
therefore has abelian representations characterized by $\theta\in[0,2\pi)$.
For $\theta\neq 0$, the wave function considered as a map
$\psi:{\mathcal{Q}}_{2}\to\mathbb{C}$ is multi-valued. Alternatively, one can
define single-valued wave-functions the universal cover
$\widetilde{{\mathcal{Q}}_{2}}$, which the the product of a plane and a half-
plane. On the universal cover $\widetilde{{\mathcal{Q}}_{2}}$ particle
exchanges in $\pi_{1}({\mathcal{Q}}_{2})$ are represented by translations. The
Hilbert space on $\widetilde{{\mathcal{Q}}_{2}}$ decomposes into abelian
representations of $\pi_{1}({\mathcal{Q}}_{2})$ labeled by the quasi-momentum
and $\theta$.

Besides working on ${\mathcal{Q}}_{2}$ or $\widetilde{{\mathcal{Q}}_{2}}$,
there is a third option: to work with single-valued functions on
${\mathcal{X}}_{2}$ and incorporate the $\theta$ exchange statistics as a
gauge interaction potential for either bosons or fermions [11]. This gauge
potential is singular at the particle coincidence, but because that point is
removed from configuration space ${\mathcal{Q}}_{2}$ this singularity presents
no difficulties. For $N=2$, the gauge potential is the vector potential of a
delta-function ‘flux tube’ at the two-body coincidence [12, 6]. This works for
abelian braid group anyons because topological exchange statistics derive from
flat connections on fiber bundles over configuration spaces. Therefore, they
can be absorbed into a gauge potential on a covering space, and a trivial
gauge on the universal cover [71]. As a result, free particles with fractional
exchange statistics can be modeled with bosons (or fermions) where the
statistics are ‘absorbed’ into a statistical gauge interaction.

This strategy of absorbing statistics into a gauge potential motivated Kundu
[46] to define a one-dimensional anyon model (also called the anyon Lieb-
Liniger model [41]). Starting from a bosonic model defined on ${\mathcal{X}}$
with singular interactions described by $\delta$, $\delta^{\prime}$ and
double-$\delta$ functions on the $\Delta_{2}$ and $\Delta_{3}$ coincidence
loci, Kundu performs a density-dependent gauge transformation 222Although
discrete models are outside the scope of this analysis, a similar technique
defines the anyon-Hubbard model [80]. Density-dependent interactions for
bosons are ‘transmuted’ by the gauge transformation into site-dependent phase
slips. See also [54] for the relation of the anyon Hubbard model to the
Kundu/Lieb-Liniger model.. The gauge-transformed model retains
$\delta$-interactions on the $\Delta_{2}$ locus on which the creation and
annihilation operators no longer satisfy bosonic relations. The wave function
that is constructed on ${\mathcal{X}}$ jumps by $\exp(\pm i\theta)$ across
$\Delta_{2}$. Similarly, Girardeau defined a model with similar ‘phase slips’
of $\exp(\pm i\theta)$ from a gauge transformation of a hard-core, fermion
model which does not require singular interactions [44].

The anyon Lieb-Liniger model has been shown to possess generalized _exclusion_
statistics [47]. However, the phase slips $\exp(\pm i\theta)$ on
${\mathcal{X}}$ in either the boson-based Lieb-Liniger model or fermion-based
Girardeau model are the result of a particle-label dependent gauge
transformation and do not have the same topological interpretation as
fractional _exchange_ statistics of indistinguishable particles. For example,
in the simplest case of two particles, unlike the braid anyon case described
above, there are not an infinite number of possible exchange phases
multiplying the entire wave function depending on a winding number. Instead,
there is a single phase difference between the two different particle
orderings of the same wave function defined on ${\mathcal{X}}$. The non-
topological interpretation of Kundu/Lieb-Liniger anyons presented here agrees
with several previous analyses that argue that fractional exchange statistics
cannot be absorbed into a gauge potential for one-dimensional systems [82,
53]. In contrast, because of their similar topological origin from excluded
co-dimension $\tilde{d}=2$ coincidences, we hypothesize a ‘transmutation’ from
statistics to dynamics is possible for particles obeying traid or fraid
exchange statistics, and is an avenue of future research.

Two other models sometimes identified as one-dimensional anyon models also
have trivial topological exchange statistics: Leinaas-Myrhaim anyons and
Calogero-Sutherland anyons. In the Leinaas-Myrheim analysis of one-dimensional
particles systems, the model is built on the exchange-trivial underlying space
$|{\mathcal{Q}}|$ [7]. To make the Hamiltonian self-adjoint on the domain
$|{\mathcal{Q}}|$, boundary conditions must be imposed for the wave function
on $\Delta_{2}$. These self-adjoint extensions are characterized by an ‘anyon-
like’ parameter that effectively interpolates between Neumann boundary
conditions with bosonic symmetry and Dirichlet boundary conditions with
fermionic antisymmetry. The Leinaas-Myrheim model can be lifted to the Lieb-
Liniger bosonic model on ${\mathcal{X}}$ with two-body delta-interactions,
except the wave functions and all observables are restricted to the underlying
space $|{\mathcal{Q}}|$ [41]. Like the Kundu model, here the interpolating
statistics are again from the dynamics and not the topology, which is trivial.
Further, we hypothesize that the Leinaas-Myrheim model on $|{\mathcal{Q}}|$
can also be lifted to ${\mathcal{X}}$ with fermionic antisymmetry or Kundu-
like phase shifts using statistical mapping techniques [83, 53, 55, 25, 84].

