11institutetext:  Institute for Solid State Physics, University of Tokyo,
Kashiwa, Chiba 277-8581, Japan 22institutetext:  Kavli Institute for the
Physics and Mathematics of the Universe (WPI),
University of Tokyo, Kashiwa, Chiba 277-8583, Japan 33institutetext: Yukawa
Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan

# Topological terms of (2+1)d flag-manifold sigma models

Ryohei Kobayashi 2 Yasunori Lee 3 Ken Shiozaki 3 Yuya Tanizaki
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We examine topological terms of $(2+1)$d sigma models and their consequences
in the light of classifications of invertible quantum field theories utilizing
bordism groups. In particular, we study the possible topological terms for the
$U(N)/U(1)^{N}$ flag-manifold sigma model in detail. We argue that the Hopf-
like term is absent, contrary to the expectation from a nontrivial homotopy
group $\pi_{3}(U(N)/U(1)^{N})=\mathbb{Z}$, and thus skyrmions cannot become
anyons with arbitrary statistics. Instead, we find that there exist
${N(N-1)\over 2}-1$ types of Chern-Simons terms, some of which can turn
skyrmions into fermions, and we write down explicit forms of effective
Lagrangians.

††preprint: YITP-21-17

##  1 Introduction and Summary

Nonlinear sigma models appear as low-energy effective descriptions in broad
areas of modern physics, ranging from high-energy to condensed-matter physics.
For example, they have allowed systematic analyses of strong-interaction
phenomena Coleman:1969sm ; Callan:1969sn  even before the establishment of
quantum chromodynamics (QCD), while they have also provided useful tools to
study the long-range physics of anti-ferromagnetic spin systems Haldane:1983ru
.

In nonlinear sigma models, Lagrangian consists not only of the usual kinetic
term but also of topological terms. Schematically, the kinetic term can be
written as

$\mathcal{L}_{\mathrm{kin}}=\int
g_{ab}(\sigma)\,\mathrm{d}\sigma^{a}\wedge\star\mathrm{d}\sigma^{b},$ (1)

where $\sigma:M_{d}\to X$ is a map from the spacetime $M_{d}$ to the target
space $X$, and $g_{ab}(\sigma)$ is the metric of $X$, which defines the
effective coupling constant. In practical use, we can add higher-derivative
corrections to this Lagrangian, but (1) is often sufficient for our purpose.

Let us consider what kind of topological terms can be added to this
Lagrangian. According to the standard lore, they are classified by the
homotopy of the target space $\pi_{*}(X)$ Coleman198802 . In the path
integral, we are interested in the configuration with the finite Boltzmann
weight $\exp(-\mathcal{L}_{\mathrm{kin}}[\sigma])<\infty$. Taking the
infinite-volume limit $M_{d}\to\mathbb{R}^{d}$, in order to have a finite
action, the sigma-model field must be constant as it approaches the
infinities,

$\lim_{|x|\to\infty}\sigma(x)=\sigma_{*}\in X.$ (2)

As a result, we can identify the infinities of the spacetime with a single
point, and we may regard $\sigma:S^{d}\to X$. By definition, the topological
nature of such maps is classified by $\pi_{d}(X)$. Based on this kind of
argument, it has been considered that there is a theta term corresponding to
nontrivial $\pi_{d}(X)$, and also that there is a Wess-Zumino term Wess:1971yu
; Witten:1983tw  when $\pi_{d+1}(X)=\mathbb{Z}$ while $\pi_{d}(X)=0$, as it
allows us to extend the sigma-model configuration on $S^{d}$ to a disk
$D^{d+1}$ to define the well-defined phase factor mod $2\pi$.

On the other hand, recent developments have led to more refined understanding
of topological terms. Assuming quantum field theories (QFTs) can be defined on
any spacetime manifolds (with given structures), the principle of locality
requires their partition functions to be consistent under cutting and gluing
of spacetime manifolds. Along this line together with the unitarity,
(exponentiated) topological terms should be regarded as partition functions of
the so-called invertible QFTs Freed:2004yc . As a result, for the case at hand
where the sigma-model target space is $X$, topological terms are classified up
to continuous ones by a group $\mathrm{Inv}_{\text{}}^{d}(X)$ Freed:2006mx ;
Freed:2016rqq , which sits in the middle of the short exact sequence

$0\longrightarrow\mathrm{Ext}(\Omega^{\text{}}_{d}(X),\mathbb{Z})\longrightarrow\mathrm{Inv}_{\text{}}^{d}(X)\longrightarrow\mathrm{Hom}(\Omega^{\text{}}_{d+1}(X),\mathbb{Z})\longrightarrow
0.$ (3)

In addition, ordinary theta angles invisible in
$\mathrm{Inv}_{\text{}}^{d}(X)$ are classified by
$\mathrm{Hom}(\Omega_{d}^{\text{}}(X),U(1))$ along with discrete theta angles
Yonekura:2018ufj . An important point is that these classifications use
bordism groups $\Omega_{\ast}^{\text{}}(X)$111 In this paper, we consider spin
bordism groups $\Omega^{\mathrm{spin}}_{\ast}(X)$ since the underlying
microscopic theories are assumed to contain fermions.  rather than homotopy
groups $\pi_{\ast}(X)$, and correspondingly the traditional arguments relying
on homotopy groups might miss some subtleties. This indeed turns out to be the
case, and one of the examples regarding the Hopf term in 3d $\mathbb{C}P^{1}$
sigma model was recently pointed out by Freed, Komargodski, and Seiberg
Freed:2017rlk . In this paper, we will further explore 3d sigma models with
target spaces $\mathbb{C}P^{N-1}$ and flag manifolds $U(N)/U(1)^{N}$, both of
which can be viewed as a generalization of the simplest $\mathbb{C}P^{1}$
case.222 Generalized (co)homologies including bordism have been also
effectively used to analyze the nonlinear sigma model descriptions of 4d QCD-
like theories Freed:2006mx ; Lee:2020ojw ; Yonekura:2020upo . 333 Analyses on
the sigma-model dynamics from the bordism perspective have also been carried
out in Thorngren:2017vzn .

In passing, we mention another motivation to consider the flag-manifold sigma
models. In $(1+1)$d, the sigma models with flag-manifold target space have
appeared as effective descriptions of the $SU(N)$-analogue of
antiferromagnetic Heisenberg-like chains Bykov:2011ai ; Bykov:2012am ;
Bykov:2015pka ; Lajko:2017wif , and are known to show a rich phase diagram
thanks to various theta terms Tanizaki:2018xto ; Ohmori:2018qza ;
Hongo:2018rpy ; Wamer:2020inf  (see Affleck:2021ypq  for a review). Similarly,
the $(2+1)$d flag-manifold sigma models describe the $2$d quantum $SU(3)$
Heisenberg model on several lattices PhysRevB.88.184430 ; PhysRevB.95.184427 ,
and some of them are equivalent to the Tsunetsugu-Arikawa point tsunetsugu2006
of the $SU(2)$ spin-$1$ model. It is expected that the $SU(N)$ anti-
ferromagnetic Heisenberg systems may be realized with the ultracold atoms
using alkaline earth metals PhysRevLett.92.170403 ; 1367-2630-11-10-103033 ;
gorshkov2010two ; taie20126 ; Zhang1467 ; 0034-4885-77-12-124401 ;
CAPPONI201650 , and thus theoretical predictions on these systems may be able
to be verified by such experiments. Therefore, it is an interesting question
to ask what kind of topological terms can enrich those quantum spin
systems.444Furthermore, $4$d flag-manifold sigma models have appeared in the
study of $SU(N)$ gauge theories Faddeev:1998yz ; Amari:2018gbq , and our
analysis can be used to identify possible theta terms in those models.

Let us summarize our result on the $U(N)/U(1)^{N}$ flag-manifold sigma model,
which shall be discussed in detail in Section 4. Although the homotopy
argument suggests the presence of Hopf-like term because of the nontrivial
homotopy $\pi_{3}(U(N)/U(1)^{N})=\mathbb{Z}$, we conclude that such a term
does not exist since

$\Omega^{\mathrm{spin}}_{3}(U(N)/U(1)^{N})=0$ (4)

for $N\geq 3$, assuming that the spacetime manifold is equipped with spin
structure. We also find that there are ${N(N-1)\over 2}-1$ types of Chern-
Simons-like terms according to the bordism computation, and we try to write
down their explicit forms using the gauged sigma-model description. In such a
description, we introduce $U(1)$ gauge fields $a_{1},\ldots,a_{N-1}$, which
indeed allows us to write down the Chern-Simons coupling, but there still
exists a puzzle. When naively writing down possible Chern-Simons term, we get

$\sum_{I=1}^{N-1}{k_{I}\over 4\pi}a_{I}\mathrm{d}a_{I}+\sum_{I<J}{k_{IJ}\over
2\pi}a_{I}\mathrm{d}a_{J},$ (5)

and there are ${N(N-1)\over 2}$ of them, which overcounts the number of
independent Chern-Simons couplings compared to the bordism computation. It
turns out that there is a nontrivial relation, which allows us to eliminate
one of the off-diagonal Chern-Simons couplings, and we find there exist
globally well-defined $3$-forms, which can be added to the effective
Lagrangian, as a byproduct. For example for $N=3$, the topological terms of
the effective Lagrangian are given as

${k_{1}\over 4\pi}a_{1}\mathrm{d}a_{1}+{k_{2}\over
4\pi}a_{2}\mathrm{d}a_{2}+{h\over 2\pi}a_{12}a_{23}a_{31},$ (6)

where the first two terms are diagonal Chern-Simons terms, and the last one is
a non-quantized topological coupling with $a_{ij}$ defined as (61), while the
off-diagonal Chern-Simons term is absent. In particular, when the diagonal
Chern-Simons terms have odd levels, skyrmions are transformed as fermions
under spacetime rotations.

The rest of the paper is organized as follows. In Section 2, we review the
case of 3d $\mathbb{C}P^{1}$ sigma model. We first directly compute bordism
groups to identify the topological (“Hopf”) term, and then discuss its
consequence for the statistics of skyrmions from the physics point of view,
using gauged realization of the model. In Sections 3 and 4, we further go on
to the case of 3d sigma model with target spaces $\mathbb{C}P^{N-1}$ and flag
manifolds $U(N)/U(1)^{N}$, respectively. Appendix A serves as a supplement
involving some subtleties arising in the flag manifold case. In Appendix B,
various relevant cohomology groups are computed via Leray-Serre spectral
sequence (LSSS). In Appendix C, we review the realization of the Hopf term of
the $\mathbb{C}P^{1}$ model as a partition function of a fermionic invertible
phase Abanov:1999qz .

