$\displaystyle I_{4}$ $\displaystyle=$ $\displaystyle(2\pi
S^{\prime})^{2}|\psi|^{2}\sum_{n,m}\int d\xi_{1}^{(m)}d\xi_{1}^{(n)}d\eta$
$\displaystyle\times\epsilon_{rijk}\,\frac{\partial
r_{i}^{(m)}}{\partial\xi_{1}^{(m)}}\frac{\partial
r_{j}^{(n)}}{\partial\xi_{1}^{(n)}}\left(\frac{\partial
r_{k}^{(n)}}{\partial\eta}-\frac{\partial
r_{k}^{(m)}}{\partial\eta}\right)\,\delta^{3}(\Delta{\bf r}_{S^{3}})\ .$

We tacitly assumed $\partial r_{r}^{(m)}/\partial\xi_{2}^{(m)}>0$, $\partial
r_{r}^{(n)}/\partial\xi_{2}^{(n)}>0$, as well as $\partial
r_{r}^{(m)}/\partial\xi_{1}^{(m)}=\partial
r_{r}^{(n)}/\partial\xi_{1}^{(n)}=0$, which can be always arranged by the
choice of parametrization (e.g., the $S^{3}$ shells are indeed the equal-
radius 3-spheres, etc.). Next, we rewrite $\delta^{3}({\bf r}-{\bf
r}^{\prime})$ as

$\delta^{3}({\bf r}-{\bf r}^{\prime})=-\frac{1}{4\pi}\nabla^{2}\frac{1}{|{\bf
r}-{\bf r}^{\prime}|}=\frac{1}{4\pi}\boldsymbol{\nabla}\frac{{\bf r}-{\bf
r}^{\prime}}{|{\bf r}-{\bf r}^{\prime}|^{3}}\ .$ (133)

This allows us to apply Gauss’ theorem to the last integral over a
3-dimensional domain. The boundary of this 3-dimensional manifold is precisely
the 2-dimensional manifold of $\Delta{\bf r}_{S^{3}}\subset S^{3}$ probed by
the linking number integral. We have:

$\displaystyle I_{4}$ $\displaystyle=$ $\displaystyle-\frac{(2\pi
S^{\prime})^{2}}{4\pi}|\psi|^{2}\sum_{n,m}\\!\int\\!\epsilon_{ijk}\,dr_{i}^{(m)}dr_{j}^{(n)}d\eta\,\frac{\partial(r_{k}^{(m)}\\!-\\!r_{k}^{(n)})}{\partial\eta}$
$\displaystyle\quad\times\partial_{p}\frac{r_{p}^{(m)}-r_{p}^{(n)}}{\left|{\bf
r}^{(m)}-{\bf r}^{(n)}\right|^{3}}$ $\displaystyle=$
$\displaystyle-\frac{(2\pi
S^{\prime})^{2}}{4\pi}|\psi|^{2}\sum_{n,m}\int\\!\epsilon_{ijk}\,dr_{i}^{(m)}dr_{j}^{(n)}\,\frac{r_{k}^{(m)}-r_{k}^{(n)}}{\left|{\bf
r}^{(m)}-{\bf r}^{(n)}\right|^{3}}\ .$

Note that only the upper bound of the $\eta$ integral

$\displaystyle I_{\eta}$ $\displaystyle=$ $\displaystyle\int
d\eta\frac{\partial\delta r_{k}}{\partial\eta}\partial_{k}\frac{\delta
r_{k}}{|\delta{\bf r}|^{3}}=\int d\delta r_{k}\frac{\partial}{\partial\delta
r_{k}}\frac{\delta r_{k}}{|\delta{\bf r}|^{3}}$ (135) $\displaystyle=$
$\displaystyle\frac{\delta r_{k}}{|\delta{\bf
r}|^{3}}\Biggr{|}_{\eta}-\frac{\delta r_{k}}{|\delta{\bf
r}|^{3}}\Biggr{|}_{0}$

contributes to (IV.4), while the lower bound disappears for physical reasons.
At the singularity ($\eta=0$), we have $\delta{\bf r}={\bf r}^{(m)}-{\bf
r}^{(n)}\to 0$, so $\delta r_{k}/|\delta{\bf r}|^{3}$ is not convergent or
well-defined. However, this divergence must be regularized away upon the
coarse-graining of microscopic degrees of freedom, which goes into the
construction of the field theory. Placing the lower bound of $I_{\eta}$ at
some small non-zero value of $\eta\to\epsilon>0$ would ensure an exact
cancellation with the upper bound, i.e. $I_{4}=\int d^{2}r\,I_{\eta}\to 0$;
each $\eta$ shell picks the same topologically protected Hopf index $N$.
However, the lower bound is strictly at the $\eta\to 0$ singularity since we
formulated the original integral over the 4-dimensional volume $B^{4}$ which
includes the singularity (the purpose of this was precisely to develop a
prescription for integrating out a singular Hopf index density in $d=4$
dimensions). It is not possible to extract the value of the Hopf index by
integrating out the skyrmion loop structure within a single point in space.
The lower bound of $I_{\eta}$ should average out to zero because the Hopf
index transforms non-trivially under spatial inversions and no information
about such transformations can be packed into a point using the existing
degrees of freedom.

Since (IV.4) reduces the integral (IV.4) to the Gauss’ linking number for
skyrmions, we can read the Hopf index $N\to N^{\textrm{s}}$ of the spin
texture directly from the spinor field $\psi$. Taking (123) into account,

$I_{4}\xrightarrow{\varphi\to\textrm{const.}}-(2\pi)^{2}|\psi|^{2}N^{\textrm{s}}.$
(136)

The same holds in the charge sector,

$I_{4}\xrightarrow{\theta,\phi\to\textrm{const.}}-|\psi|^{2}\\!\int\limits_{S^{3}}\\!d^{3}x\,\epsilon_{ijk}a_{i}\partial_{j}a_{k}=-(2\pi)^{2}|\psi|^{2}N^{\textrm{c}}\
.$ (137)

The Hopf invariants of charge (c) and spin (s) currents are represented by
different gauge fields

$\displaystyle N^{\textrm{c}}$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{2}}\int\limits_{S^{3}}d^{3}x\,\epsilon_{ijk}a_{i}\partial_{j}a_{k}$
(138) $\displaystyle N^{\textrm{s}}$ $\displaystyle=$
$\displaystyle\frac{1}{(4\pi)^{2}}\int\limits_{S^{3}}d^{3}x\,\epsilon_{ijk}A_{i}\partial_{j}A_{k}\
,$

and define separate Hopf index current densities

$\displaystyle\mathcal{J}_{\mu}^{\textrm{c}}$ $\displaystyle=$
$\displaystyle\epsilon_{\mu\nu\alpha\beta\gamma}\partial_{\nu}(a_{\alpha}\partial_{\beta}a_{\gamma})$
(139) $\displaystyle\mathcal{J}_{\mu}^{\textrm{s}}$ $\displaystyle=$
$\displaystyle\epsilon_{\mu\nu\alpha\beta\gamma}\partial_{\nu}(A_{\alpha}\partial_{\beta}A_{\gamma})$

normalized to the corresponding flux quanta. But, it is easy to see from
(IV.4) and (124) that combining the charge ($\varphi$) and spin ($\phi$)
vorticity embeds into the $I_{4}$ integral the gauge invariant combination
$a_{\mu}+qA_{\mu}$ of the two gauge fields, with $q=1/2$. Hence,