Similarly, in the Calogero-Sutherland model, interactions prevent two-body
coincidences and a parameter interpolates between BE and FD exclusion
statistics. For indistinguishable particles, the inverse square interaction is
sufficiently singular to exclude $\Delta_{2}$ and therefore the configuration
space is ${\mathcal{Q}}_{2}$. Lifting the model to ${\mathcal{X}}_{2}$, one is
free to define arbitrary phases to different orderings of particles and define
a Calogero-Sutherland _anyon_ model with fractional exclusion statistics and
order-dependent phase slips [21, 23, 42]. However, since there is no exchange
possible, such phases are a gauge symmetry that only has consequences for
observables defined on the (non-universal) covering space ${\mathcal{X}}_{2}$
and as such do not constitute a statistical gauge interaction.

##  V Conclusions

To summarize, for indistinguishable particles on a one-dimensional interval we
find the following possibilities for the group describing topological exchange
statistics:

$\displaystyle\pi_{1}^{*}({\mathcal{Q}})$ $\displaystyle=$ $\displaystyle
S_{N}$ $\displaystyle\pi_{1}({\mathcal{Q}}_{2})$ $\displaystyle=$
$\displaystyle 1$ $\displaystyle\pi_{1}^{*}({\mathcal{Q}}_{3})$
$\displaystyle=$ $\displaystyle T_{N}$
$\displaystyle\pi_{1}^{*}({\mathcal{Q}}_{2,2})$ $\displaystyle=$
$\displaystyle F_{N}$ $\displaystyle\pi_{1}^{*}({\mathcal{Q}}_{\\{3;2,2\\}})$
$\displaystyle=$ $\displaystyle W_{N}.$ (19)

The novel strand groups $T_{N}$, $F_{N}$ and $W_{N}$ result when three-body
and certain four-body coincidences are removed from the configuration space
orbifold ${\mathcal{Q}}$. Unlike the braid group, these co-dimension
$\tilde{d}=2$ exclusions preserve the self-inverse property of pairwise
exchanges (9a), but break the other defining relations of the symmetric group
(9b) and (9c). These groups provide the possibility of novel abelian and non-
abelian anyons, but their irreducible representations have not been classified
or explored.

For indistinguishable particles on a circle, the underlying topology of the
base space $S^{1}$ gets ‘mixed up’ with the symmetric group in the same way
the braid group on (for example) $S^{2}$ allows different topological exchange
statistics than the more familiar braid group on $\mathbb{R}^{2}$. The
equivalent groups to (19) expressed in two alternate forms are

$\displaystyle\pi_{1}^{*}({\mathcal{Q}})$ $\displaystyle=$
$\displaystyle{\mathbb{Z}}^{N}\rtimes
S_{N}={\mathbb{Z}}_{\zeta}\ltimes\tilde{S}_{N}$
$\displaystyle\pi_{1}({\mathcal{Q}}_{2})$ $\displaystyle=$
$\displaystyle{\mathbb{Z}}_{\zeta}$
$\displaystyle\pi_{1}^{*}({\mathcal{Q}}_{3})$ $\displaystyle=$
$\displaystyle{\mathbb{Z}}^{N}\rtimes
T_{N}={\mathbb{Z}}_{\zeta}\ltimes\tilde{T}_{N}$
$\displaystyle\pi_{1}^{*}({\mathcal{Q}}_{2,2})$ $\displaystyle=$
$\displaystyle{\mathbb{Z}}^{N}\rtimes
F_{N}={\mathbb{Z}}_{\zeta}\ltimes\tilde{F}_{N}$
$\displaystyle\pi_{1}^{*}({\mathcal{Q}}_{\\{3;2,2\\}})$ $\displaystyle=$
$\displaystyle{\mathbb{Z}}^{N}\rtimes
W_{N}={\mathbb{Z}}_{\zeta}\ltimes\tilde{W}_{N}.$ (20)

The affine strand groups $\tilde{T}_{N}$, $\tilde{F}_{N}$ and $\tilde{W}_{N}$
arise from broken relations of $\tilde{S}_{N}$ in the same way as the non-
affine versions. As far as we know, this is the first time these hyperbolic
groups have been identified in a physical system and their group structures
and irreducible representations are largely unexplored. Also, except for the
case of ${\mathcal{Q}}_{2}$, all of these groups provide the possibility for
novel abelian and non-abelian anyons.