##  2 Review on the topological term of 3d $\mathbb{C}P^{1}$ model

$3$d $\mathbb{C}P^{1}$ sigma model gives a low-energy description of the $2$d
anti-ferromagnetic quantum spin systems. Because of
$\pi_{3}(\mathbb{C}P^{1})=\mathbb{Z}$ which represents the Hopf fibration, it
has been suspected for more than three decades that the model accommodates a
topological theta term, which can be schematically written as

$\mathrm{i}\theta\,\mathrm{Hopf}[n].$ (7)

The paper by Wilczek and Zee Wilczek:1983cy  has shown that the (magnetic)
skyrmion acquires the fractional spin $\theta/2\pi$ and becomes an anyon due
to this Hopf term. The Hopf term does not have a local expression in the
original spin fields, but can be written as a Chern-Simons form using the
$U(1)$ gauge field in the gauged sigma-model description Wu:1984kd .

Meanwhile, starting from the $(2+1)$d $SU(2)$ anti-ferromagnetic quantum
Heisenberg spin system, it is known that the Hopf term does not appear in the
low-energy theory Haldane:1988zz . Instead, the discrete value of the theta
term $\theta=\pi$ arises when the spin variable is coupled to fermions by
Yukawa interaction, after integrating out fermions Abanov:1999qz .

In a recent paper Freed:2017rlk , it has been elucidated that only these
values, $\theta=0$ and $\theta=\pi$, are consistent as local and unitary QFTs.
Indeed, $k=\theta/\pi$ behaves as the level of $U(1)$ spin Chern-Simons term,
and the magnetic skyrmion becomes fermion if $k$ is odd.

In the following of this section, we will give a review on these facts, as it
is a basic ingredient in order to extend these results for general flag-
manifold sigma models.

###  2.1 Formal and explicit descriptions of topological terms of
$\mathbb{C}P^{1}$

Here, we first give a formal description on the possible topological terms in
the $\mathbb{C}P^{1}$ sigma model, and then write down the explicit forms for
the topological term. Unless explicitly stated, we always assume that the
spacetime manifolds are equipped with spin structure.

As briefly explained in Sec. 1, while topological terms have been
conventionally classified using homotopy, the correct framework to be adopted
is bordism rather than homotopy when we require the generalized locality (i.e.
consistency under cutting and gluing rules) for QFTs, so that they can be
defined on any manifolds beyond spheres. Accordingly, topological terms of the
$3$d $\mathbb{C}P^{1}$ sigma model should be captured by spin bordism groups
$\Omega^{\mathrm{spin}}_{*}(\mathbb{C}P^{1})$ from this modern point of
view555 Strictly speaking, the first two statements are established as
theorems, while the last statement is still better to be regarded as a
conjecture. Note, however, that since it can be viewed as a formal way to
present the Stora-Zumino descent equation Stora:1983ct ; Zumino:1983ew , the
authors think it quite reasonable. :

  * •

$U(1)$ factors in
$\mathrm{Hom}(\Omega^{\text{spin}}_{3}(\mathbb{C}P^{1}),U(1))$ classifies the
ordinary theta angles Yonekura:2018ufj ,

  * •

$\mathrm{Ext}(\Omega^{\text{spin}}_{3}(\mathbb{C}P^{1}),\mathbb{Z})\simeq\mathrm{Tors}(\Omega^{\text{spin}}_{3}(\mathbb{C}P^{1}))$
classifies the discrete theta angles Freed:2006mx ; Freed:2016rqq ,

  * •

$\mathrm{Hom}(\Omega^{\text{spin}}_{4}(\mathbb{C}P^{1}),\mathbb{Z})\simeq\mathrm{Free}(\Omega^{\mathrm{spin}}_{4}(\mathbb{C}P^{1}))$
classifies the Chern-Simons or Wess-Zumino terms Freed:2006mx ; Freed:2016rqq
.

These groups can be computed using the Atiyah-Hirzebruch spectral sequence
(AHSS)

$E^{2}_{p,q}=H_{p}(\mathbb{C}P^{1};\Omega^{\mathrm{spin}}_{q}(\mathrm{pt}))\Longrightarrow\Omega^{\mathrm{spin}}_{p+q}(\mathbb{C}P^{1}),$
(8)

where the $E^{2}$-page on the left hand side converges to the desired group on
the right hand side. The explicit form of the $E^{2}$-page is given as
follows:

$\sseqpage[homologicalSerregrading,grid=chess,classes={draw=none},yscale=0.7,xlabel={H_{p}(\mathbb{C}P^{1})},ylabel={\Omega^{\mathrm{spin}}_{q}(\mathrm{pt})}]\class["\mathbb{Z}"](0,0)\class["\mathbb{Z}_{2}"](0,1)\class["\mathbb{Z}_{2}"](0,2)\class["\mathbb{Z}"](0,4)\class["\mathbb{Z}"](2,0)\class["\mathbb{Z}_{2}"](2,1)\class["\mathbb{Z}_{2}"](2,2)\class["\mathbb{Z}"](2,4)\d{[}dashed]2(2,1)\class[""](3,0)\class[""](4,0)$
(9)

It is known that, if the space $X$ is connected, bordism groups split as

$\Omega_{\ast}^{\text{spin}}(X)=\Omega_{\ast}^{\text{spin}}(\mathrm{pt})\oplus\widetilde{\Omega}_{\ast}^{\text{spin}}(X),$
(10)

and therefore differentials going into the $p=0$ column (e.g. the dashed
arrow) are all trivial. As a result,
$E^{2}_{2,1}=H_{2}(\mathbb{C}P^{1};\Omega^{\mathrm{spin}}_{1}(\mathrm{pt}))=\mathbb{Z}_{2}$
survives to the $E^{\infty}$-page, and one has

$\begin{array}[]{ccc}\mathrm{Free}(\Omega_{3}^{\text{spin}}(\mathbb{C}P^{1}))&=&0,\\\
\mathrm{Tors}(\Omega_{3}^{\text{spin}}(\mathbb{C}P^{1}))&=&\mathbb{Z}_{2},\\\
\mathrm{Free}(\Omega_{4}^{\text{spin}}(\mathbb{C}P^{1}))&=&\mathbb{Z}.\end{array}$
(11)

The free part of $\Omega_{4}^{\mathrm{spin}}(\mathbb{C}P^{1})$ comes from
$\Omega_{4}^{\mathrm{spin}}(\mathrm{pt})$ describing the gravitational Chern-
Simons term, which is not of our interest; our interest will only be in
$\widetilde{\Omega}_{4}^{\text{spin}}(\mathbb{C}P^{1})$. The Hopf term with
continuous $\theta\in\mathbb{R}/2\pi\mathbb{Z}$ parameter is in fact absent as
the free part of $\Omega_{3}^{\mathrm{spin}}(\mathbb{C}P^{1})$ is trivial. We
identify the torsion part of $\Omega_{3}^{\mathrm{spin}}(\mathbb{C}P^{1})$ as
the “Hopf” term of 3d $\mathbb{C}P^{1}$ model, but its theta angle should be
discretized as it is $\mathbb{Z}_{2}$-valued.

Let us move on to explicit computations in order to write down the discrete
theta term. We can realize the $\mathbb{C}P^{1}$ sigma model as a $U(1)$ gauge
theory according to the fibration $U(1)\to SU(2)\to\mathbb{C}P^{1}$, that is,
we introduce the $SU(2)$-valued scalar field $U(x)$ and the $U(1)$ gauge field
$a=a_{\mu}\mathrm{d}x^{\mu}$. We note that $SU(2)$ matrix $U$ can be written
as

$U=\begin{pmatrix}z_{1}&-z_{2}^{*}\\\ z_{2}&z_{1}^{*}\end{pmatrix},$ (12)

with $|z_{1}|^{2}+|z_{2}|^{2}=1$, and thus the $SU(2)$ scalar field is
equivalent to the two-component complex scalar field
$z=(z_{\alpha})_{\alpha=1,2}$ with a unit norm. Assigning $U(1)$ gauge charge
$1$ to this scalar $z$, the kinetic term in the Lagrangian is given by

${1\over g^{2}}|(\partial_{\mu}+\mathrm{i}a_{\mu})z|^{2}={1\over
g^{2}}\left(|\partial_{\mu}z|^{2}+\mathrm{i}a_{\mu}(-z^{\dagger}\partial_{\mu}z+(\partial_{\mu}z^{\dagger})z)+a_{\mu}^{2}\right).$
(13)

Since this is quadratic in $a_{\mu}$, the equation of motion can be solved as

$a=\mathrm{i}z^{\dagger}\mathrm{d}z.$ (14)

With this $U(1)$ gauge field, we define the Hopf term as the Chern-Simons
term, following the idea of Wu:1984kd . That is, we extend the spacetime
manifold $M_{3}$ to the $4$d spin manifold $W_{4}$ with $\partial W_{4}=M_{3}$
and define

$\exp\Bigl{(}\mathrm{i}(\pi
k)\mathrm{Hopf}[M_{3},(z,a)]\Bigr{)}:=\exp\left(\mathrm{i}{k\over
4\pi}\int_{W_{4}}\mathrm{d}\tilde{a}\wedge\mathrm{d}\tilde{a}\right),$ (15)

where $\tilde{a}$ is the $4$d extension of $a$ as $U(1)$ gauge fields.

We shall give concrete computations to show that this gives the correct
discrete theta term in the next subsection. Before that, let us make several
remarks, which would be confusing at first sight.

In the definition (15), we only extend the $U(1)$ gauge field $a$, while the
$\mathbb{C}P^{1}$ field needs not be extended to $W_{4}$, and therefore (14)
holds only on the boundary $M_{3}$ but not on the $4$d bulk.666In the work of
Abanov Abanov:2000ea , the similar idea has been used to find the Hopf term by
integrating out the fermion when the spin variable
$\bm{n}=z^{\dagger}\bm{\sigma}z$ couples to it. Instead of directly evaluating
the fermion determinant, they consider an infinitesimal variation of the spin
variable, and try to integrate it to obtain the result. Within the
$\mathbb{C}P^{1}$ model, however, the Hopf term is a total derivative
Wu:1984kd , and the infinitesimal variation identically vanishes. They
circumvented this issue by embedding $\mathbb{C}P^{1}\subset\mathbb{C}P^{N-1}$
with $N>2$. As $\mathrm{B}U(1)=\mathbb{C}P^{\infty}$, our prescription (15)
can be formally regarded as the $N\to\infty$ limit of it. We note that
$\Omega^{\mathrm{spin}}_{3}(\mathbb{C}P^{1})=\mathbb{Z}_{2}$ shows that the
$\mathbb{C}P^{1}$ field may not have a $4$d extension when the spin structure
is taken into account: Only after taking the double copy, $M_{3}\sqcup M_{3}$,
we can take the $4$d bounding manifold $M^{\prime}_{4}$ such that the
extension of the $\mathbb{C}P^{1}$ field is possible.