$I_{4}=-|\psi|^{2}\int\limits_{S^{3}}d^{3}x\,\epsilon_{ijk}(a_{i}+qA_{i})\partial_{j}(a_{k}+qA_{k})\
.$ (140)

The topological Lagrangian density (117) can be related to (IV.4) with

$\displaystyle\int\limits_{-\infty}^{\tau_{0}}d\tau\int\limits_{B^{4}}d^{4}x\,\mathcal{L}_{t}$
$\displaystyle\propto$
$\displaystyle\int\limits_{-\infty}^{\tau_{0}}d\tau\int\limits_{B^{4}}d^{4}x\,\psi^{\dagger}\epsilon_{0ijkl}\partial_{0}\partial_{i}\partial_{j}\partial_{k}\partial_{l}\psi$
$\displaystyle=$
$\displaystyle\int\limits_{B^{4}}d^{4}x\,\psi^{\dagger}\epsilon_{ijkl}\partial_{i}\partial_{j}\partial_{k}\partial_{l}\psi=I_{4}\
,$

so we can use the structure of (140) and (139) to deduce

$\mathcal{L}_{t}=iK_{t}\,j_{\mu}\mathcal{J}_{\mu}$ (141)

in imaginary time, where the combined Hopf index current is

$\mathcal{J}_{\mu}=\frac{1}{q^{2}}\,\epsilon_{\mu\nu\alpha\beta\gamma}\partial_{\nu}(a_{\alpha}+qA_{\alpha})\partial_{\beta}(a_{\gamma}+qA_{\gamma})\
.$ (142)

The full Lagrangian density of non-relativistic particles may be

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\psi^{\dagger}D_{0}\psi+\frac{1}{2m}|D_{i}\psi|^{2}-\mu|\psi|^{2}+u|\psi|^{4}$
$\displaystyle+\frac{1}{12e^{2}}(\epsilon^{\mu\nu\lambda\alpha\beta}\partial_{\alpha}a_{\beta})^{2}+\mathcal{L}_{\textrm{int}}+\mathcal{L}_{t}\
,$

where $D_{\mu}=\partial_{\mu}+ia_{\mu}$. Some additional interactions
$\mathcal{L}_{\textrm{int}}$, such as the spin-orbit coupling or a Kondo
coupling to localized magnetic moments, might be needed to stabilize the
phases with a non-trivial Hopf index dynamics. The topological Lagrangian
density $\mathcal{L}_{t}$ is written in its purely spinor form (117), so there
is no need to include the auxiliary spin-chirality gauge field $A_{\mu}$ in
this Lagrangian; the spinor kinematics automatically implements a topological
Hall or magnetoelectric effect. This theory is a continuum-limit effective
theory where the topological term arises from some microscopic Berry’s phase.
The earlier Lagrangian (IV.1) is the next low-energy approximation obtained
from (IV.4) when the spinor retains a sufficient phase stiffness, i.e. the
particles are not localized. Apart from driving topological order,
$\mathcal{L}_{t}$ also implements a Hopf variant of the topological
magnetoelectric effect. Since $\mathcal{L}_{t}\to 0$ in conventional phases,
the resulting relationship $a_{\mu}+qA_{\mu}\to 0$ neutralizes any spin Hopf
singularity by binding a charge Hopf singularity to it. Then, electrons moving
through a spin-texture background that features interlinked skyrmions would
exhibit charge current flow as if they were moving through the equivalent set
of interlinked magnetic field loops.

The accurate picture of charge fractionalization needs to be derived from the
spinor theory like (IV.4). The charge current of non-relativist particles is
given by (109) with $j^{0}=\rho$. When we extract the U(1) gauge field from
$\psi\sim e^{i\varphi}$ by a singular gauge transformation
$a_{\mu}\to\partial_{\mu}\varphi$, the Lagrangian density variation with
respect to $a_{\mu}$ gives

$j^{\mu}-\frac{1}{e^{2}}\partial_{\nu}f^{\mu\nu}=3K_{t}|\psi|^{2}\mathcal{J}^{\mu}\
.$ (144)

A finite density of Hopf singularities in the topologically ordered ground
state needs to be driven by a Berry curvature field

$\mathcal{B}=\langle\mathcal{J}^{0}\rangle\neq 0$ (145)

which can be generated with a combination of the external electromagnetic
field and spin texture according to (142). This field, and not the particle
density, determines the coupling constant of the topological Lagrangian
density (117)

$K_{t}=\frac{1}{3\mathcal{B}}$ (146)

according to (144) and $j^{0}=\rho=|\psi|^{2}$. At the same time, the Hopf
index of the ground state is quantized

$N=\frac{1}{(4\pi)^{2}}\int\limits_{B^{4}}d^{4}x\,\mathcal{J}^{0}\in\mathbb{Z}$
(147)

and captures the mutual inter-linking of both magnetic and skyrmion flux loops
in three-dimensional manifolds. Since $\mathcal{B}/(4\pi)^{2}$ is the density
of topological defects, the number of particles per topological defect is

$\nu=\frac{|\psi|^{2}}{\mathcal{B}/(4\pi)^{2}}\ ,$ (148)

The imaginary-time topological Lagrangian density of the effective theory can
now be expressed as

$\displaystyle\mathcal{L}_{t}$ $\displaystyle=$ $\displaystyle
i\frac{\nu}{3(2\pi)^{2}}\epsilon_{\mu\alpha\beta\gamma\delta}(\partial_{\mu}\theta+a_{\mu}+qA_{\mu})$
$\displaystyle\quad\times\partial_{\alpha}(a_{\beta}+qA_{\beta})\partial_{\gamma}(a_{\delta}+qA_{\delta})\
.$

This is tailored to the particle density $\rho=|\psi|^{2}$ of a particular
incompressible quantum liquid, since the density determines the filling factor
$\nu$. In contrast, the master theory (IV.4) contains the topological term
(117) whose coupling constant (146) does not depend on the density.