In none of these cases does the group giving topological exchange statistics
furnish an abelian representation with fractional exchange statistics for an
arbitrary $\theta$. Our conclusion agrees with [82, 53] that if a one-
dimensional model exhibits fractional exchange statistics, these are of a
dynamical origin and cannot be absorbed into a consistent gauge potential. In
contrast, because they have a topological origin, the alternate strand groups
like $T_{N}$ and $\tilde{T}_{N}$ described above should be ‘transmutable’ into
a gauge potential, and this looks to be a promising avenue for investigating
there phenomenological signatures. Because the traid and fraid group also
derive from co-dimension two topological defects in configuration space, we
hypothesize there could be similarities to braid group anyons where
connections among conformal field theory, fusion rules, and quantum groups is
a productive line of research; c.f. [85]. A field theoretical formulation of
these alternate strand groups is certainly required for applications to many-
body systems.

Are these alternate strand groups feasible to realize physically? Ultracold
atoms can be confined to effectively one-dimensional traps, and in principle
hard-core three body interactions can be engineered in cold atoms systems [29,
30, 31, 32]. Density-induced interactions in Floquet-driven lattice models
also show promise [86, 87]. Hard-core three body interactions would effective
exclude $\Delta_{3}$ from $\mathcal{Q}$, and dynamical models with three body
interactions have shown signs of novel statistics and other thermodynamic
properties [88, 80, 89, 90]. In Ref. [28], we have described and depicted the
lowest energy wave functions obeying abelian traid exchange statistics for
three particles in a harmonic trap, but exploring these solutions for more
particles and classifying how they transform under discrete symmetries is an
ongoing project.

In contrast, the paired two-body interactions necessary to exclude
$\Delta_{2,2}$ are non-local. A single two-body coincidence would need to
prevent other two-body coincidences anywhere in the system. As a fundamental
interaction, non-local interactions are typically excluded, but as an
effective theory for long-range interactions in a many-body system, it may
have interest. More generally, because these novel strands groups arise
‘naturally’ as degenerations of the ubiquitous symmetric group, one can image
that they could emerge in a variety of non-particle model contexts.

As a final note, building an ‘intrinsic’ quantum theory for indistinguishable
particles directly on the quotient space, without reference to a covering
space of distinguishable particles, is an incomplete project that requires
mathematical and conceptual definitions. Operators that are self-adjoint on
${\mathcal{X}}$, such as the Hamiltonian and the single-particle position and
momentum, no longer have that property when restricted to ${\mathcal{Q}}$ [22,
71, 91]. For relative momentum and other operators that are not symmetric
under particle exchange, self-adjointness cannot be restored and their
interpretation does not seem to extend unambiguously to indistinguishable
particles.

###### Acknowledgements.

We would like to thank Franscesca Ark, André Eckardt, Andreas Bock Michelsen,
Zachary Tomares, Philip Johnson, Maxim Olshanii, Thore Posske, and Thomas
Schmidt for useful discussions.

##  Appendix A Topological quantization

The topological approach to quantization developed from several independent
directions [92, 56, 93, 7, 94, 95, 3, 96, 97]. Possible quantizations of a
configuration space ${\mathcal{X}}$ are classified using irreducible unitary
representations of the fundamental group $\pi_{1}({\mathcal{X}})$. The
fundamental group $\pi_{1}({\mathcal{X}})$ describes equivalence classes of
based loops on on a connected manifold ${\mathcal{X}}$. Whenever
${\mathcal{X}}$ is connected, the isomorphism type of $\pi_{1}({\mathcal{X}})$
is independent of the base point.

Abelian representations of $\pi_{1}({\mathcal{X}})$ characterize flat $U(1)$
connections on flat line bundles over ${\mathcal{X}}$ and record the holonomy
of the scalar wave function around closed loops. For non-trivial
representations, the corresponding wave functions transform by a phase under
holonomy and are therefore considered multi-valued wave functions on
${\mathcal{X}}$, also called $\theta$-structures [96, 98]. Similarly, non-
abelian representations, considered up to conjugation, characterize flat
$U(\mu)$ connections on higher-rank fiber bundles leading to multi-valued,
multi-component wave functions [57, 71, 99].

If working with multi-valued wave functions on ${\mathcal{X}}$ is distasteful
or inconvenient, one can instead work with single-valued wave functions on the
simply-connected universal cover of ${\mathcal{X}}$, denoted
$\widetilde{{\mathcal{X}}}$ with covering map
$p_{\mathcal{X}}:\widetilde{{\mathcal{X}}}\to\mathcal{X}$. This covering map
also defines the original configuration space as a quotient space

${\mathcal{X}}=\widetilde{{\mathcal{X}}}/D$ (21)

where $D$ is a group of deck transformations (diffeomorphisms on
$\widetilde{{\mathcal{X}}}$ compatible with the covering map
$p_{\mathcal{X}}$) isomorphic to $\pi_{1}({\mathcal{X}})$. Some basic examples
include the universal cover of a circle $S^{1}$ by the real line
${\mathbb{R}}$ and the universal cover of the $n$-torus by ${\mathbb{R}}^{n}$.
Another famous example is the quantum rotor, whose configuration space
$\mathrm{SO}(3)$ is not simply-connected. The fundamental group
$\pi_{1}(\mathrm{SO}(3))=Z_{2}$ has two irreducible representations. The non-
trivial irreducible representations corresponds to wave functions with half-
integer spin that are double-valued on $\mathrm{SO}(3)$, but single-valued on
its universal cover $\mathrm{SU}(2)$ [92]. In these cases, working with
single-valued functions on the universal cover can be simpler than multi-
valued functions on the base manifold, but that is certainly not always the
case. For example, the universal cover of the configuration space whose
fundamental group is the braid group $B_{N}$ is quite complicated for $N\geq
3$.