Next, let us prove that the Hopf term defined in (15) is a bordism invariant
of $3$d spin manifolds equipped with a map $\sigma=(z,a)$ to
$\mathbb{C}P^{1}$. To see this, suppose the pair $(M_{3},\sigma)$ is bordant
to $(M_{3}^{\prime},\sigma^{\prime})$, i.e., there exists a 4d spin manifold
$Y_{4}$ and a map $\widetilde{\sigma}:Y_{4}\to\mathbb{C}P^{1}$, such that
$\partial Y_{4}=M_{3}\sqcup\overline{M^{\prime}_{3}}$ and

$\widetilde{\sigma}\bigr{|}_{M_{3}}=\sigma,\quad\widetilde{\sigma}\bigr{|}_{\overline{M^{\prime}_{3}}}=\sigma^{\prime}.$
(16)

Then we have

$\displaystyle\frac{\exp\Bigl{(}{\mathrm{i}(\pi
k)\mathrm{Hopf}[M_{3},\sigma]}\Bigr{)}}{\exp\Bigl{(}{\mathrm{i}(\pi
k)\mathrm{Hopf}[M^{\prime}_{3},\sigma^{\prime}]}\Bigr{)}}=\exp\left(\mathrm{i}{k\over
4\pi}\int_{Y_{4}}\mathrm{d}\tilde{a}\wedge\mathrm{d}\tilde{a}\right)=1.$ (17)

Here, unlike in the definition (15), the $4$d extension to $Y_{4}$ is taken as
a $\mathbb{C}P^{1}$ field, and thus the $U(1)$ gauge field is subject to the
constraint (14), $\tilde{a}=\mathrm{i}\tilde{z}^{\dagger}\mathrm{d}\tilde{z}$.
As a result, $\mathrm{d}\tilde{a}\wedge\mathrm{d}\tilde{a}=0$ identically.
Hence, we can see that

$\displaystyle\exp\Bigl{(}{\mathrm{i}(\pi
k)\mathrm{Hopf}[M_{3},\sigma]}\Bigr{)}=\exp\Bigl{(}{\mathrm{i}(\pi
k)\mathrm{Hopf}[M^{\prime}_{3},\sigma^{\prime}]}\Bigr{)},$ (18)

which means that the Hopf term (15) gives a bordism invariant.

###  2.2 Statistics of skyrmions

Let us illustrate how the level $k$ of the Hopf term is quantized. Since the
3d spin bordism group equipped with a map to $\mathbb{C}P^{1}$ is given by
$\Omega_{3}^{\mathrm{spin}}(\mathbb{C}P^{1})=\mathbb{Z}_{2}$, the Hopf term
must take a $\mathbb{Z}_{2}$ valued phase,
$\exp\Bigl{(}{\mathrm{i}\pi\,\mathrm{Hopf}[M_{3},\sigma]}\Bigr{)}=\pm 1$.
Therefore, we should identify

$k\sim k+2,$ (19)

in the path-integral weight, $\exp\Bigl{(}{\mathrm{i}(\pi
k)\mathrm{Hopf}[M_{3},\sigma]}\Bigr{)}$. This shows that we can verify the
definition (15) of the Hopf term by constructing a $\mathbb{C}P^{1}$
configuration $\sigma$ on a $3$-manifold $M_{3}$ which gives
$\exp\Bigl{(}{\mathrm{i}\pi\mathrm{Hopf}[M_{3},\sigma]}\Bigr{)}=-1$ for $k=1$.

We can construct such a configuration by taking the spacetime $M_{3}$ to be
$S^{2}\times S^{1}$, introducing a magnetic skyrmion on the space $S^{2}$, and
then performing the adiabatic $2\pi$ rotation along the imaginary time
$S^{1}$. This is exactly the way to measure the statistics of the skyrmion in
a semi-classical manner Wilczek:1983cy ; Wu:1984kd . A skyrmion can be put on
$S^{2}$ by setting

$z(\theta,\phi)=\begin{pmatrix}\cos{\theta\over 2}\\\
\mathrm{e}^{\mathrm{i}\phi}\sin{\theta\over 2}\end{pmatrix},$ (20)

where $(\theta,\phi)$ is the spherical coordinate of $S^{2}$ with
$0\leq\theta\leq\pi$ and $0\leq\phi<2\pi$. We can perform the adiabatic $2\pi$
rotation along the imaginary time $0\leq\tau<2\pi$ as

$z(\tau,\theta,\phi):=\mathrm{e}^{\mathrm{i}\tau/2}\exp\left(\mathrm{i}{\tau\over
2}\sigma_{3}\right)z(\theta,\phi)=\begin{pmatrix}\mathrm{e}^{\mathrm{i}\tau}\cos{\theta\over
2}\\\ \mathrm{e}^{\mathrm{i}\phi}\sin{\theta\over 2}\end{pmatrix}.$ (21)

Here, $\exp\left(\mathrm{i}{\tau\over 2}\sigma_{3}\right)$ describes the
adiabatic rotation, and we further multiply the overall $U(1)$ phase
$\mathrm{e}^{\mathrm{i}\tau/2}$ so that the configurations at $\tau=0$ and
$\tau=2\pi$ are in the same gauge. We then obtain the $U(1)$ gauge field $a$
on $S^{2}\times S^{1}$ from (21) as

$a=-\cos^{2}{\theta\over 2}\,\mathrm{d}\tau-\sin^{2}{\theta\over
2}\,\mathrm{d}\phi.$ (22)

In order to evaluate (15), we need to extend (22) from $S^{2}\times S^{1}$ to
a $4$-manifold. We can take $W_{4}=S^{2}\times D^{2}$ as an extension, where
the polar coordinate of $D^{2}$ is given by $(\rho,\tau)$ and the extension of
$a$ is

$\tilde{a}=-\cos^{2}{\theta\over
2}\,\rho^{2}\mathrm{d}\tau-\sin^{2}{\theta\over 2}\,\mathrm{d}\phi.$ (23)

We can readily check that

${1\over 4\pi}\int_{S^{2}\times
D^{2}}\mathrm{d}\tilde{a}\wedge\mathrm{d}\tilde{a}=\pi,$ (24)

which confirms that $\exp\left(\mathrm{i}\pi\mathrm{Hopf}[S^{2}\times
S^{1},(z,a)]\right)=-1$. This means that the skyrmion becomes fermionic due to
the Hopf term when we set $k=1$.

One should be able to confirm the statistics of the skyrmion also by checking
the statistics of its creation/annihilation operator. Since a single skyrmion
in the space $S^{2}$ introduces the configuration of the $U(1)$ gauge field
(22), we identify its creation/annihilation operator as the monopole operator
$m(x)$ for the $U(1)$ gauge field. The monopole operator $m(x)$ is defined as
a defect operator. Roughly speaking, we remove a point $x$ from the spacetime
$3$-manifold $M_{3}$, and require the $U(1)$ gauge field $a$ to behave as

$\int_{S^{2}_{x}}{1\over 2\pi}\mathrm{d}a=\pm 1,$ (25)

where $S^{2}_{x}$ is a sufficiently small two-sphere surrounding the removed
point $x$. When the monopole operator $m(x)$ is inserted, we assert that the
path integral is performed over the gauge fields with this boundary condition.

We note that the monopole operator is not $U(1)$ gauge invariant when the
Chern-Simons term exists Pisarski:1986gr ; Affleck:1989qf : Under the gauge
transformation, $a\mapsto a+\mathrm{d}\varepsilon$ and
$z\mapsto\mathrm{e}^{-\mathrm{i}\varepsilon}z$,

$m(x)\mapsto\mathrm{e}^{\mathrm{i}\,k\,\varepsilon(x)}m(x).$ (26)

As a result, the gauge-invariant monopole operator should be defined as

$m(x)z_{\alpha_{1}}(x)\cdots z_{\alpha_{k}}(x).$ (27)

The quantum number of this operator should be identical with that of the
corresponding skyrimon. In order to find the spin of the gauge-invariant
monopole operator, note that the spherical coordinate dependence of the scalar
field $z$ is described by the monopole harmonics, $Y_{q,\ell,m}$, near the
monopole singularity Borokhov:2002ib . For the minimal monopole charge,
$q=1/2$, and the allowed values of $\ell$ are $\ell\in|q|+\mathbb{Z}_{\geq 0}$
Wu:1976ge ; Wu:1977qk . Therefore, the spin of the monopole operator is given
by $k/2$ mod $\mathbb{Z}$, which is consistent with the above computations.

##  3 Topological term of 3d $\mathbb{C}P^{N-1}$ sigma model with $N\geq 3$

Let us move on to the $3$d $\mathbb{C}P^{N-1}$ nonlinear sigma model, and
consider its topological terms. Even though the expressions for the
topological terms of $\mathbb{C}P^{1}$ and $\mathbb{C}P^{N-1}$ are
superficially similar, we shall emphasize the difference between them in this
section. This understanding turns out to be useful to find the correct
description of topological terms for the flag-manifold model in the next
section.

###  3.1 Conflict between homotopy argument and the gauged-sigma model

The $\mathbb{C}P^{N-1}$ sigma model can be realized using a
$\mathbb{C}^{N}$-valued scalar field $z=(z_{\alpha})_{\alpha=1,\ldots,N}$ with
a unit norm, with a $U(1)$ gauge field $a$. This corresponds to the fibration

$S^{1}\to S^{2N-1}\to\mathbb{C}P^{N-1},$ (28)

which is identical with the Hopf fibration for $N=2$. In the following, we
consider the case with $N\geq 3$. As a consequence of the homotopy exact
sequence, for $k\geq 3$,

$\pi_{k}(S^{2N-1})=\pi_{k}(\mathbb{C}P^{N-1})$ (29)

as $\pi_{k}(S^{1})=\pi_{k-1}(S^{1})=0$, and we immediately find that

$\pi_{3}(\mathbb{C}P^{N-1})=\pi_{4}(\mathbb{C}P^{N-1})=0.$ (30)

Therefore, in the conventional view of the homotopy, we do not expect the $3$d
$\mathbb{C}P^{N-1}$ model to have any topological terms, neither theta terms
nor Wess-Zumino terms.