Apart from the Hopf singularities, spinor fields in $d=4$ spatial dimensions
can support monopoles and hedgehogs as topological defects. Hedgehogs in $d=4$
require a spinor representation of the Spin(4) group instead of Spin(3)=SU(2),
but monopoles are derived from the same charge currents as the hopfions. The
topological Lagrangian density

$\mathcal{L}_{t}=-K_{t}\,\psi^{\dagger}\epsilon_{\mu\nu\alpha\beta\gamma}\partial_{\mu}\partial_{\nu}\partial_{\alpha}\partial_{\beta}\partial_{\gamma}\psi$
(150)

is sensitive to all of these topological defects, and all can be handled with
a complex coupling $K_{t}$ in imaginary time. The real part
$\textrm{Re}(K_{t})$ is the coupling for hopfions as discussed here; apart
from generating a current $j_{\mu}$ factor, two partial derivatives are
converted into gauge fields as $\partial_{\mu}\sim ia_{\mu}$. The imaginary
part $\textrm{Im}(K_{t})$ is the coupling for monopoles or hedgehogs; beside
the current factor, three derivatives are converted into the antisymmetric
rank-3 gauge field $\partial_{\mu}\partial_{\nu}\partial_{\lambda}\to
i\mathcal{A}_{\mu\nu\lambda}$ whose flux over a 3-sphere is the enclosed
monopole/hedgehog topological invariant in the $\pi_{3}(S^{3})$ homotopy
group. The mismatch between the collected factors of $i$ is compensated by the
corresponding real and imaginary parts of $K_{t}$ to give the proper (real)
field equations in real time. Note that the symmetry transformations and even
the Hermitian property of (150) depend on the character of extracted
singularities on top of the transformations of $\psi$. It is unlikely that
monopoles and hedgehogs could coexist with Hopf singularities, but the ability
to construct the same topological term for both implies the possibility of
driving phase transitions between different types of topological order by
adjusting the particle density $|\psi|^{2}$. Phase transitions between
incompressible quantum liquids can also be independently driven by the Hopf
singularities of charge or spin currents.

Physical models which could possibly host incompressible quantum liquids of
hopfions need to break the mirror inversion symmetries and remain invariant
under time reversal. This is how the Hopf index, the topological Lagrangian
density, and the driving agent (145) of Hopf topological orders transform,

$\mathcal{B}\xrightarrow{x_{0}\to-
x_{0}}\mathcal{B}\quad,\quad\mathcal{B}\xrightarrow{x_{i}\to-
x_{i}}-\mathcal{B}\ .$ (151)

This is immediately compatible with transformations of the U(1) gauge field
$a_{\mu}$. However, the equivalent transformations of the spin-chirality gauge
field $A_{\mu}$

$\displaystyle A_{0}\xrightarrow{x_{0}\to-x_{0}}A_{0}$
$\displaystyle\quad,\quad$ $\displaystyle A_{0}\xrightarrow{x_{i}\to-
x_{i}}A_{0}$ (152) $\displaystyle A_{j}\xrightarrow{x_{0}\to-x_{0}}-A_{j}$
$\displaystyle\quad,\quad$ $\displaystyle A_{j}\xrightarrow{x_{i}\to-
x_{i}}(-1)^{\delta_{ij}}A_{j}$

impose a constraint on the transformations of the spin vector $\hat{\bf n}$.
The relationship

$\epsilon_{\cdots ij}\partial_{i}A_{j}\sim\epsilon_{\cdots
ij}\epsilon^{abc}\hat{n}^{a}(\partial_{i}^{\phantom{x}}\hat{n}^{b})(\partial_{j}^{\phantom{x}}\hat{n}^{c})$
(153)

implies

$\displaystyle\partial_{i}A_{j}\xrightarrow[i\neq j]{x_{0}\to-
x_{0}}-\partial_{i}A_{j}$ $\displaystyle\quad\Rightarrow\quad$
$\displaystyle\hat{n}^{a}\xrightarrow{x_{0}\to-x_{0}}-\hat{n}^{a}$
$\displaystyle\partial_{i}A_{j}\xrightarrow[i\neq j]{x_{k}\to-
x_{k}}(-1)^{\delta_{ik}+\delta_{jk}}\partial_{i}A_{j}$
$\displaystyle\quad\Rightarrow\quad$
$\displaystyle\hat{n}^{a}\xrightarrow{x_{k}\to-x_{k}}+\hat{n}^{a}$

This is reminiscent of the pseudovector behavior in $d=3$, but the
transformation under mirror inversion is trivial and insensitive to the spin
direction $a$. In fact, the non-trivial $d=3$ pseudovector behavior under
inversion cannot be naturally generalized to $d=4$ because the spin indices
$a\in\\{1,2,3\\}$ cannot be correlated with the spatial directions
$i\in\\{1,2,3,4\\}$.

##  V Conclusions

We developed a field theory description of generic systems whose states can be
characterized by a Hopf index. The basic ingredient is a real or emergent
vector gauge field, generally coupled to matter in $d=3$ or $d=4$ spatial
dimensions. Extensions to higher dimensions are possible but not of interest
in the present study. We observed that electrons can host independent Hopf
invariants in their charge and spin sectors, and that the dynamics of the two
is normally correlated via a type of magnetoelectric effect. Furthermore, the
coupling of matter to an independent gauge field promotes by itself two
distinct Hopf invariants, one per degree of freedom. The conservation of a
Hopf index means that closed loops of vorticity or gauge flux can maintain
their topologically non-trivial pattern of interlinks despite quantum and
thermal fluctuations.

One of our main findings is that a “pseudogap” phase of interlinked loops can
be thermodynamically distinguished from trivial disordered states in $d=3$
dimensions with a generalization of the Wilson loop correlator. This phase can
exist at finite temperatures below a loop-unlinking transition or an
alternative monopole-driven transition in which the loops fall apart. It need
not spontaneously break a symmetry or have topological order. Instead, it
features a sharp suppression of the topological index fluctuations at low
frequencies, which is expected to leave an imprint in the specific heat as
well as the quantum noise of the currents coupled to the Hopf index. The
latter is manifested through the chiral quantum anomaly in systems with a
Dirac or Weyl electron spectrum.

In $d=4$ spatial dimensions, the Hopf index conservation can produce
topological order. Even though it is irrelevant to materials, we characterized
its basic properties as a part of the effort to classify topological orders.
This Hopf topological order can exist only at zero temperature and has many
similarities to the fractional quantum Hall effect. However, it differs from
the Hall effect by the precise topological ground state degeneracy for the
same filling factor. It also features a direct mechanism for angular momentum
fractionalization, due to the ability of fractional charge to move along a
flux loop which is interlinked with another loop. Furthermore, a fractional
braiding statistics is topologically protected only between point-
quasiparticles and 2-sphere excitations. Loop-loop braiding is not quantized.
Interestingly, the classical Hopf index can be conserved on a topologically
trivial manifold at finite temperatures, so additional “pseudogap” states can
exist as in $d=3$ dimensions.