##  Appendix B Particle statistics in quantum mechanics

Two main approaches to characterizing particle statistics are _exchange
statistics_ and _exclusion statistics_. Exchange statistics considers how wave
functions or operators transform when indistinguishable particles are
exchanged. Further, there are two main approaches to exchange statistics in
quantum mechanical systems: the _symmetrization postulate_ and _topological
exchange statistics_ , and we briefly compare them in this appendix. We focus
on systems of particles on manifolds and do not consider the interesting case
of particles on graphs [100, 101, 19] which may have novel applications to
tight-binding lattice models. We also restrict ourselves to quantum mechanics
in configuration spaces and Hilbert spaces with a fixed number of particles
and do not consider the important case of field operators and Fock spaces. For
completeness, exclusion statistics is briefly described at the end of this
appendix.

The first and oldest approach to exchange statistics of indistinguishable
particles in quantum mechanics is the symmetrization postulate. For the sake
of simplicity, consider a single-component, single-particle Hilbert space
${\mathcal{H}}=L^{2}({\mathcal{M}})$ of square-integrable wave functions on
the manifold ${\mathcal{M}}$. For $N$ distinguishable but identical particles,
the total Hilbert space ${\mathcal{H}}^{N}$ can be constructed as the $N$-fold
tensor product
${\mathcal{H}}^{N}={\mathcal{H}}\otimes\cdots\otimes{\mathcal{H}}$. This space
is equivalent to the Hilbert space ${\mathcal{H}}^{N}=L^{2}({\mathcal{X}})$ of
square-integrable functions on the total configuration space
${\mathcal{X}}={\mathcal{M}}^{N}$. Note that the symmetrization postulate also
applies to multi-component single-particle wave functions living in
${\mathcal{H}}=L^{2}({\mathcal{M}})\otimes\mathbb{C}^{\mu}$; see [59] for
examples with spin components.

Indistinguishability is imposed by symmetrizing a system of identical
distinguishable particles. This is accomplished using representations of
particle permutations $s\in S_{N}$ as unitary operators $U(s)$ that act on
${\mathcal{H}}^{N}$. The action of operators $U(s)$ on ${\mathcal{H}}^{N}$ is
induced from the realization of passive particle permutations $s\in S_{N}$ as
diffeomorphisms $O_{s}$ acting on ${\mathcal{X}}$. Note that when
${\mathcal{M}}$ is an interval in ${\mathbb{R}}$ then ${\mathcal{X}}$ is a box
in ${\mathbb{R}}^{d}$ and the diffeomorphisms are a representation of $s\in
S_{N}$ by orthogonal matrices $O_{s}\in\mathrm{O}(Nd)$.

The two abelian representations of $S_{N}$ provide the technology relevant to
constructing the state space for bosons and fermions. Wave functions obeying
BE exchange statistics are elements of ${\mathcal{H}}^{N}_{+}$, the projection
of ${\mathcal{H}}^{N}$ onto the symmetric subspace, and wave functions in the
antisymmetric subspace ${\mathcal{H}}^{N}_{-}$ have FD exchange statistics.
After symmetrization, each choice of particle labels is equivalent to a choice
of gauge within ${\mathcal{H}}^{N}_{\pm}$ [23, 22].

The symmetrization postulate can also be extended to the non-abelian
irreducible representations of $S_{N}$ for $N\geq 3$. Wave functions belonging
to these representations obey so-called parastatistics [102, 103]. These
multi-component wave functions are useful for constructing states of
indistinguishable fermions or bosons that have internal degrees of freedom
[104, 23, 59].

However, the symmetrization postulate approach to indistinguishability
presumes single-valued wave functions on ${\mathcal{X}}$ with $S_{N}$ exchange
statistics. As we discuss in Sect. IV, non-$S_{N}$ exchange statistics can be
mimicked by adding interactions to fermions or bosons. Alternatively, multi-
valued wave functions with non-$S_{N}$ statistics arise ‘kinematically’ (i.e.,
without interactions) in the topological approach to exchange statistics when
the configuration space is not simply-connected [7, 71]. Instead of treating
exchanges as global transformations, exchanges are specific active continuous
processes [103] which can be visualized as strand diagrams, i.e. $N$ paths on
${\mathcal{M}}$ that connect particle configurations differing by at most a
permutation of identical particles. When the $N$ paths do not coincide on
${\mathcal{M}}$, they can be lifted unambiguously to one path in
${\mathcal{M}}^{N}$, up to a label permutation gauge.