However, if we regard the $\mathbb{C}P^{N-1}$ model as an example of $U(1)$
gauge-Higgs systems, we can write down a Chern-Simons coupling for the $U(1)$
gauge field,

${\mathrm{i}k\over
4\pi}\int_{W_{4}}\mathrm{d}\tilde{a}\wedge\mathrm{d}\tilde{a},$ (31)

where $W_{4}$ is a four-dimensional extension of the spacetime $3$-manifold
along with the spin structure. We would like to resolve the discrepancy
between these two arguments, and also find the physical consequence of the
topological terms.

###  3.2 Spin bordism of $\mathbb{C}P^{N-1}$ and the topological term

To find possible topological terms, we again consider AHSS

$E^{2}_{p,q}=H_{p}(\mathbb{C}P^{N-1};\Omega^{\mathrm{spin}}_{q}(\mathrm{pt}))\Longrightarrow\Omega^{\mathrm{spin}}_{p+q}(\mathbb{C}P^{N-1}).$
(32)

The $E^{2}$ page is given by the following figure:

$\sseqpage[homologicalSerregrading,grid=chess,classes={draw=none},yscale=0.7,xlabel={H_{p}(\mathbb{C}P^{N-1})},ylabel={\Omega^{\mathrm{spin}}_{q}(\mathrm{pt})}]\class["\mathbb{Z}"](0,0)\class["\mathbb{Z}_{2}"](0,1)\class["\mathbb{Z}_{2}"](0,2)\class["\mathbb{Z}"](0,4)\class["\mathbb{Z}"](2,0)\class["\mathbb{Z}_{2}"](2,1)\class["\mathbb{Z}_{2}"](2,2)\class["\mathbb{Z}"](2,4)\class["\mathbb{Z}"](4,0)\class["\mathbb{Z}_{2}"](4,1)\class["\mathbb{Z}_{2}"](4,2)\class["\mathbb{Z}"](4,4)\class["\mathbb{Z}"](6,0)\class["\mathbb{Z}_{2}"](6,1)\class["\mathbb{Z}_{2}"](6,2)\class["\mathbb{Z}"](6,4)\d{2}(4,0)\d{2}(4,1)$
(33)

where the homology of $\mathbb{C}P^{N-1}$ is

$H_{*}(\mathbb{C}P^{N-1})=\left\\{\begin{array}[]{ccc}\mathbb{Z}&\quad*=0,2,\ldots,2(N-1),\\\
0&\quad\mathrm{else},\end{array}\right.$ (34)

(see Appendix B.1 to confirm this result from the Leray-Serre spectral
sequence (LSSS)). Important difference from the $N=2$ case is that
$H_{4}(\mathbb{C}P^{N-1})=\mathbb{Z}$ for $N\geq 3$ and this gives rise to a
possibly-nontrivial differential $d^{2}:E^{2}_{4,0}\to E^{2}_{2,1}$. This map
is known to be given by a dual of Steenrod square composed with mod $2$
reduction Teichner:MR1214960 , and here the generator $x\in
H^{2}(\mathbb{C}P^{N-1};\mathbb{Z}_{2})$ is mapped to
$\mathrm{Sq}^{2}(x)=x\cup x\in H^{4}(\mathbb{C}P^{N-1};\mathbb{Z}_{2})$.
Correspondingly $d^{2}$ is nontrivial, and thus $E^{2}_{2,1}=\mathbb{Z}_{2}$
is eliminated in the $E^{\infty}$-page, leading to

$\Omega^{\mathrm{spin}}_{3}(\mathbb{C}P^{N-1})=0\qquad(N\geq 3).$ (35)

Therefore, (not only the ordinary ones but also) the discrete theta angles do
not exist in the $3$d $\mathbb{C}P^{N-1}$ sigma model for $N\geq 3$. Instead,
we have

$\widetilde{\Omega}^{\mathrm{spin}}_{4}(\mathbb{C}P^{N-1})=2\mathbb{Z}$ (36)

which is nothing but the Chern-Simons coupling (31) in our description. Here,
the map $H_{4}(\mathbb{C}P^{N-1};\mathbb{Z})=\mathbb{Z}\to
2\mathbb{Z}\subset\Omega^{\mathrm{spin}}_{4}(\mathbb{C}P^{N-1})$ is a
multiplication by two,777We would like to note that this factor $2$ appears
due to the spin structure of the spacetime. When we only assume the
orientation structure, we need to compute the oriented bordism, instead. As
(reduced) oriented bordism is exactly identical to the homology at low degrees
without this extra factor, we can classify the topological terms using
ordinary (co)homology in such cases Davighi:2020vcm . Use of generalized
(co)homology allows us to deal with the subtle factor related to the spacetime
structure.  since the latter is the kernel of $d^{2}:E^{2}_{4,0}\to
E^{2}_{2,1}$. To be more explicit,888We thank the anonymous referee for
suggestion of giving the following explicit explanation.  we can take
$\mathbb{C}P^{2}\subset\mathbb{C}P^{N-1}$ as the generator of
$H_{4}(\mathbb{C}P^{N-1},\mathbb{Z})$, which satisfies
$\int_{\mathbb{C}P^{2}}\left({\mathrm{d}a\over 2\pi}\right)^{2}=1$. On spin
$4$-manifolds $X$, however, the Atiyah-Singer index theorem tells that
$\int_{X}\left({\mathrm{d}a\over 2\pi}\right)^{2}$ is always an even integer,
so there is a factor-$2$ difference between the integral homology and the spin
bordism of degree $4$.

Let us make several remarks. Although the definition (31) looks very similar
to the one (15) in the $\mathbb{C}P^{1}$ model, there are important
differences. As $\Omega^{\mathrm{spin}}_{3}(\mathbb{C}P^{N-1})=0$ for $N\geq
3$, the $\mathbb{C}P^{N-1}$ configurations in $3$ dimensions are always null-
bordant within the spin manifold. Therefore, unlike in the case of
$\mathbb{C}P^{1}$, we can take the $4$d extension of the fields
$(\tilde{z},\tilde{a})$ as $\mathbb{C}P^{N-1}$ fields in the definition (31).
A related remark is that, for $\mathbb{C}P^{N-1}$ models, we do not have any
identifications on $k$, although the Hopf tem for the $\mathbb{C}P^{1}$ model
has (19). For such an identification to exist, the phase factor
$\exp\left({\mathrm{i}\over 4\pi}\int a\mathrm{d}a\right)$ needs to be
quantized. This indeed occurs for the $\mathbb{C}P^{1}$ model because
$\mathrm{d}\tilde{a}\wedge\mathrm{d}\tilde{a}=0$ when $\tilde{a}$ is
restricted to $\tilde{a}=\mathrm{i}\tilde{z}^{\dagger}\mathrm{d}\tilde{z}$.
However, for $\mathbb{C}P^{N-1}$ with $N\geq 3$, one can easily check that
$\mathrm{d}\tilde{a}\wedge\mathrm{d}\tilde{a}\not=0$ even with the constraint
$\tilde{a}=\mathrm{i}\tilde{z}^{\dagger}\mathrm{d}\tilde{z}$, so such
quantization of phases does not occur. This explains the fact that the
topological term (31) comes from
$\Omega^{\mathrm{spin}}_{4}(\mathbb{C}P^{N-1})$, not from
$\Omega^{\mathrm{spin}}_{3}(\mathbb{C}P^{N-1})$, in an intuitive way.

###  3.3 Statistics of skyrmions

Let us discuss the statistics of skyrmions. We add the level-$1$ Chern-Simons
coupling to $\mathbb{C}P^{N-1}$ sigma model for $N\geq 3$.

In order to discuss the quantum number of skyrmions, we can look at the
statistics of corresponding monopole operators, as we have briefly explained
in Sec. 2.2. Because of the Chern-Simons coupling (31), the naive monopole
operator is not $U(1)$ gauge invariant, and the gauge-invariant monopole
operator is given as

$m(x)z_{\alpha}(x).$ (37)

Therefore, the skyrmion is in the defining representation of $SU(N)$ flavor
symmetry. In order to discuss its spin, we again note that the $z$ field in
the vicinity of the monopole singularity should be decomposed with the
monopole harmonics $Y_{1/2,\ell,m}$. Thus, the $z$ field has a half-integer
spin, and so does the monopole operator $m(x)z_{\alpha(x)}$. As a result, the
corresponding skyrmion is a fermion.

Let us confirm this result by directly evaluating the topological term (31)
for the adiabatic $2\pi$-rotation of a skyrmion. To be specific, let us set
$N=3$ and put a $\mathbb{C}P^{1}$ skyrmion configuration in the first two
components of $\mathbb{C}P^{2}$ fields:

$z(\theta,\phi)=\begin{pmatrix}\cos{\theta\over 2}\\\
\mathrm{e}^{\mathrm{i}\phi}\sin{\theta\over 2}\\\ 0\end{pmatrix}.$ (38)

Here, we take the spherical coordinate of the space $S^{2}$, and make $\theta$
and $\phi$ run over $[0,\pi]$ and $[0,2\pi)$ respectively. Then, we can
realize the $2\pi$ rotation of this skyrmion by multiplying the following
$SU(3)$ matrix,

$\tilde{z}(\rho,\alpha,\theta,\phi)=\begin{pmatrix}1&0&0\\\
0&\rho\,\mathrm{e}^{\mathrm{i}\alpha}&-\sqrt{1-\rho^{2}}\\\
0&\sqrt{1-\rho^{2}}&\rho\,\mathrm{e}^{-\mathrm{i}\alpha}\end{pmatrix}z(\theta,\phi).$
(39)

Here, $\alpha\sim\alpha+2\pi$ is regarded as the imaginary-time circle at
$\rho=1$, and it is extended to the two-dimensional disk $D^{2}$. Importantly,
unlike the case of $\mathbb{C}P^{1}$ model, we here take a 4d extension of
both the $U(1)$ gauge field and the scalar field.999 We note that the same is
true for skyrmions of $4$d $N_{f}$-flavor QCD; for $N_{f}=2$, we cannot extend
the skyrmion configuration with $2\pi$ rotation into 5d, but it becomes
possible if we embed it into larger flavors $N_{f}\geq 3$ Witten:1983tx .