This work is meant only to open the problem of topological Hopf dynamics for
further investigation. Many interesting questions remain. The detailed nature
of the correlated phases with conserved Hopf index is not yet known. We
observed that the lowest-temperature phases of this kind could perhaps protect
some other knot invariants, but this requires a comprehensive further study
because quantum fluctuations have a general tendency to create and annihilate
small loops by tunneling. Field theory can be used to calculate the
correlation functions for charge and spin currents in these phases, and also
characterize the loop-unlinking phase transition. It would be also important
to analyze microscopic models where the loop-unlinking transitions captured by
the field theory can be explicitly demonstrated. In terms of relating to
experiments, we should look for materials which host emergent U(1) gauge
fields and admit correlated low-temperature phases without obvious long-range
order. Correlated superconductors such as cuprates and chiral quantum magnets
such as Pr2Ir2O7 are perhaps currently the best candidates. If strong
interactions can be induced in a Dirac or Weyl semimetal, then such a system
might also be able to realize a Hopf “pseudogap” state and provide a window
into its dynamics via the chiral anomaly.

It is known that various topological states, quantum Hall liquids in
particular Thouless  _et al._ (1982); Niu  _et al._ (1985), can be
equivalently described in real and momentum space. The momentum space
description is built upon a Berry curvature, while the real space description
is conveniently given by a topological Lagrangian term. Both predict the same
response functions. A similar relationship exists for the Hopf invariant.
Presently, we studied the Hopf index dynamics in the real space. The relevant
topological Lagrangian term in $d=3$ spatial dimensions is actually the
$\theta$-term of the chiral quantum anomaly Wang  _et al._ (2023). We also
constructed a more impactful topological term for $d=4$, which leads to a
topological order with charge fractionalization. The equivalent momentum-space
description of hopfion dynamics is found in Hopf insulators Moore  _et al._
(2008); Deng  _et al._ (2013, 2014, 2018); Chang  _et al._ (2017); Volovik and
Zhang (2017); Schuster  _et al._ (2019); Ünal  _et al._ (2019);
Alexandradinata  _et al._ (2021); Lapierre  _et al._ (2021); Graf and Piéchon
(2023); Wang  _et al._ (2023). This suggests another approach to the possible
realizations of the correlated Hopf quantum liquids – by opening an
interaction gap inside a Hopf insulator’s band, and relating the Hopf quantum
liquids to symmetry-protected topological (SPT) phases.

There are also some bigger theoretical questions. An emerging picture from
this study is that topological orders are members of a broader family of
correlated phases which are unified by instanton confinement, i.e. the
conservation of a classical topological index. We developed a basic
renormalization group assessment of instanton confinement in Ref.Nikolic
(2023). However, it is not clear yet if further universal characterization is
possible, apart from the basic notion that delocalized topological defects
play the leading role in the dynamics. Recent studies of superconductors
Nikolić  _et al._ (2013); Nikolić (2014, 2016) and magnets Nikolić (2020b)
with strong spin-orbit coupling have found various mean-field phases with
ordered arrays of topological defects and anti-defects (vortices and
hedgehogs, respectively). Quantum and thermal fluctuations can in principle
melt such lattices of topological defects, while the spin-orbit coupling
protects against a massive defect annihilation. The spin-orbit coupling also
provides an independent “magnetic length” scale, which can define the extent
of short-range coherence for a quantum liquid with delocalized topological
defects. Hence, this is a possible route for obtaining unconventional
“pseudogap” or SPT phases. Instanton confinement here indicates that the total
zero topological charge is conserved. There is evidence Mays and Nikolić
(2023) that such phases could possess a topologically non-trivial excitation
spectrum, featuring nodes in the bulk and gapless states on the system
boundary.

The relationship between the Hopf index and chiral anomaly is also
fundamentally interesting. The QED ground state on $\mathbb{R}^{4}$ is in the
confined-instanton phase with a conserved Hopf index, while no topological
order exists on non-simply connected manifolds. This is a hallmark of Abelian
gauge fields; the action cost $\sim\log(R)$ of an uncompensated instanton
which alters the Hopf index scales at least as a logarithm of the system size
$R$. In contrast, non-Abelian gauge fields allow instantons which evolve into
a pure-gauge configuration at infinite distances, and hence enable a finite
action cost Belavin  _et al._ (1975); ’t Hooft (1976); Schäfer and Shuryak
(1998); ’t Hooft and Bruckmann (2000); Ogilvie (2012). This leads to quark
confinement, but removes the confined-instanton phase. If the spatial manifold
is topologically non-trivial, even the Abelian gauge fields are not able to
hold on to their Hopf index due to vacuum instantons. This translates into a
non-degenerate ground state and the absence of hopfion topological orders in
$d=3$. Nevertheless, the hopfion dynamics of Abelian gauge fields is not-
trivial. The low-temperature confined-instanton phase is a subtle replacement
for topological order, where the particle-flux attachment is replaced with the
attachment of certain particle creation/annihilation events to the equivalent
topological defect events, i.e. instantons. The latter is a quantum anomaly.
We have envisioned a scenario in which the anomaly becomes fractionalized just
like its topological order counterpart, and this idea can be explored further.

##  VI Acknowledgements

This research was partly supported by the Department of Energy, Basic Energy
Sciences, Materials Sciences and Engineering Award DE-SC0019331. Support was
also provided by the Quantum Science and Engineering Center at George Mason
University.

##  Appendix A Topological current conservation from the topological
Lagrangian term

The Lagrangian density with a topological term couples the charge current
$j_{\mu}\sim\partial_{\theta}+a_{\mu}$ to the current $\mathcal{J}_{\mu}$ of
topological defects. Here we show that abundant phase $\theta$ fluctuations
lead to the conservation $\partial_{\mu}\mathcal{J}_{\mu}=0$ of the
topological current. We will work in the continuum limit. The alternative
would be to construct a compact lattice theory with $\theta\in(0,2\pi)$ and
integer-valued topological current $\mathcal{J}_{\mu}$, but this would be
plagued by an inconsistency of the topological coupling $\mathcal{L}_{t}\sim
ij_{\mu}\mathcal{J}_{\mu}$ on the lattice. Since the charge current $j_{\mu}$
lives on the physical lattice and $\mathcal{J}_{\mu}$ lives on the dual
lattice, a topological coupling would have to explicitly break some lattice
symmetry or have a fairly complicated form.