Topological exchange statistics also begins from the configuration space
${\mathcal{X}}$ for $N$ distinguishable particles on a connected manifold
${\mathcal{M}}$ (4). If the base manifold has dimension $\dim{\mathcal{M}}=d$
and connectivity described by the fundamental group $\pi_{1}({\mathcal{M}})$,
then ${\mathcal{X}}$ is a manifold with dimension $\dim{\mathcal{X}}=Nd$ and
fundamental group $\pi_{1}({\mathcal{X}})=\pi_{1}({\mathcal{M}})^{N}$.

On ${\mathcal{X}}$, paths that start and end at the same place realize
particle ‘exchanges’ where every particle returns to its original location,
but if ${\mathcal{M}}$ is not simply connected, then even these paths can be
topologically non-trivial and wave functions on ${\mathcal{M}}$ need not be
single-valued. For each irreducible representation of
$\pi_{1}({\mathcal{M}})^{N}$ there is a class of multi-valued solutions. If
the particles on ${\mathcal{M}}$ are distinguishable but _identical_ , then
one can restrict to a diagonal representation of $\pi_{1}({\mathcal{M}})^{N}$
that factors into the $N$-fold product of the same irreducible representation
of $\pi_{1}({\mathcal{M}})$.

The configuration space for indistinguishable particles,
${\mathcal{Q}}={\mathcal{X}}/S_{N}$, is the topological quotient of
${\mathcal{X}}$ by the symmetric group [56, 7]. However, as described in Sect.
II, the space ${\mathcal{Q}}$ has singular orbifold points with a non-trivial
topological structure. The standard solution removes these singular points
(and therefore the path ambiguity) from configuration space entirely [56, 7,
22, 8, 15]. The motivation for the orbifold approach is to resolve this
ambiguity.

The orbifold fundamental group allows a derivation of symmetric group
statistics (i.e., bosons, fermions and parastatistics) for simply-connected
base spaces and generalized parastatistics for multiply-connected base spaces
in a unified fashion and without needing to exclude path-ambiguous two-body
collisions. For fermions and bosons on simply-connected manifolds with non-
singular interactions, the orbifold approach to topological exchange
statistics gives the same results as the symmetrization postulate in any
dimension, whereas the standard approach to topological exchange statistics
does not for $d\leq 2$ without introducing additional constraints or dynamics
[7, 22].

Finally, in contrast to either form of exchange statistics, exclusion
statistics [105, 106] is a more recent approach to understanding particle
statistics. It employs a fractional statistics parameter $g$ to describe how
the effective dimension of the single particle Hilbert space changes as the
number of particles increases 333A related notion is Gentile statistics
characterized by the maximum occupation number of a state [113, 23, 114]..
This parameter is $g=0$ for BE statistics because any number of bosons can
occupy the same state, and $g=1$ for FD statistics because the Pauli exclusion
principle requires an additional state for each fermion. This parameter is
consistent with the fractional exchange statistics parameter for abelian
representations of the braid group [108]. Unlike the topological approach to
exchange statistics, formulating indistinguishability in terms of exclusion
statistics does not depend on the dimension of the underlying physical space
${\mathcal{M}}$. As a result, it has been especially useful for analyzing one-
dimensional dynamical models [21, 47, 109] for which the standard (non-
orbifold) topological approach gives trivial results.