Now, let us evaluate the Chern-Simons term. The $U(1)$ gauge field is obtained
as

$\tilde{a}=\mathrm{i}\tilde{z}^{\dagger}\mathrm{d}\tilde{z}=-\sin^{2}{\theta\over
2}\,\mathrm{d}\phi-\sin^{2}{\theta\over 2}\,\rho^{2}\mathrm{d}\alpha.$ (40)

The phase factor acquired by the skyrmion is given by

$\displaystyle\exp\left({\mathrm{i}\over 4\pi}\int_{S^{2}\times
D^{2}}\mathrm{d}\tilde{a}\wedge\mathrm{d}\tilde{a}\right)$ (41)
$\displaystyle=$ $\displaystyle\exp\left({\mathrm{i}\over
4\pi}2\int_{0}^{\pi}\sin^{2}{\theta\over 2}\mathrm{d}\sin^{2}{\theta\over
2}\int_{0}^{2\pi}\mathrm{d}\phi\int_{0}^{1}\mathrm{d}\rho^{2}\int_{0}^{2\pi}\mathrm{d}\alpha\right)$
$\displaystyle=$ $\displaystyle-1,$

which shows that the skyrmion with the Chern-Simons coupling becomes a
fermion.

##  4 Topological terms of 3d flag-manifold sigma model

We finally consider sigma models having flag manifolds

$\dfrac{U(N)}{U(1)^{N}}$ (42)

as a target space, which are another type of generalizations of
$\mathbb{C}P^{1}=U(2)/U(1)^{2}$.

###  4.1 Spin bordism of flag manifolds

As before, the classification of topological terms is given in terms of
$\Omega^{\mathrm{spin}}_{*}(U(N)/U(1)^{N})$. The AHSS we consider is

$E^{2}_{p,q}=H_{p}(U(N)/U(1)^{N};\Omega^{\mathrm{spin}}_{q}(\mathrm{pt}))\Longrightarrow\Omega^{\mathrm{spin}}_{p+q}(U(N)/U(1)^{N})$
(43)

and the $E^{2}$ page is as follows:

$\sseqpage[homologicalSerregrading,grid=chess,classes={draw=none},xscale=2.5,yaxisgap=1.5cm,xaxisextendend=1.5cm]\par\class["\mathbb{Z}"](0,0)\class["\mathbb{Z}_{2}"](0,1)\class["\mathbb{Z}_{2}"](0,2)\class["\mathbb{Z}"](0,4)\par\class["\mathbb{Z}^{\oplus(N-1)}"](2,0)\class["\mathbb{Z}_{2}^{\oplus(N-1)}"](2,1)\class["\mathbb{Z}_{2}^{\oplus(N-1)}"](2,2)\class["\ast"](2,4)\par\class["\mathbb{Z}^{\oplus\left({N(N-1)\over
2}-1\right)}"](4,0)\class["\mathbb{Z}_{2}^{\oplus\left({N(N-1)\over
2}-1\right)}"](4,1)\class["\ast"](4,2)\class["\ast"](4,4)\par\d{2}(4,0)\d{2}(4,1)$
(44)

See Appendix B.2 for the computation of integral cohomology of flag manifolds,
which is necessary for filling in the above table. Here, note that the
$\mathbb{Z}_{2}$ cohomology ring of flag manifolds is known to be
Borel:MR51508

$\begin{array}[]{ccl}H^{\ast}\left(\dfrac{U(N)}{U(1)^{N}};\mathbb{Z}_{2}\right)&\simeq&H^{\ast}\left(BU(1)^{N};\mathbb{Z}_{2}\right)\big{/}I\\\
&=&\mathbb{Z}_{2}[x_{1},\dots,x_{N}]\Big{/}\Big{(}\Lambda_{\mathbb{Z}}(x_{1},\dots,x_{N})\otimes\mathbb{Z}_{2}\Big{)}\end{array}$
(45)

where $\Lambda_{\mathbb{Z}}(x_{1},\dots,x_{N})$ is a ring of symmetric
functions of generators $x_{1},\dots,x_{N}\in H^{2}(BU(1)^{N})$. This implies
that the differential $d^{2}:E^{2}_{4,0}\to E^{2}_{2,1}$ given by a dual
Steenrod square composed with mod $2$ reduction Teichner:MR1214960  is
nontrivial, and none of the $\mathbb{Z}_{2}$’s involved survive to the
$E^{\infty}$-page. This shows that

$\Omega^{\mathrm{spin}}_{3}(U(N)/U(1)^{N})=0,$ (46)

and thus both ordinary and discrete theta angles do not exist in the $3$d
$U(N)/U(1)^{N}$ sigma model for $N\geq 3$. We also find that

$\widetilde{\Omega}^{\mathrm{spin}}_{4}(U(N)/U(1)^{N})=(2\mathbb{Z})^{\oplus(N-1)}\oplus\mathbb{Z}^{\oplus\left({(N-1)(N-2)\over
2}-1\right)},$ (47)

each factor corresponding to a Chern-Simons-like term, and there are
${N(N-1)\over 2}-1$ of them in total.101010The factor $2$ in front of
$\mathbb{Z}$ appears in the same way as we have discussed in (36). Here, it
means that the diagonal Chern-Simons terms are quantized with the coefficient
${1\over 4\pi}$, while the off-diagonal ones are quantized with the
coefficient ${1\over 2\pi}$, which is twice of that of diagonal ones.  In this
formula, we have already neglected the purely gravitational term by
considering the reduced bordism.

As we can trivially rewrite $U(N)/U(1)^{N}\simeq SU(N)/U(1)^{N-1}$, we expect
that there are $N-1$ types of $U(1)$ gauge fields, $(a_{I})_{I=1,\ldots,N-1}$,
in the gauged sigma-model description. This would naively allow us to write
down the $(N-1)$ diagonal Chern-Simons terms and ${(N-1)(N-2)\over 2}$ off-
diagonal Chern-Simons terms,

$\sum_{I=1}^{N-1}\frac{k_{I}}{4\pi}\int_{W_{4}}f_{I}^{2}+\sum_{I<J}\frac{k_{IJ}}{2\pi}\int_{W_{4}}f_{I}\wedge
f_{J},$ (48)

where $f_{I}=\mathrm{d}a_{I}$. However, this overcounts the number of
independent Chern-Simons couplings according to the bordism computations (47).
We shall resolve this issue in the next subsection by closely looking at the
constraint on $U(1)$ gauge fields.

In passing, we note that this is again in conflict with the homotopy argument,
where the homotopy exact sequence on the fibration

$U(1)^{N}\to U(N)\to\dfrac{U(N)}{U(1)^{N}}$ (49)

leads to

$\begin{array}[]{ccc}\pi_{3}\left(\dfrac{U(N)}{U(1)^{N}}\right)&=&\mathbb{Z},\\\
\pi_{4}\left(\dfrac{U(N)}{U(1)^{N}}\right)&=&0.\\\ \end{array}$ (50)

Thus, the homotopy argument would imply the existence of one continuous theta
angle. However, the bordism suggests that the $3$d theta terms (both ordinary
and discrete) do not exist. Instead, there should to be many Chern-Simons
terms, which are also invisible from the homotopy group.

###  4.2 Effective Lagrangian of $3$d flag-manifold sigma models

Let us consider the form of the effective Lagrangian for $3$d flag-manifold
sigma models. We take an approach of the gauged nonlinear sigma model, by
introducing an $SU(N)$ scalar field $U$ and $U(1)$ gauge fields
$a_{I=1,\ldots,N-1}$. We denote the $SU(N)$-valued scalar field $U$ as

$U=\left[z_{1},z_{2},\ldots,z_{N}\right],$ (51)

where $z_{i}$ are $N$-component complex vectors with the constraint

$z^{\dagger}_{i}\cdot z_{j}=\delta_{ij},$ (52)

and also with $\det[z_{1},\ldots,z_{N}]=1$. We can then express $z_{N}$ in
terms of the other scalars $\\{z_{I}\\}_{I=1,\ldots,N-1}$ as

$z_{N,\alpha}=\sum_{\alpha_{1},\ldots,\alpha_{N-1}}\varepsilon_{\alpha_{1}\cdots\alpha_{N-1}\alpha}z^{*}_{1,\alpha_{1}}\cdots
z^{*}_{N-1,\alpha_{N-1}}.$ (53)

Under the $U(1)$ gauge transformations

$a_{I}(x)\mapsto a_{I}(x)+\mathrm{d}\lambda_{I}(x),$ (54)

the scalar fields transform as

$z_{I}\mapsto\mathrm{e}^{-\mathrm{i}\lambda_{I}}z_{I}$ (55)

and

$z_{N}\mapsto\mathrm{e}^{\mathrm{i}(\lambda_{1}+\cdots+\lambda_{N-1})}z_{N}.$
(56)

We can write down the $SU(N)$-symmetric kinetic term as

${1\over g^{2}}\sum_{i=1}^{N}|(\mathrm{d}+\mathrm{i}a_{i})z_{i}|^{2},$ (57)

where we have set $a_{N}:=-(a_{1}+\cdots+a_{N-1})$ to emphasize the
$\mathbb{Z}_{N}$ permutation symmetry of this Lagrangian. We can solve the
equation of motion for the gauge fields to find the constraint

$a_{I}=\mathrm{i}z^{\dagger}_{I}\mathrm{d}z_{I},$ (58)

for $I=1,\ldots,N-1$. With this setup, we try to find the possible topological
terms that can be added to this Lagrangian.