The insight from the continuum limit is still valuable even though we lose
access to Mott insulator phases. Integrating out $\theta$ in the imaginary-
time path integral yields

$\displaystyle\int\mathcal{D}\theta\,e^{-\int
d^{d}x\left(\frac{\kappa_{c}}{2}j_{\mu}j_{\mu}+iK_{d}j_{\mu}\mathcal{J}_{\mu}\right)}$
(154) $\displaystyle\qquad\propto\int\mathcal{D}\theta\,e^{-\int
d^{d}x\left[\frac{\kappa_{c}}{2}(\partial_{\mu}\theta)^{2}+\kappa_{c}(\partial_{\mu}\theta)a_{\mu}+iK_{d}(\partial_{\mu}\theta)\mathcal{J}_{\mu}\right]}$
$\displaystyle\qquad\to\int\mathcal{D}\theta\,e^{\int
d^{d}x\,\left[\frac{\kappa_{c}}{2}\theta\partial_{\mu}\partial_{\mu}\theta-i\,\theta\,K_{d}\partial_{\mu}\mathcal{J}_{\mu}\right]}$
$\displaystyle\qquad=\int\mathcal{D}\theta\,\exp\biggl{\\{}\int
d^{d}x\biggl{[}\frac{1}{2}\Bigl{(}\theta-i(K_{d}\partial_{\mu}\mathcal{J}_{\mu})(\kappa_{c}\partial_{\mu}\partial_{\mu})^{-1}\Bigr{)}$
$\displaystyle\qquad\qquad\qquad\times(\kappa_{c}\partial_{\mu}\partial_{\mu})\Bigl{(}\theta-i(\kappa_{c}\partial_{\mu}\partial_{\mu})^{-1}(K_{d}\partial_{\mu}\mathcal{J}_{\mu})\Bigr{)}$
$\displaystyle\qquad\qquad+(K_{d}\partial_{\mu}\mathcal{J}_{\mu})(\kappa_{c}\partial_{\mu}\partial_{\mu})^{-1}(K_{d}\partial_{\mu}\mathcal{J}_{\mu})\biggr{]}\biggr{\\}}$
$\displaystyle\qquad\propto\exp\left[\int
d^{d}x\,(K_{d}\partial_{\mu}\mathcal{J}_{\mu})(\kappa_{c}\partial_{\mu}\partial_{\mu})^{-1}(K_{d}\partial_{\mu}\mathcal{J}_{\mu})\right]\
.$

We gauged-away $\partial_{\mu}a_{\mu}\to 0$. The indices in different brackets
are independent even if the same symbol is used, and $\theta$ was integrated
out naively on the $(-\infty,\infty)$ interval as appropriate for large
$\kappa_{c}$. We see that the current non-conserving
$\partial_{\mu}\mathcal{J}_{\mu}\neq 0$ fluctuations would be strongly
suppressed in the resulting action if $\kappa_{c}$ were small (the operator
$\partial_{\mu}\partial_{\mu}\to-k^{2}$ has only negative eigenvalues).

The last effective action has a more transparent form in momentum space,

$S_{\textrm{eff}}=\int\frac{d^{d}k}{(2\pi)^{d}}\,\frac{|\Delta({\bf
k})|^{2}}{k^{2}}\ ,$ (155)

where $\Delta({\bf k})$ is the Fourier transform of
$K_{d}\partial_{\mu}\mathcal{J}_{\mu}$. This translates into an interaction
potential $V(\delta r)$

$S_{\textrm{eff}}=\int d^{d}r\,d^{d}r^{\prime}\,\Delta({\bf r})\Delta({\bf
r}^{\prime})\,V(|{\bf r}-{\bf r}^{\prime}|)$ (156)

which naively behaves as

$V(\delta r)\sim\frac{1}{\delta r^{d-2}}$ (157)

by dimensional analysis. Its ultra-violet divergence in the $\delta r\to 0$
limit is the agent that suppresses the local fluctuations of
$\Delta\propto\partial_{\mu}\mathcal{J}_{\mu}$. In practice, the amount of
suppression is limited by an ultra-violet cut-off length $\delta r\to a$. The
continuum limit regime holds as long as we are justified approximating $a\to
0$. Incompressible quantum liquids are still consistently captured in the
continuum limit, requiring the topological current conservation despite the
loss of long-range phase coherence (the stability of such phases is confirmed
by the renormalization group of instanton confinement). If the extremely
abundant fluctuations push the system into a Mott insulator phase, then we are
no longer justified in regarding the cut-off length $a$ small; constructing a
compact lattice theory with a lattice constant $a$ is in order, and the
topological currents gain a new freedom to fluctuate without being conserved.
This amounts to a dual picture featuring the condensation of topological
defects.

##  Appendix B Topological invariant in the $\pi_{2n-1}(S^{n})$ homotopy group

A generalization of the Hopf index to the $\pi_{2n-1}(S^{n})$ homotopy groups
involves a Spin($n+1$) unit-magnitude vector field $\hat{{\bf n}}({\bf
x})=(\hat{n}^{a_{1}},\cdots,\hat{n}^{a_{n+1}})$ which lives in a $d=2n-1$
dimensional sphere manifold ${\bf x}\in S^{2n-1}$. The flux of the vector
field chirality on any $n$-sphere $S^{n}$ embedded in $S^{2n-1}$ is a
quantized topological invariant of the $\pi_{n}(S^{n})$ homotopy group:

$\frac{1}{n!\,S_{n}}\oint\limits_{S^{n}}d^{n}x\,\epsilon_{j_{1}\cdots
j_{n}}\epsilon^{a_{0}\cdots
a_{n}}\hat{n}^{a_{0}}(\partial_{j_{1}}\hat{n}^{a_{1}})\cdots(\partial_{j_{n}}\hat{n}^{a_{n}})\in\mathbb{Z}\
.$ (158)

$S_{n}$ is the “area” of the unit $n$-sphere. In our notation, if the Levi-
Civita tensor has fewer indices than $d$ as in the last formula, it is
understood that all directions spanned by the indices live on the integration
manifold. The standard mathematical notation utilizes the wedge product:

$\frac{1}{n!\,S_{n}}\oint\limits_{S^{n}}\epsilon^{a_{0}\cdots
a_{n}}\hat{n}^{a_{0}}\,(d\hat{n}^{a_{1}})\wedge\cdots\wedge(d\hat{n}^{a_{n}})\in\mathbb{Z}$
(159)

where $d\hat{n}^{a}=dx^{i}\partial_{i}\hat{n}^{a}$, but we will emphasize all
indices here because they are helpful in the bookkeeping of the multiplicity
factors like $n!$. Let us define the chirality tensor

$\displaystyle\chi_{i_{1}\cdots i_{n-1}}$ $\displaystyle=$
$\displaystyle\frac{1}{n!}\epsilon_{i_{1}\cdots i_{n-1}j_{1}\cdots
j_{n}}\epsilon^{a_{0}\cdots a_{n}}$
$\displaystyle\qquad\times\hat{n}^{a_{0}}(\partial_{j_{1}}\hat{n}^{a_{1}})\cdots(\partial_{j_{n}}\hat{n}^{a_{n}})\
.$

Since $|\hat{{\bf n}}|=1$ and $\hat{\bf n}\in S^{n}$, all linearly independent
directions that $\hat{\bf n}$ and its derivatives can span are exhausted in
the chirality expression. Hence,

$\partial_{i_{1}}\chi_{i_{1}\cdots
i_{n-1}}=\cdots=\partial_{i_{n-1}}\chi_{i_{1}\cdots i_{n-1}}=0\ .$ (161)