## References

  * Bartolomei  _et al._ [2020] H. Bartolomei, M. Kumar, R. Bisognin, A. Marguerite, J.-M. Berroir, E. Bocquillon, B. Plaçais, A. Cavanna, Q. Dong, U. Gennser, Y. Jin, and G. Fève, Science 368, 173 (2020).
  * Nakamura  _et al._ [2020] J. Nakamura, S. Liang, G. C. Gardner, and M. J. Manfra, Nat. Phys. 16, 931 (2020).
  * Wilczek [1982a] F. Wilczek, Phys. Rev. Lett. 49, 957 (1982a).
  * Halperin [1984] B. I. Halperin, Phys. Rev. Lett. 52, 1583 (1984).
  * Arovas  _et al._ [1984] D. Arovas, J. R. Schrieffer, and F. Wilczek, Phys. Rev. Lett. 53, 722 (1984).
  * Wilczek [1990] F. Wilczek,  _Fractional statistics and anyon superconductivity_ (World Scientific, Singapore, Teaneck, NJ, 1990).
  * Leinaas and Myrheim [1977] J. M. Leinaas and J. Myrheim, Il Nuovo Cimento B (1971-1996) 37, 1 (1977).
  * Wu [1984a] Y.-S. Wu, Phys. Rev. Lett. 52, 2103 (1984a).
  * Wu [1984b] Y.-S. Wu, Phys. Rev. Lett. 53, 111 (1984b).
  * Forte [1992] S. Forte, Rev. Mod. Phys. 64, 193 (1992).
  * Khare [2005] A. Khare,  _Fractional Statistics And Quantum Theory_, 2nd ed. (World Scientific, Singapore, 2005).
  * Wilczek [1982b] F. Wilczek, Phys. Rev. Lett. 48, 1144 (1982b).
  * Moore and Read [1991] G. Moore and N. Read, Nuc. Phys. B 360, 362 (1991).
  * Wen [1991] X. G. Wen, Phys. Rev. Lett. 66, 802 (1991).
  * Nayak  _et al._ [2008] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Rev. Mod. Phys. 80, 1083 (2008).
  * Kitaev [2003] A. Y. Kitaev, Ann. Phys. 303, 2 (2003).
  * Freedman  _et al._ [2003] M. Freedman, A. Kitaev, M. Larsen, and Z. Wang, Bull. Am. Math. Soc. 40, 31 (2003).
  * Alicea  _et al._ [2011] J. Alicea, Y. Oreg, G. Refael, F. von Oppen, and M. P. A. Fisher, Nat. Phys. 7, 412 (2011).
  * Macia̧żek and Sawicki [2019] T. Macia̧żek and A. Sawicki, Comm. Math. Phys. 371, 921 (2019).
  * Minguzzi and Vignolo [2022] A. Minguzzi and P. Vignolo, Strongly interacting trapped one-dimensional quantum gases: an exact solution (2022), arXiv:2201.02362 [cond-mat.quant-gas] .
  * Ha [1995] Z. N. C. Ha, Nuc. Phys. B 435, 604 (1995).
  * Bourdeau and Sorkin [1992] M. Bourdeau and R. D. Sorkin, Phys. Rev. D 45, 687 (1992).
  * Polychronakos [1999] A. P. Polychronakos, in  _Aspects topologiques de la physique en basse dimension. Topological aspects of low dimensional systems_ , edited by A. Comtet, T. Jolicœur, S. Ouvry, and F. David (Springer Berlin Heidelberg, Berlin, Heidelberg, 1999) pp. 415–471.
  * Landsman [2016] N. P. K. Landsman, Reviews in Mathematical Physics 28, 1650019 (2016).
  * Ohya [2021] S. Ohya, Annals of Physics 434, 168657 (2021).
  * Thurston [2002] W. Thurston,  _Geometry and topology of three-manifolds_ , electronic version 1.1 ed. (unpublished, 2002).
  * Adem  _et al._ [2007] A. Adem, J. Leida, and Y. Ruan,  _Orbifolds and Stringy Topology_ , 1st ed. (Cambridge University Press, Cambridge ; New York, 2007).
  * Harshman and Knapp [2020] N. L. Harshman and A. C. Knapp, Ann. Phys. 412, 168003 (2020).
  * Büchler  _et al._ [2007] H. P. Büchler, A. Micheli, and P. Zoller, Nat. Phys. 3, 726 (2007).
  * Daley and Simon [2014] A. J. Daley and J. Simon, Phys. Rev. A 89, 053619 (2014).
  * Mahmud  _et al._ [2014] K. W. Mahmud, E. Tiesinga, and P. R. Johnson, Phys. Rev. A 90, 041602 (R) (2014).
  * Valiente [2019] M. Valiente, Phys. Rev. A 100, 013614 (2019).
  * Cisneros  _et al._ [2020] B. Cisneros, M. Flores, J. Juyumaya, and C. Roque-Márquez, An alexander type invariant for doodles (2020), arXiv:2005.06290 [math.GT] .
  * Khovanov [1997] M. Khovanov, Trans. Am. Math. Soc. 349, 2297 (1997).
  * Bartholomew  _et al._ [2018] A. Bartholomew, R. Fenn, N. Kamada, and S. Kamada, J. Knot Theory Ramifications 27 (2018).
  * González  _et al._ [2021] J. González, J. L. León-Medina, and C. Roque-Márquez, Homology, Homotopy and Applications 23, 275 (2021).
  * Bardakov  _et al._ [2019] V. Bardakov, M. Singh, and A. Vesnin, Geometriae Dedicata 203, 135 (2019).
  * Naik  _et al._ [2020] T. Naik, N. Nanda, and M. Singh, Forum Mathimaticum 32, 1095 (2020).