We have observed that there are potentially ${N(N-1)\over 2}$ Chern-Simons
type terms listed in (48) for the 3d sigma model on the flag manifold.
Meanwhile, due to

$E^{2}_{4,0}=H_{4}(SU(N)/U(1)^{N-1})=\mathbb{Z}^{\oplus\left({N(N-1)\over
2}-1\right)},$ (59)

the $E^{2}$ page of the AHSS (44) implies the existence of ${N(N-1)\over 2}-1$
of them. This suggests that one specific linear combination of Chern-Simons
terms in (48) somehow becomes trivial, which enables us to identify
${N(N-1)\over 2}-1$ independent Chern-Simons terms modded out by such a
relation. In Appendix A, we actually find that the $U(1)$ gauge fields are
subject to the relation given by

$\sum_{I=1}^{N-1}f_{I}^{2}+\sum_{I<J}f_{I}f_{J}=-{1\over
2}\mathrm{d}\left(\sum_{i\not=j<k}a_{ij}a_{jk}a_{ki}\right),$ (60)

where we have introduced

$a_{ij}=z^{\dagger}_{i}\mathrm{d}z_{j}$ (61)

for $i\not=j$. Under the $U(1)$ gauge transformations, they behave as

$\displaystyle
a_{ij}\mapsto\mathrm{e}^{+\mathrm{i}\lambda_{i}}z^{\dagger}_{i}\mathrm{d}(\mathrm{e}^{-\mathrm{i}\lambda_{j}}z_{j})=\mathrm{e}^{\mathrm{i}(\lambda_{i}-\lambda_{j})}\left(z^{\dagger}_{i}\mathrm{d}z_{j}+(-\mathrm{i})(z^{\dagger}_{i}\cdot
z_{j})\mathrm{d}\lambda_{j}\right)=\mathrm{e}^{\mathrm{i}(\lambda_{i}-\lambda_{j})}a_{ij},$
(62)

where we have used $z^{\dagger}_{j}\cdot z_{i}=0$ for $i\not=j$. Therefore,
although $a_{ij}$ are not $U(1)$ gauge invariant, they transform covariantly.
Accordingly, following combinations

$a_{ij}\wedge a_{jk}\wedge a_{ki},$ (63)

for $i\not=j\not=k$, are gauge invariant111111$a_{ij}a_{ji}$ is also gauge
invariant, but it is not a Lorentz scalar in $3$d as it is a $2$-form. We can
construct Lorentz scalars by considering
$(a_{ij}a_{ji})\wedge\star(a_{k\ell}a_{\ell k})$ or
$a_{ij}a_{ji}\wedge\star(\mathrm{d}a_{k})$, but they are higher-derivative
corrections. We note that, in $2$d, $a_{ij}a_{ji}$ is dual to the Lorentz
scalar, and it indeed appears in the effective Lagrangian Lajko:2017wif ;
Tanizaki:2018xto . Therefore, we may be able to regard this new term,
$a_{ij}a_{jk}a_{ki}$, as its $3$d analogue. , which define globally well-
defined $3$-forms. This shows that the right-hand-side of (60) is an exact
$4$-form, and we have reproduced the result of the cohomology group by an
explicit computation. We note that

$(a_{ij}a_{jk}a_{ki})^{*}=a_{ji}a_{ik}a_{kj},$ (64)

so the right-hand-side of (60) is real-valued as it should be.

According to this relation (60), the following combination involving Chern-
Simons terms

$\sum_{I=1}^{N-1}\frac{1}{2\pi}\int_{W_{4}}f_{I}\wedge
f_{I}+\sum_{I<J}\frac{1}{2\pi}\int_{W_{4}}f_{I}\wedge
f_{J}-\sum_{i\not=j<k}\frac{1}{4\pi}\int_{M_{3}}a_{ij}a_{jk}a_{ki}$ (65)

potentially provides a bordism invariant of 3d spin manifolds equipped with a
map to $U(N)/U(1)^{N}$, as in the case of the Hopf term in the
$\mathbb{C}P^{1}$ model. Here, $M_{3}$ is a 3d spacetime manifold where
$\partial W_{4}=M_{3}$. However, since the spin bordism
$\Omega^{\mathrm{spin}}_{3}(U(N)/U(1)^{N})=0$, we can take a $4$d extension as
a flag-manifold field, not as $U(1)$ gauge fields. Hence, the above candidate
term turns out to be trivial

$\frac{\mathrm{i}}{2\pi}\int_{W_{4}}\left(\sum_{I=1}^{N-1}f_{I}\wedge
f_{I}+\sum_{I<J}f_{I}\wedge f_{J}-{1\over
2}\sum_{i\not=j<k}\mathrm{d}(a_{ij}a_{jk}a_{ki})\right)=0,$ (66)

and we can simply eliminate one of the off-diagonal Chern-Simons term by using
(60).121212The relation (60) has been used to construct the Hopf invariant for
the $SU(3)/U(1)^{2}$ flag manifold Kisielowski:2013ina  (see also
Amari:2018gbq ). In order to detect $\pi_{3}(U(N)/U(1)^{N})=\mathbb{Z}$, as a
special feature of the sphere $S^{3}$, we can take the globally defined $U(1)$
gauge fields, $a_{1},\ldots,a_{N-1}$ on $S^{3}$, and this fact plays a crucial
role to have the Hopf invariant in Kisielowski:2013ina . Discussion in this
paragraph has shown that the corresponding $\theta$ terms do not exist for the
$3$d sigma model including discrete ones, when we adopt the bordism
classification of topological terms in order to define QFTs on general
$3$-manifolds.

In the following, let us explicitly work on the case $N=3$, i.e. on the $3$d
$SU(3)/U(1)^{2}$ flag-manifold sigma model. Topological terms of the effective
Lagrangian can be written as

${\mathrm{i}k_{1}\over
4\pi}\int_{W_{4}}\mathrm{d}\tilde{a}_{1}\wedge\mathrm{d}\tilde{a}_{1}+{\mathrm{i}k_{2}\over
4\pi}\int_{W_{4}}\mathrm{d}\tilde{a}_{2}\wedge\mathrm{d}\tilde{a}_{2}+\mathrm{i}{h\over
2\pi}\int_{M_{3}}a_{12}a_{23}a_{31}.$ (67)

Here, we note that

$(a_{12}a_{23}a_{31})^{*}=a_{12}a_{23}a_{31}-\mathrm{d}(a_{12}a_{21}),$ (68)

and thus it gives a real-valued $3$-form up to a total derivative. Therefore,
we do not need to introduce coupling constants separately for
$a_{12}a_{23}a_{31}$ and $(a_{12}a_{23}a_{31})^{*}=a_{21}a_{13}a_{32}$. The
coupling constant $h$ for $a_{12}a_{23}a_{31}$ does not need to be quantized
unlike the Chern-Simons couplings $k_{1},k_{2}$, and it does not have a
periodicity unlike the theta terms, even though the term is topological. Even
if we add the off-diagonal Chern-Simons term, ${k_{12}\over
2\pi}\mathrm{d}\tilde{a}_{1}\wedge\mathrm{d}\tilde{a}_{2}$, its effect can be
taken into account with the above Lagrangian by rewriting the couplings
$k^{\prime}_{1}=k_{1}-2k_{12}$, $k^{\prime}_{2}=k_{2}-2k_{12}$, and
$h^{\prime}=h+k_{12}$.

Let us give a quick observation on how we could introduce the Chern-Simons
coupling in view of symmetry. Our kinetic term (57) is constructed so that it
has $\mathbb{Z}_{3}$ permutation symmetry, $z_{i}\mapsto z_{i+1\bmod 3}$,
which is often a consequence of the underlying lattice symmetry of quantum
spin systems. However, the Chern-Simons coupling cannot be introduced in a
$\mathbb{Z}_{3}$-symmetric way. Indeed, the only $\mathbb{Z}_{3}$ symmetric
combination of the diagonal Chern-Simons term is

$f_{1}^{2}+f_{2}^{2}+f_{3}^{2}=2(f_{1}^{2}+f_{2}^{2}+f_{1}f_{2}),$ (69)

since $f_{3}=-(f_{1}+f_{2})$, and because of the relation (60), this is a
trivial element of the cohomology. This would suggest that, if we try to
construct a $SU(3)/U(1)^{2}$ model with nontrivial Chern-Simons terms from the
$SU(3)$ Heisenberg-like model on triangular lattices, we need to break the
full wallpaper group, $p6m$, to some smaller subgroup explicitly.

Lastly, we consider the properties of skyrmions for the $SU(3)/U(1)^{2}$ sigma
model. As there are two independent $U(1)$ gauge fields, there are two
different skyrmions charges defined by $Q_{I}={1\over
2\pi}\int_{M_{2}}\mathrm{d}a_{I}$ for $I=1,2$. Let us introduce a skyrmion for
$Q_{1}$ and discuss its statistics as we have done in Sec. 3. Such a skyrmion
can be constructed as

$U(\theta,\phi)=[z_{1},z_{2},z_{3}]=\begin{pmatrix}\cos{\theta\over
2}&0&\mathrm{e}^{-\mathrm{i}\phi}\sin{\theta\over 2}\\\
\mathrm{e}^{\mathrm{i}\phi}\sin{\theta\over 2}&0&-\cos{\theta\over 2}\\\
0&1&0\end{pmatrix}.$ (70)

We can further construct a $4$d extension of the adiabatically $2\pi$-rotated
skyrmions by multiplying an $SU(3)$ matrix from the left, exactly as we did in
(39). One can readily check that the skyrmion becomes a fermion when $k_{1}$
is odd. Note that the non-quantized topological term, $a_{12}a_{23}a_{31}$,
does not affect the result since it vanishes identically for this
configuration on $S^{2}\times S^{1}$.

###### Acknowledgements.

The authors thank Shuhei Ohyama and Mayuko Yamashita for valuable discussions,
and also Yuji Tachikawa for comments on the manuscript. R. K. and Y. L. are
grateful to the hospitality of YITP during their stays. The authors thank the
YITP workshop YITP-W-21-11 on “Theoretical studies of topological phases of
matter”, which was useful to complete this work. The work of R. K. was
supported by Japan Society for the Promotion of Science (JSPS) through Grant
No. 19J20801. The work of Y. L. was partially supported by JSPS Research
Fellowship for Young Scientists. The work of Y. T. was partially supported by
JSPS KAKENHI Grant-in-Aid for Research Activity Start-up, 20K22350. The work
of K. S. was supported by PRESTO, JST (Grant No. JPMJPR18L4) and CREST, JST
(Grant No. JPMJCR19T2).

##  Appendix A Derivation of constraint relations for the flag sigma model

In this appendix, we prove the relation for the flag-manifold sigma model,

$\sum_{i=1}^{N}f_{i}^{2}=-\mathrm{d}\left(\sum_{i\not=j<k}a_{ij}a_{jk}a_{ki}\right),$
(71)

where $\sum_{i\not=j<k}$ means the summation over all $1\leq i,j,k\leq N$
satisfying $i\not=j$ and both $i,j<k$. Combined with the relation
$f_{N}=-f_{1}-\cdots-f_{N-1}$, we find that

$\sum_{I=1}^{N-1}f_{I}^{2}+\sum_{I<J}f_{I}f_{J}=-{1\over
2}\mathrm{d}\left(\sum_{i\not=j<k}a_{ij}a_{jk}a_{ki}\right),$ (72)

where capital $I,J$ run over $1,\ldots,N-1$, while $i,j,k$ run over
$1,\ldots,N$.