This can be generally solved by introducing a smooth antisymmetric tensor
gauge field $A_{i_{1}\cdots i_{n-1}}$ of rank $n-1$ in our $d=2n-1$ space:

$\chi_{i_{1}\cdots i_{n-1}}=\epsilon_{i_{1}\cdots i_{n-1}jk_{1}\cdots
k_{n-1}}\partial_{j}A_{k_{1}\cdots k_{n-1}}\ .$ (162)

Now, we set up a system of $n$-dimensional open manifolds $D_{p}$ whose
boundaries are $n-1$ dimensional interlinked “flux loop” manifolds $C_{p}$.
The cross-sections of $C_{p}$ “tubes” are also $n$-dimensional; this is
intentional, and in fact fixes the space dimensionality to $d=2n-1$. The flux
carried by each “loop” is a quantized “current” computed through the $S^{n}$
cross-section of the “loop”,

$I_{p}=n_{p}I_{0}=\frac{1}{(n-1)!}\oint\limits_{S^{n}}d^{n}x\,\hat{\eta}_{i_{1}\cdots
i_{n-1}}\chi_{i_{1}\cdots i_{n-1}}\ ,$ (163)

where

$I_{0}=S_{n}\quad,\quad n_{p}\in\mathbb{Z}\ .$ (164)

The last integral employs another convenient custom notation:
$\hat{\eta}_{i_{1}\cdots i_{n-1}}({\bf x})$ is defined on the integration
manifold ${\bf x}\in S^{n}$ as the unit antisymmetric tensor which is locally
orthogonal to $S^{n}$, pointing “outwards”. The integral (163) is then the
chirality “flux” through the “surface” $S^{n}$, a multi-dimensional
generalization of the electric flux $\oint d{\bf A}\,{\bf E}=\oint
d^{2}x\,\hat{\eta}_{i}E_{i}$ from the $d=3$ Gauss’ law of electrodynamics. The
total “current” passing through the “loop” $C_{p}$ is:

$\displaystyle\sum_{q(C_{p})}I_{q}$ $\displaystyle=$
$\displaystyle\sum_{q(C_{p})}I_{0}n_{q}=\frac{1}{(n-1)!}\int\limits_{D_{p}}d^{n}x\,\hat{\eta}_{i_{1}\cdots
i_{n-1}}\chi_{i_{1}\cdots i_{n-1}}$ (165) $\displaystyle=$
$\displaystyle\oint\limits_{C_{p}}d^{n-1}x\,\epsilon_{k_{1}\cdots
k_{n-1}}A_{k_{1}\cdots k_{n-1}}\ .$

We used Stokes-Cartan theorem in the last step. Next, we multiply this by
$I_{0}n_{p}$ and sum over $p$ (all “loops”), noting that the cross-section
$S^{n}(p)$ of a “loop” is orthogonal to its boundary $C_{p}$,

$\displaystyle\sum_{p,q}I_{0}^{2}n_{p}^{\phantom{x}}n_{q}^{\phantom{x}}=\frac{1}{(n-1)!}\sum_{p}\int\limits_{S^{n}_{\phantom{x}}(p)}\\!\\!d^{n}x\,(\hat{\eta}_{i_{1}\cdots
i_{n-1}}\chi_{i_{1}\cdots i_{n-1}})$
$\displaystyle\qquad\qquad\qquad\times\oint\limits_{C_{p}}d^{n-1}x\,\epsilon_{k_{1}\cdots
k_{n-1}}A_{k_{1}\cdots k_{n-1}}$
$\displaystyle\qquad=\int\\!d^{2n-1}x\,\epsilon_{i_{1}\cdots
i_{n-1}jk_{1}\cdots k_{n-1}}A_{i_{1}\cdots i_{n-1}}\partial_{j}A_{k_{1}\cdots
k_{n-1}}\ .$

Recalling $n_{p},n_{q}\in\mathbb{Z}$ and (164), the last equation reveals the
integer quantization $N\in\mathbb{Z}$ of the Hopf index

$N=\frac{1}{I_{0}^{2}}\\!\oint\limits_{S^{2n-1}}\\!\\!\\!\\!d^{2n-1}x\,\epsilon_{i_{1}\cdots
i_{n-1}jk_{1}\cdots k_{n-1}}A_{i_{1}\cdots i_{n-1}}\partial_{j}A_{k_{1}\cdots
k_{n-1}}$ (166)

This derivation parallels the one from Section II.1. It applies to the spin
sector of the Spin($n+1$) group in $d=2n-1$ spatial dimensions, where the
chirality gauge field is generally a higher-rank tensor derived from the
vector field $\hat{\bf n}$ configuration

$\displaystyle\epsilon_{i_{1}\cdots i_{n-1}jk_{1}\cdots
k_{n-1}}\partial_{j}A_{k_{1}\cdots k_{n-1}}=$ (167)
$\displaystyle\qquad=\frac{1}{n!}\epsilon_{i_{1}\cdots i_{n-1}j_{1}\cdots
j_{n}}\epsilon^{a_{0}\cdots
a_{n}}\hat{n}^{a_{0}}(\partial_{j_{1}}\hat{n}^{a_{1}})\cdots(\partial_{j_{n}}\hat{n}^{a_{n}})\
,$

and $I_{0}=S_{n}$.

The expression (166) reveals a constraint on $n$ which was not apparent
before: if $n$ is odd, then an integration by parts yields $N=-N$, i.e. $N=0$.
Non-trivial integer values of the Hopf index are obtained only for even $n$.
Indeed, the homotopy groups $\pi_{2n-1}(S^{n})$ have an integer-valued
topological invariant, the Hopf index, only for even $n$, with an exception of
$\pi_{1}(S^{1})$ whose integer invariant is not the Hopf index; see Table 1.
Any additional structure of the homotopy invariants is not contained in (166).

$k$ |  $n=2k-1$ |  $n=2k$
---|---|---
$1$ |  $\pi_{1}(S^{1})=\mathbb{Z}$ |  $\pi_{3}(S^{2})=\mathbb{Z}$
$2$ |  $\pi_{5}(S^{3})=\mathbb{Z}_{2}$ |  $\pi_{7}(S^{4})=\mathbb{Z}\times\mathbb{Z}_{12}$
$3$ |  $\pi_{9}(S^{5})=\mathbb{Z}_{2}$ |  $\pi_{11}(S^{6})=\mathbb{Z}$
$4$ |  $\pi_{13}(S^{7})=\mathbb{Z}_{2}$ |  $\pi_{15}(S^{8})=\mathbb{Z}\times\mathbb{Z}_{120}$
Table 1: Homotopy groups of the $\pi_{2n-1}(S^{n})$ kind.

Based on this, we anticipate the existence of topological orders in $d=4k$
($k\in\mathbb{N}$) spatial dimensions, supported by currents that transform
according to a representation of the Spin($2k+1$) group. We will argue in the
next Appendix that charge currents and other types of currents may also be
able to support these generalized Hopf topological orders. Topological defects
characterized by the $\pi_{4k-1}(S^{2k})$ homotopy invariant bind a fractional
amount of charge to form elementary quasiparticles.