  * Note [1] Our analysis does not consider the anyon-Hubbard model [110, 80, 49, 111, 86, 87, 112] because our topological methods do not extend to configuration spaces with discrete topology, although we note that the continuum limit of the anyon-Hubbard model has recently been derived [54].
  * Hansson  _et al._ [1992] T. Hansson, J. Leinaas, and J. Myrheim, Nuclear Physics B 384, 559 (1992).
  * Posske  _et al._ [2017] T. Posske, B. Trauzettel, and M. Thorwart, Phys. Rev. B 96, 195422 (2017).
  * Sree Ranjani  _et al._ [2009] S. Sree Ranjani, P. Panigrahi, A. Kapoor, and A. Khare, Annals of Physics 324, 1176 (2009).
  * Zhu and Wang [1996] J.-X. Zhu and Z. D. Wang, Phys. Rev. A 53, 600 (1996).
  * Girardeau [2006] M. D. Girardeau, Phys. Rev. Lett. 97, 100402 (2006).
  * del Campo [2008] A. del Campo, Phys. Rev. A 78, 045602 (2008).
  * Kundu [1999] A. Kundu, Phys. Rev. Lett. 83, 1275 (1999).
  * Batchelor  _et al._ [2006] M. T. Batchelor, X.-W. Guan, and N. Oelkers, Phys. Rev. Lett. 96, 210402 (2006).
  * Pâţu  _et al._ [2007] O. I. Pâţu, V. E. Korepin, and D. V. Averin, Journal of Physics A: Mathematical and Theoretical 40, 14963 (2007).
  * Hao and Chen [2012] Y. Hao and S. Chen, Phys. Rev. A 86, 043631 (2012).
  * Zinner [2015] N. T. Zinner, Phys. Rev. A 92, 063634 (2015).
  * Stouten  _et al._ [2018] E. Stouten, P. W. Claeys, M. Zvonarev, J.-S. Caux, and V. Gritsev, Journal of Physics A: Mathematical and Theoretical 51, 485204 (2018).
  * Pâţu [2019] O. I. Pâţu, Phys. Rev. A 100, 063635 (2019).
  * Valiente [2020] M. Valiente, Phys. Rev. A 102, 053304 (2020).
  * Bonkhoff  _et al._ [2021] M. Bonkhoff, K. Jägering, S. Eggert, A. Pelster, M. Thorwart, and T. Posske, Phys. Rev. Lett. 126, 163201 (2021).
  * Valiente [2021] M. Valiente, Phys. Rev. A 103, L021302 (2021).
  * Laidlaw and DeWitt [1971] M. G. G. Laidlaw and C. M. DeWitt, Phys. Rev. D 3, 1375 (1971).
  * Imbo  _et al._ [1990] T. D. Imbo, C. Shah Imbo, and E. C. G. Sudarshan, Physics Letters B 234, 103 (1990).
  * James and Kerber [1984] G. James and A. Kerber,  _The Representation Theory of the Symmetric Group_, Encyclopedia of Mathematics and its Applications (Cambridge University Press, Cambridge, 1984).
  * Harshman [2016a] N. L. Harshman, Few-Body Systems 57, 45 (2016a).
  * Juhász  _et al._ [2018] A. Juhász, D. P. Thurston, and I. Zemke, Naturality and mapping class groups in heegaard floer homology (2018), arXiv:1210.4996 [math.GT] .
  * Boileau  _et al._ [2003] M. Boileau, S. Maillot, and J. Porti,  _Three-Dimensional Orbifolds and Their Geometric Structures_ (Société Mathématique de France, Paris, 2003).
  * Birman [1969] J. S. Birman, Comm. Pure App. Math. 22, 41 (1969).
  * Thouless and Wu [1985] D. J. Thouless and Y.-S. Wu, Phys. Rev. B 31, 1191 (1985).
  * Einarsson [1990] T. Einarsson, Phys. Rev. Lett. 64, 1995 (1990).
  * Hatsugai  _et al._ [1991] Y. Hatsugai, M. Kohmoto, and Y.-S. Wu, Phys. Rev. B 43, 2661 (1991).
  * Harshman and Knapp [2018] N. L. Harshman and A. C. Knapp, Few-Body Systems 59, 79 (2018).
  * Harshman [2016b] N. L. Harshman, Few-Body Systems 57, 11 (2016b).
  * Harshman  _et al._ [2017] N. Harshman, M. Olshanii, A. Dehkharghani, A. Volosniev, S. G. Jackson, and N. Zinner, Phys. Rev. X 7, 041001 (2017).
  * Girardeau [1960] M. Girardeau, J. Math. Phys. 1, 516 (1960).
  * Harshman [2017] N. L. Harshman, Phys. Rev. A 95, 053616 (2017).
  * Balachandran [1991] A. Balachandran, Int. J. Mod. Phys. B 05, 2585 (1991).
  * Humphreys [1992] J. E. Humphreys,  _Reflection Groups and Coxeter Groups_ (Cambridge University Press, 1992).
  * Mostovoy and Roque-Márquez [2020] J. Mostovoy and C. Roque-Márquez, J. Knot Theory Ramifications 29 (2020).
  * Mostovoy [2021] J. Mostovoy (2021), pre-print arXiv:2006.08007.
  * Artin [1947] E. Artin, Annals of Mathematics 48, 101 (1947).
  * Altmann [1977] S. L. Altmann,  _Induced Representations in Crystals and Molecules: Point, Space, and Nonrigid Molecule Groups_ (Academic Press, 1977).