Let $U$ be a $U(N)$-valued scalar field and define the Maurer-Cartan form as

$A=U^{\dagger}\mathrm{d}U.$ (73)

Let $P_{i}$ be the projector to $i$-th component and denote

$A_{i}=P_{i}AP_{i}$ (74)

and131313Since $A_{i}$ is a $U(1)$ gauge field, $A_{i}^{2}$-term in the
definition of $F_{i}$ is not necessary, as it automatically vanishes. However,
the proof in this appendix can be generalized to the case where $A_{i}$ is a
$U(n)$ gauge field, which corresponds to more general flag manifolds,
$U(n_{1}+\ldots+n_{N})/(U(n_{1})\times\cdots\times U(n_{N}))$. Therefore, in
the proof, we do not use any special properties for $U(1)$ gauge fields, and
keep the $A_{i}^{2}$-term in the field strength.

$F_{i}=\mathrm{d}A_{i}+A_{i}^{2}.$ (75)

We note that, in this appendix, the gauge fields, $A$ and $A_{i}$, are taken
to be anti-Hermitian, and thus there is an extra fator of $\mathrm{i}$ to make
the connection with the main text of the paper:

$a_{i}=\mathrm{tr}(\mathrm{i}A_{i}),\quad f_{i}=\mathrm{tr}(\mathrm{i}F_{i}).$
(76)

Let us prove the following relations,

$\displaystyle\sum_{i}\mathrm{tr}(F_{i})$ $\displaystyle=$ $\displaystyle 0,$
(77) $\displaystyle\sum_{i}\mathrm{tr}(F_{i}^{2})$ $\displaystyle=$
$\displaystyle\mathrm{d}\left\\{\sum_{i\not=j<k}\mathrm{tr}(P_{i}AP_{j}AP_{k}A)\right\\}.$
(78)

The first relation (77) can be shown very easily. Since $\mathrm{d}A=-A^{2}$,
we find that

$F_{i}=P_{i}(-A^{2})P_{i}+P_{i}AP_{i}AP_{i}=-P_{i}A\sum_{j\not=i}P_{j}AP_{i}.$
(79)

Therefore,

$\displaystyle\sum_{i}\mathrm{tr}(F_{i})$ $\displaystyle=$
$\displaystyle-\sum_{i}\sum_{j\not=i}\mathrm{tr}(P_{i}AP_{j}A)=0.$ (80)

Here, we use the cyclic property of the trace, which gives
$\mathrm{tr}(P_{i}AP_{j}A)=-\mathrm{tr}(P_{j}AP_{i}A)$.

Let us show the second relation (78).

$\displaystyle\sum_{i=1}^{N}\mathrm{tr}(F_{i}^{2})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N}\sum_{j,k\not=i}\mathrm{tr}(P_{i}AP_{j}AP_{i}AP_{k}A)$
(81) $\displaystyle=$
$\displaystyle\sum_{I=1}^{N-1}\mathrm{tr}(P_{I}A(\bm{1}-P_{I})AP_{I}A(\bm{1}-P_{I})A)$
$\displaystyle+\sum_{J,K}\mathrm{tr}\left(\left(\bm{1}-\sum_{L}P_{L}\right)AP_{J}A\left(\bm{1}-\sum_{M}P_{M}\right)AP_{K}A\right).$

In order to obtain the last expression, we replace the summations $i,j,k$ over
$1,\ldots,N$ into the summations $I,J,K$ over $1,\ldots,N-1$ and $N$ is
treated separately, and we eliminate $P_{N}$ by substituting
$P_{N}=\bm{1}-P_{1}-\cdots-P_{N-1}$. By cyclic property,
$\mathrm{tr}((P_{I}A)^{4})=0$ and
$\sum_{J,K,L,M}\mathrm{tr}(P_{J}AP_{K}AP_{L}AP_{M}A)=0$. We therefore find

$\displaystyle\sum_{i}\mathrm{tr}(F_{i}^{2})$ $\displaystyle=$
$\displaystyle\sum_{I}\left\\{\mathrm{tr}(P_{I}\mathrm{d}AP_{I}\mathrm{d}A)+2\mathrm{tr}((P_{I}A)^{2}P_{I}\mathrm{d}A)\right\\}$
(82)
$\displaystyle-\sum_{J,K}\mathrm{tr}(P_{J}\mathrm{d}AP_{K}\mathrm{d}A)+2\sum_{J,K,L}\mathrm{tr}(P_{J}AP_{K}AP_{L}\mathrm{d}A)$
$\displaystyle=$
$\displaystyle\mathrm{d}\left\\{\sum_{J\not=K}\mathrm{tr}(P_{J}AP_{K}A^{2})+{2\over
3}\left(\sum_{I}\mathrm{tr}((P_{I}A)^{3})-\sum_{J,K,L}\mathrm{tr}(P_{J}AP_{K}AP_{L}A)\right)\right\\}$
$\displaystyle=$ $\displaystyle\mathrm{d}\left\\{{1\over
3}\sum_{J\not=K\not=L}\mathrm{tr}(P_{J}AP_{K}AP_{L}A)+\sum_{J\not=K}\mathrm{tr}(P_{J}AP_{K}AP_{N}A)\right\\}$
$\displaystyle=$
$\displaystyle\mathrm{d}\left\\{\sum_{i\not=j<k}\mathrm{tr}(P_{i}AP_{j}AP_{k}A)\right\\}.$

The first equation on the right-hand-side is obtained by replacing
$A^{2}=-\mathrm{d}A$ in (81) with the above note. As it turns out to be a
total derivative, we extract the exterior derivative to find the second
equality, and replace $\mathrm{d}A$ by $-A^{2}$ inside the overall
differential. In order to go from the second line to the third line on the
right-hand-side, we have replaced $A^{2}=A(\sum_{L}P_{L}+P_{N})A$ and then
consider the combinatorics for the domain of summations. To obtain the last
line from the third one, we note that $\mathrm{tr}(P_{J}AP_{K}AP_{L}A)$ is
cyclically symmetric in $J,K,L$, so we can replace
$\sum_{J\not=K\not=L}=3\sum_{J\not=K<L}$ in the first term. Then, we can
combine the first and second terms in the third line by reintroducing the
dummy indices $i,j,k$ running over $1,\ldots,N$, which gives the last line,
and this completes the proof.

##  Appendix B Leray-Serre spectral sequence for cohomologies of sigma models

Leray-Serre spectral sequence (LSSS) provides us a systematic way to find
cohomologies from the fibration,

$F\to E\to B.$ (83)

When $\pi_{1}(B)$ acts trivially on the cohomology of the fiber $H^{\ast}(F)$,
LSSS gives

$H^{p}(B;H^{q}(F))\Longrightarrow H^{p+q}(E),$ (84)

that is, the $E_{2}$ page can be written down by knowing the cohomologies of
the base space $B$ and the fiber $F$, and it converges to the cohomology of
the total space $E$. LSSS is a first-quadrant spectral sequence, so if $B$ or
$F$ is finite dimensional, then LSSS collapses at some finite page. There are
two typical ways to use LSSS. The first one is to obtain $H^{*}(E)$ when we
already know about $H^{*}(B)$ and $H^{*}(F)$. The another one is the
“opposite” way: knowing $H^{*}(E)$ and one of cohomologies of $B$ or $F$, we
constrain the possible cohomologies of the last one.

In the context of the sigma model, we are interested in the cohomology of the
target space $X$. In many cases, the target space is given by the homogeneous
Riemannian space $G/H$, because this appears when a global symmetry $G$ is
spontaneously broken to its subgroup $H$ by some condensates. Then, there is a
principal $H$-bundle,

$H\to G\to G/H.$ (85)

When both $G$ and $H$ are continuous and connected Lie groups, and $G$ is
moreover simply connected, then the homotopy exact sequence tells

$\pi_{1}(G/H)\simeq 0.$ (86)

Therefore, LSSS works nicely, which gives

$H^{p}(G/H;H^{q}(H))\Longrightarrow H^{p+q}(G).$ (87)

Since (85) gives a principal bundle, which is not a general fibration, we have
another fibration out of (85);

$G\to G/H\to\mathrm{B}H,$ (88)

where $\mathrm{B}H$ is the classifying space of $H$. When $H$ is connected,
$\pi_{1}(\mathrm{B}H)\simeq\pi_{0}(H)=0$, and thus LSSS again works nicely. We
obtain

$H^{p}(\mathrm{B}H;H^{q}(G))\Longrightarrow H^{p+q}(G/H).$ (89)

Therefore, this spectral sequence makes a relationship between topological
terms of $H$ gauge fields and topological terms of $G/H$-valued sigma models.

In the following of this appendix, we first study the cohomology of
$\mathbb{C}P^{N-1}$ as an exercise, and then compute and reinterpret the
cohomology of $SU(N)/U(1)^{N-1}$.

###  B.1 Cohomology of $\mathbb{C}P^{N-1}$

Let us apply the LSSS to the fibration (28), which gives

$H^{p}(\mathbb{C}P^{N-1};H^{q}(U(1)))\Longrightarrow H^{p+q}(S^{2N-1}).$ (90)

We note that $H^{*}(S^{n})=\mathbb{Z}$ for $*=0,n$ and others vanish.
Therefore, the cohomologies of the total space and the fiber is known, and
thus we can constrain $H^{*}(\mathbb{C}P^{N-1})$ from consistency. Let us
explicitly work on the case $N=3$, i.e. for the case of $\mathbb{C}P^{2}$. The
$E_{2}$ page is given by

$\sseqpage[cohomologicalSerregrading,grid=chess,classes={draw=none},yscale=0.65,xrange={0}{4},xlabel={H^{p}(\mathbb{C}P^{2})},ylabel={H^{q}(S^{1})}]\class["\mathbb{Z}"](0,0)\class["\mathbb{Z}"](0,1)\class["\mathbb{Z}"](2,0)\class["\mathbb{Z}"](2,1)\class["\mathbb{Z}"](4,0)\class["\mathbb{Z}"](4,1)\d{2}(0,1)\d{2}(2,1)$
(91)

Since $\mathbb{C}P^{N-1}$ is connected, $H^{0}(\mathbb{C}P^{N-1})=\mathbb{Z}$,
and then we can fill the $0$th column. Since the total-space cohomology
$H^{*}(S^{5})$ does not have the nontrivial cohomology until $*=5$,
$\mathbb{Z}$ in $(p,q)=(0,1)$ must be eliminated by the $d_{2}$ differential.
Let $y\in H^{1}(S^{1})$ be the generator, and we denote $d_{2}(y)=x$ and
$H^{2}(\mathbb{C}P^{N-1})\simeq\mathbb{Z}$ is generated by $x$. Then,
$(p,q)=(2,1)$ is generated by their tensor products $xy$, which again should
be eliminated by $d_{2}$, and the Leibniz rule gives
$d_{2}(xy)=d_{2}(x)y+xd_{2}(y)=x^{2}$. Thus, $x^{2}$ is the generator of
$H^{4}(\mathbb{C}P^{2})\simeq\mathbb{Z}$. Since $(p,q)=(4,1)$ must not be
eliminated by differentials because $H^{5}(S^{5})=\mathbb{Z}$, we find that
$d_{2}(x^{2}y)=0$, and then we obtain
$H^{*}(\mathbb{C}P^{2})\simeq\mathbb{Z}[x]/(x^{3})$ as a ring isomorphism.
This reproduces (34).