##  Appendix C Hopf index of the U(1) gauge field and the axial vector anomaly
in higher dimensions

The general Hopf index (166) is applicable to the charge sector as well. To
begin with, the tensor gauge field $A_{i_{1}\cdots i_{n-1}}$ can directly
describe the special monopoles of a U(1) gauge field $A_{i}$ whose quantized
topological charge $I_{0}=2\pi$ obtains as the “magnetic” (162) flux through
an $n$-sphere submanifold of the $S^{2n-1}$ space,

$\Phi=\frac{1}{(n-1)!}\oint\limits_{S^{n}}d^{n}x\,\hat{\eta}_{i_{1}\cdots
i_{n-1}}\chi_{i_{1}\cdots i_{n-1}}=I_{0}\times\textrm{integer}\ .$ (168)

These special monopoles are singular on $(d-1)-(n-1)=n-1$ submanifolds instead
of being point singularities in the $d=2n-1$ dimensional space. This is
determined by the rank of the gauge field. Nevertheless, the flux quantization
is derived from the lowest rank of $A_{i}$ through a hierarchical construction
Nikolić (2020a). Other kinds of monopole structures, if featured in the
dynamics, may also be able to carry a Hopf index.

Of particular interest are the elementary monopole structures, i.e.
“vortices”. These are $d-2$ singular domains in the $d$ dimensional space.
Their topological charge is quantized in the units of $2\pi$, and obtains from
the curl of the U(1) gauge field $A_{i}$ in the $2$-plane manifold orthogonal
to the singularity. If no other monopole structures are present, then we can
construct the tensor $A_{i_{1}\cdots i_{n-1}}$ only from the curls of
$A_{\mu}$,

$\displaystyle A_{i_{1}\cdots i_{n-1}}$ $\displaystyle=$
$\displaystyle\frac{1}{n!}\epsilon_{i_{1}\cdots i_{n-1}j_{1}\cdots
j_{n}}\epsilon_{k_{1}\cdots k_{n-1}j_{1}\cdots j_{n}}$ $\displaystyle\times
A_{k_{1}}(\partial_{k_{2}}A_{k_{3}})\cdots(\partial_{k_{n-2}}A_{k_{n-1}})\ .$

Note that we are specializing to the cases of even $n$, i.e. odd $n-1$, which
provide a non-trivial Hopf index. Using (162), we find the specialized form of
the flux (168)

$\displaystyle\Phi=\frac{1}{2^{n/2}}\int d^{n}x\,\hat{\eta}_{i_{1}\cdots
i_{n-1}}\epsilon_{i_{1}\cdots i_{n-1}k_{1}\cdots k_{n}}$
$\displaystyle\qquad\qquad\times F_{k_{1}k_{2}}F_{k_{3}k_{4}}\cdots
F_{k_{n-1}k_{n}}\ ,$ (170)

where

$F_{ij}=\partial_{i}A_{j}-\partial_{j}A_{i}\ .$ (171)

The flux quantum $I_{0}$ is different than before and we must compute it from
scratch. Consider a magnetic field with singular structures of quantized flux.
These must be $d-2$ dimensional “vortex” manifolds $V_{p}$ of singular points
in a $d$-dimensional world: an intersection between such a manifold and an
orthogonal $2$-plane is a singular point, so that the line integral of $A_{i}$
on the closed loop on the $2$-plane around the singularity yields $2\pi$. If
the $2$-plane is spanned by the coordinates $x_{i},x_{j}$ and the orthogonal
singular $d-2$ manifold by $x_{k_{1}},\dots,x_{k_{d-2}}$, then

$F_{ij}({\bf
x})=2\pi\sum_{p}\int\limits_{V_{p}}d^{d-2}y\,\epsilon_{ijk_{1}\cdots
k_{d-2}}\delta^{d}({\bf x}-{\bf y})\ .$ (172)

Substituting into (C) tells us that only the $d-n=n-1$ dimensional overlaps of
$n/2$ “vortices” matter; each overlap manifold is orthogonal to all
$k_{1},\dots,k_{n}$, which means it is orthogonal to the integration manifold
of ${\bf x}$. Therefore, the manifold of ${\bf x}$ intersects each $V_{p}$
overlap manifold at a single point. The product of all Dirac
$\delta$-functions integrates out to $1$, and we have

$\displaystyle\Phi=\frac{(2\pi)^{k}}{2^{k}}\sum_{p_{1}\cdots
p_{k}}\;\int\limits_{V_{p_{1}}}\\!d^{d-2}y_{1}\cdots\int\limits_{V_{p_{k}}}\\!\\!d^{d-2}y_{k}\int
d^{n}x$ $\displaystyle\quad\times\hat{\eta}_{i_{1}\cdots
i_{n-1}}\epsilon_{i_{1}\cdots i_{n-1}k_{1}\cdots
k_{n}}\epsilon_{k_{1}k_{2}l_{1}^{1}\cdots
l_{d-2}^{1}}\cdots\epsilon_{k_{n-1}k_{n}l_{1}^{k}\cdots l_{d-2}^{k}}$
$\displaystyle\quad\times\delta^{d}({\bf x}-{\bf y}_{1})\cdots\delta^{d}({\bf
x}-{\bf y}_{k})$ $\displaystyle\quad=(2\pi)^{n/2}(n-1)!\times\textrm{integer}$
(173)

Note $n=2k$. The flux quantum is

$I_{0}=(2\pi)^{n/2}(n-1)!$ (174)

because the Levi-Civita tensors in the last integral make integer multiples of
$2^{n/2}(n-1)!$, due to the $n/2$ pair-wise $k_{i}$ permutations and the
permutations of $i_{1},\dots,i_{n-1}$. The resulting Hopf index (166)

$\displaystyle N^{\prime}$ $\displaystyle=$
$\displaystyle\left[\frac{(n-1)!}{I_{0}}\right]^{2}\\!\\!\oint\limits_{S^{2n-1}}\\!\\!d^{2n-1}x\,\epsilon_{i_{1}\cdots
i_{2n-1}}$ $\displaystyle\qquad\times
A_{i_{1}}(\partial_{i_{2}}A_{i_{3}})\cdots(\partial_{i_{2n-2}}A_{i_{2n-1}})$
$\displaystyle=$
$\displaystyle\frac{2}{(4\pi)^{n}}\\!\\!\oint\limits_{S^{2n-1}}\\!\\!d^{2n-1}x\,\epsilon_{i_{1}\cdots
i_{2n-1}}A_{i_{1}}F_{i_{2}i_{3}}\cdots F_{i_{2n-2}i_{2n-1}}$

is integer-valued.