  * Sutherland [2004] B. Sutherland,  _Beautiful Models: 70 Years of Exactly Solved Quantum Many-Body Problems_ , 1st ed. (World Scientific Publishing Company, River Edge, N.J, 2004).
  * Turner [1984] J. W. Turner, J. Phys. A 17, 2791 (1984).
  * Jain [2008] S. R. Jain, Phys. Lett. A 372, 1978 (2008).
  * Keilmann  _et al._ [2011] T. Keilmann, S. Lanzmich, I. McCulloch, and M. Roncaglia, Nat. Comm. 2, 361 (2011).
  * Note [2] Although discrete models are outside the scope of this analysis, a similar technique defines the anyon-Hubbard model [80]. Density-dependent interactions for bosons are ‘transmuted’ by the gauge transformation into site-dependent phase slips. See also [54] for the relation of the anyon Hubbard model to the Kundu/Lieb-Liniger model.
  * Aglietti  _et al._ [1996] U. Aglietti, L. Griguolo, R. Jackiw, S.-Y. Pi, and D. Seminara, Phys. Rev. Lett. 77, 4406 (1996).
  * Cheon and Shigehara [1999] T. Cheon and T. Shigehara, Phys. Rev. Lett. 82, 2536 (1999).
  * Ohya [2022] S. Ohya, Phys. Rev. A 105, 033312 (2022).
  * Alvarez-Gaumé  _et al._ [1990] L. Alvarez-Gaumé, C. Gomez, and G. Sierra, Nuclear Physics B 330, 347 (1990).
  * Greschner and Santos [2015] S. Greschner and L. Santos, Phys. Rev. Lett. 115, 053002 (2015).
  * Sträter  _et al._ [2016] C. Sträter, S. C. L. Srivastava, and A. Eckardt, Phys. Rev. Lett. 117, 205303 (2016).
  * Paredes  _et al._ [2007] B. Paredes, T. Keilmann, and J. I. Cirac, Phys. Rev. A 75, 053611 (2007).
  * Sowiński  _et al._ [2015] T. Sowiński, R. W. Chhajlany, O. Dutta, L. Tagliacozzo, and M. Lewenstein, Phys. Rev. A 92, 043615 (2015).
  * Arcila-Forero  _et al._ [2018] J. Arcila-Forero, R. Franco, and J. Silva-Valencia, Phys. Rev. A 97, 023631 (2018).
  * Gaveau and Schulman [2012] B. Gaveau and L. S. Schulman, Journal of Physics A: Mathematical and Theoretical 45, 462001 (2012).
  * Schulman [1968] L. Schulman, Phys. Rev. 176, 1558 (1968).
  * Dowker [1972] J. S. Dowker, J. Phys. A 5, 936 (1972).
  * Mead and Truhlar [1979] C. A. Mead and D. G. Truhlar, J. Chem. Phys. 70, 2284 (1979).
  * Goldin  _et al._ [1981] G. A. Goldin, R. Menikoff, and D. H. Sharp, J. Math. Phys. 22, 1664 (1981).
  * Isham [1983] C. J. Isham, in  _Les Houches Summer School on Theoretical Physics: Relativity, Groups and Topology_ (1983) pp. 1059–1290.
  * Berry [1984] M. V. Berry, Proc. Royal Soc. London. A 392, 45 (1984).
  * Imbo and Sudarshan [1988] T. D. Imbo and E. C. G. Sudarshan, Phys. Rev. Lett. 60, 481 (1988).
  * Lee [2018] J. Lee,  _Introduction to Riemannian Manifolds_, 2nd ed., Graduate Texts in Mathematics (Springer International Publishing, 2018).
  * Harrison  _et al._ [2011] J. M. Harrison, J. P. Keating, and J. M. Robbins, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 467, 212 (2011).
  * Harrison  _et al._ [2014] J. Harrison, J. Keating, J. Robbins, and A. Sawicki, Communications in Mathematical Physics 33, 1293–1326 (2014).
  * Green [1953] H. S. Green, Phys. Rev. 90, 270 (1953).
  * Messiah and Greenberg [1964] A. M. L. Messiah and O. W. Greenberg, Phys. Rev. 136, B248 (1964).
  * Sudarshan  _et al._ [1988] E. C. G. Sudarshan, T. D. Imbo, and T. R. Govindarajan, Physics Letters B 213, 471 (1988).
  * Haldane [1991] F. D. M. Haldane, Phys. Rev. Lett. 67, 937 (1991).
  * Polychronakos [1996] A. P. Polychronakos, Physics Letters B 365, 202 (1996).
  * Note [3] A related notion is Gentile statistics characterized by the maximum occupation number of a state [113, 23, 114].
  * Johnson and Canright [1994] M. D. Johnson and G. S. Canright, Phys. Rev. B 49, 2947 (1994).
  * Murthy and Shankar [2013] M. V. N. Murthy and R. Shankar, MatSciRep 120 (2013).
  * Amico  _et al._ [1998] L. Amico, A. Osterloh, and U. Eckern, Phys. Rev. B 58, R1703 (1998).
  * Wright  _et al._ [2014] T. M. Wright, M. Rigol, M. J. Davis, and K. V. Kheruntsyan, Phys. Rev. Lett. 113, 050601 (2014).
  * Lange  _et al._ [2017] F. Lange, S. Ejima, and H. Fehske, Phys. Rev. A 95, 063621 (2017).
  * Gentile [1940] G. Gentile, Nuovo Cim. 17, 493 (1940).
  * Dai and Xie [2004] W.-S. Dai and M. Xie, Annals of Physics 309, 295 (2004).