Unconstrained tensor gauge fields $A_{i_{1}\cdots i_{n-1}}({\bf x})$ split
into equivalence classes which are enumerated by $N\in\mathbb{Z}$ in (166).
However, the decomposition (C) is a constraint on the tensor gauge field. One
may expect that the integers $N^{\prime}$ of (C) enumerate the constrained
equivalence classes by having a quantum $q$ larger than $1$, i.e.
$N^{\prime}/q\in\mathbb{Z}$. Let us examine this by scrutinizing the
symmetries of $N$ and $N^{\prime}$. Since $A_{i_{1}\cdots i_{n-1}}$ is an
antisymmetric tensor, we can always reorder its indices as
$i_{1}<i_{2}<\cdots<i_{n-2}<i_{n-1}$. Any resulting sign change is absorbed by
the equivalent reordering of the indices in the Levi-Civita tensor. If we
denote the ordered set of indices with angle brackets, $\langle i_{1}\cdots
i_{n-1}\rangle$, and generate only non-equivalent sets of indices under
ordering, then (166) becomes:

$\displaystyle N$ $\displaystyle=$
$\displaystyle\left[\frac{(n-1)!}{I_{0}}\right]^{2}\\!\\!\oint\limits_{S^{2n-1}}\\!\\!d^{2n-1}x\,\epsilon_{\langle
i_{1}\cdots i_{n-1}\rangle j\langle k_{1}\cdots k_{n-1}\rangle}$ (176)
$\displaystyle\qquad\qquad\times A_{\langle i_{1}\cdots
i_{n-1}\rangle}\partial_{j}A_{\langle k_{1}\cdots k_{n-1}\rangle}\ .$

Furthermore, we can exchange the $i$ and $k$ index sets; an integration by
parts assures that the value of $N$ does not change. For a canonical ordering,
we can always associate the smallest of the $i_{1},k_{1}$ indices to the gauge
field factor without the derivative:

$\displaystyle N$ $\displaystyle=$ $\displaystyle
2\times\left[\frac{(n-1)!}{I_{0}}\right]^{2}\\!\\!\oint\limits_{S^{2n-1}}\\!\\!d^{2n-1}x\,\epsilon_{\langle\langle
k_{1}\cdots k_{n-1}\rangle j\langle i_{1}\cdots i_{n-1}\rangle\rangle}$ (177)
$\displaystyle\qquad\qquad\times A_{\langle i_{1}\cdots
i_{n-1}\rangle}\partial_{j}A_{\langle k_{1}\cdots k_{n-1}\rangle}\equiv 2Q\ .$

There are no further symmetries to exploit, so the quantity $Q$ represents the
irreducible (essential) information carried by the gauge field which
determines its topological equivalence class. The decomposition (C) has more
symmetry. Looking at the left-hand side of (C), we see that we can freely
permute $n-1$ index pairs of the $\partial_{i}A_{j}$ factors. Furthermore, we
can carry out $n$ different integrations by parts to apply a derivative on the
leading $A_{i_{1}}$ factor. Together, these symmetry transformations are
equivalent to all $n!$ permutations among the indices of the gauge field
factors. Hence, we have:

$\displaystyle N^{\prime}$ $\displaystyle=$ $\displaystyle
n!\times\left[\frac{(n-1)!}{I_{0}}\right]^{2}\\!\\!\oint\limits_{S^{2n-1}}\\!\\!d^{2n-1}x\,\epsilon_{\langle
j(i_{2}i_{3})\cdots(i_{2n-2}i_{2n-1})\rangle}$ (178) $\displaystyle\quad\times
A_{j}(\partial_{i_{2}}A_{i_{3}})\cdots(\partial_{i_{2n-2}}A_{i_{2n-1}})\equiv
n!\,Q^{\prime}\ .$

This exhausts the symmetry of $N^{\prime}$. The quantity $Q^{\prime}$ contains
irreducible information about the constrained field (C) which determines its
equivalence class. Now, both $Q$ and $Q^{\prime}$ are quantized in some units;
the most general Hopf index $N\in\mathbb{Z}$ requires $Q$ to be quantized in
the units of $1/2$. However, $Q^{\prime}$ cannot be equivalently quantized in
the units of $1/n!$. If that quantization were possible, then every
configuration of $A_{i_{1}\cdots i_{n-1}}$ could be smoothly deformed into a
form given by (C) because the two forms would have the same topological index.
If we generate all possible $A_{i_{1}\cdots i_{n-1}}$ configurations, we will
also cover all possible $A_{i}$ configurations by generating them from the
decompositions (C) whenever possible. The period of finding these
decompositions relative to all possible $A_{i_{1}\cdots i_{n-1}}$
configurations matches the relative quantizations of $N^{\prime}$ and $N$.
This period is also reflected in the relationships $N=2Q$ and
$N^{\prime}=n!Q^{\prime}$. Generating the “essential” information
$Q,Q^{\prime}$ at a fixed frequency gives us the frequency of symmetry-
restricted values $N,N^{\prime}$ in the respective symmetry classes. So,
taking the same quantization (frequency) for $Q$ and $Q^{\prime}$ yields:

$Q=\frac{1}{2}q\quad,\quad Q^{\prime}=\frac{1}{2}q\quad\Rightarrow\quad
N=q\quad,\quad N^{\prime}=\frac{n!}{2}q$

with $q\in\mathbb{Z}$. Consequently, we can extract an integer factor
$N^{\prime\prime}\equiv q=(2/n!)N^{\prime}\in\mathbb{Z}$ from (C) which
depends on the field configuration:

$N^{\prime\prime}=\frac{4}{n!\,(4\pi)^{n}}\\!\\!\\!\oint\limits_{S^{2n-1}}\\!\\!\\!\\!d^{2n-1}x\,\epsilon_{i_{1}\cdots
i_{2n-1}}A_{i_{1}}F_{i_{2}i_{3}}\cdots F_{i_{2n-2}i_{2n-1}}$ (179)

This is the irreducible part of the constrained Hopf index. In the light of
the quantum anomaly discussion from Section III.3, one anticipates the non-
conservation of chiral currents in the form of

$\partial_{\mu}j_{c\mu}=\frac{2}{n!\,(4\pi)^{n}}\,\epsilon_{\mu_{i}\cdots\mu_{2n}}F_{\mu_{1}\mu_{2}}F_{\mu_{3}\mu_{4}}\cdots
F_{\mu_{2n-1}\mu_{2n}}\ ,$ (180)

since the $\Gamma_{c}=i^{n-1}\Gamma_{0}\Gamma_{1}\cdots\Gamma_{d}$ matrix in
$j_{c\mu}=\bar{\psi}\Gamma_{\textrm{c}}\Gamma_{\mu}\psi$ is constructed to
have $\pm 1$ eigenvalues (i.e. we match integer changes of the chiral charge
to the integer changes of the Hopf index). This is, in fact, obtained
rigorously as the axial vector anomaly of the quantum electrodynamics
generalized to $d$ dimensions Peskin and Schroeder (1995).

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