We would like to identify this generator $x$ as a $U(1)$ field strength
$\mathrm{d}a/2\pi$ in the gauged sigma model description. Since the fibration
(28) is principal, we can obtain another fibration out of it:

$S^{2N-1}\to\mathbb{C}P^{N-1}\to\mathrm{B}U(1).$ (92)

Let us apply the LSSS to it. We note that $\pi_{1}(\mathrm{B}U(1))=0$, and
thus the complication of local coefficients does not come up. To be specific,
let us consider the case $N=3$, while the discussion for general $N$ is also
straightforward. LSSS gives

$H^{p}(\mathrm{B}U(1);H^{q}(S^{5}))\Longrightarrow H^{p+q}(\mathbb{C}P^{2}).$
(93)

Then, $E_{2}$ page looks like

$\sseqpage[cohomologicalSerregrading,grid=chess,classes={draw=none},yscale=0.65,xrange={0}{8},xlabel={H^{p}(\mathrm{B}U(1))},ylabel={H^{q}(S^{5})}]\class["\mathbb{Z}"](0,0)\class["\mathbb{Z}"](0,5)\class["\mathbb{Z}"](2,0)\class["\mathbb{Z}"](2,5)\class["\mathbb{Z}"](4,0)\class["\mathbb{Z}"](4,5)\class["\mathbb{Z}"](6,0)\class["\mathbb{Z}"](6,5)\class["\mathbb{Z}"](8,0)\class["\mathbb{Z}"](8,5)\class["\mathbb{Z}"](10,0)\class["\mathbb{Z}"](10,5)\class["\mathbb{Z}"](12,0)\class["\mathbb{Z}"](12,5)\d{[}blue]6(0,5)\d{[}blue]6(2,5)\par\structline[blue](4,5)(10,0)\structline[blue](6,5)(12,0)$
(94)

As $\mathbb{C}P^{2}$ is a $4$-manifold, $H^{*}(\mathbb{C}P^{2})$ must vanish
for $*\geq 5$. Just because of this trivial information, we find that the
$d_{6}$ differentials should be identity maps, which are drawn with blue solid
arrows. As a result, we find that, for $N\geq 3$,

$H^{4}(\mathbb{C}P^{N-1})\simeq H^{4}(\mathrm{B}U(1)),$ (95)

which is generated by

$\int{\mathrm{d}a\over 2\pi}\wedge{\mathrm{d}a\over 2\pi}.$ (96)

This elucidates that the topological term of our interest can be described as
the Chern-Simons coupling.

In this case, this argument may seem to be tautological, reminding the fact
that $\mathrm{B}U(1)\simeq\mathbb{C}P^{\infty}$. This, however, turns out to
be a good exercise for the flag-manifold sigma models, as we discuss in the
next subsection.

###  B.2 Cohomology of $SU(N)/U(1)^{N-1}$

Next, let us compute the cohomology of the flag manifold $SU(N)/U(1)^{N-1}$
using LSSS for $U(1)^{N-1}\to SU(N)\to SU(N)/U(1)^{N-1}$:

$H^{p}(SU(N)/U(1)^{N-1};H^{q}(U(1)^{N-1}))\Longrightarrow H^{p+q}(SU(N)).$
(97)

The basic strategy is as follows: We know the cohomologies $H^{*}(U(1)^{N-1})$
and $H^{*}(SU(N))=\bigwedge[x_{3},x_{5},\ldots,x_{2N-1}]$, and then we find
out $H^{*}(SU(N)/U(1)^{N-1})$ from consistency. We also note that
$\mathrm{dim}(SU(N)/U(1)^{N-1})=N(N-1)$, and the degree beyond it must vanish.

Let us take the specific example, $N=3$, in the following. The $E_{2}$ page
looks as

$\sseqpage[cohomologicalSerregrading,grid=chess,classes={draw=none},yscale=0.65,xlabel={H^{p}(SU(3)/U(1)^{2})},ylabel={H^{q}(U(1)^{2})}]\class["\mathbb{Z}"](0,0)\class["\mathbb{Z}^{\oplus
2}"](0,1)\class["\mathbb{Z}"](0,2)\class["\mathbb{Z}^{\oplus
2}"](2,0)\class["\mathbb{Z}^{\oplus 4}"](2,1)\class["\mathbb{Z}^{\oplus
2}"](2,2)\d{2}(0,1)\d{2}(0,2)\class["\mathbb{Z}^{\oplus
2}"](4,0)\class["\mathbb{Z}^{\oplus 4}"](4,1)\class["\mathbb{Z}^{\oplus
2}"](4,2)\d{2}(2,1)\d{2}(2,2)\class["\mathbb{Z}"](6,0)\class["\mathbb{Z}^{\oplus
2}"](6,1)\class["\mathbb{Z}"](6,2)\d{2}(4,1)\d{2}(4,2)$ (98)

Here, we denote $\mathbb{Z}^{\oplus n}=\oplus_{i=1}^{n}\mathbb{Z}$. As a
result, we find that

$H^{p}(SU(3)/U(1)^{2})=\left\\{\begin{array}[]{cc}\mathbb{Z}&\quad p=0,6,\\\
\mathbb{Z}^{\oplus 2}&\quad p=2,4,\\\ 0&\quad\mathrm{else}.\end{array}\right.$
(99)

Although we did not identify the cohomology ring structure, it is consistent
with (45) as a module.141414Indeed, LSSS is not necessarily useful to
determine the cohomology ring structure. In particular, the cohomology ring of
the total space cannot be determined from LSSS. For example, we can consider
the fibration, $\mathbb{C}P^{1}\to U(3)/U(1)^{3}\to\mathbb{C}P^{2}$, which can
be obtained from $U(2)\times U(1)\to U(3)\to\mathbb{C}P^{2}$ by dividing the
total space and the fiber by $U(1)^{3}$. LSSS immediately reproduces (99) as a
module, but it does not give the cohomology ring structure.  With this
information, we can start the computation of the spin bordism using AHSS, as
we did in the main text. Here, instead, let us take a close look at the
physical meaning of these nontrivial cohomologies. As in the case of
$\mathbb{C}P^{N-1}$ model, it is convenient to describe the $SU(N)/U(1)^{N-1}$
sigma model using the $SU(N)$-valued scalar field $U$ with $(N-1)$ $U(1)$
gauge fields, $a_{1},\ldots,a_{N-1}$. Then, the natural question is whether
the nontrivial cohomologies of $SU(N)/U(1)^{N-1}$ are related to the
nontrivial topological terms of these $U(1)$ gauge fields.

The answer turns out to be affirmative. To see this, we consider the following
fibration,

$SU(N)\to SU(N)/U(1)^{N-1}\to\mathrm{B}U(1)^{N-1}.$ (100)

We apply LSSS to this fibration. Let us set $N=3$ again for simplicity, then
the $E_{2}$ page is given by

$\sseqpage[cohomologicalSerregrading,grid=chess,classes={draw=none},yscale=0.65,xscale=1,xrange={0}{8},xlabel={H^{p}(\mathrm{B}U(1)^{2})},ylabel={H^{q}(SU(3))}]\class["\mathbb{Z}"](0,0)\class["\mathbb{Z}"](0,3)\class["\mathbb{Z}"](0,5)\class["\mathbb{Z}"](0,8)\class["\mathbb{Z}^{\oplus
2}"](2,0)\class["\mathbb{Z}^{\oplus 2}"](2,3)\class["\mathbb{Z}^{\oplus
2}"](2,5)\class["\mathbb{Z}^{\oplus 2}"](2,8)\class["\mathbb{Z}^{\oplus
3}"](4,0)\class["\mathbb{Z}^{\oplus 3}"](4,3)\class["\mathbb{Z}^{\oplus
3}"](4,5)\class["\mathbb{Z}^{\oplus 3}"](4,8)\class["\mathbb{Z}^{\oplus
4}"](6,0)\class["\mathbb{Z}^{\oplus 4}"](6,3)\class["\mathbb{Z}^{\oplus
4}"](6,5)\class["\mathbb{Z}^{\oplus 4}"](6,8)\class["\mathbb{Z}^{\oplus
5}"](8,0)\class["\mathbb{Z}^{\oplus 5}"](8,3)\class["\mathbb{Z}^{\oplus
5}"](8,5)\class["\mathbb{Z}^{\oplus
5}"](8,8)\class(10,0)\class(12,0)\class(10,3)\class(12,3)\class(10,5)\class(12,5)\class(10,8)\class(12,8)\par\d{[}blue]4(0,3)\d{[}blue]4(0,8)\d{[}blue]4(2,3)\d{[}blue]4(2,8)\d{[}blue]4(4,3)\d{[}blue]4(4,8)\par\structline[blue](6,3)(10,0)\structline[blue](6,8)(10,5)\structline[blue](8,3)(12,0)\par\draw[differentialstyle,red,denselydashed](0,5)--(6,0);\draw[differentialstyle,red,denselydashed](2,5)--(8,0);\structline[red,denselydashed](4,5)(10,0)\structline[red,denselydashed](6,5)(12,0)$
(101)

Here, the $d_{4}$ differentials are drawn with blue solid arrows, and the
$d_{6}$ differentials are drawn with the red dashed arrows. The spectral
sequence clashes after the $E_{6}$ page. We then find that only the $q=0$
components with $H^{p}(\mathrm{B}U(1)^{2})$ for $p=0,2,4,6$ can survive in the
$E^{\infty}$ page, and thus the nontrivial cohomology $H^{*}(SU(3)/U(1)^{2})$
has a well-defined origin in $H^{p}(\mathrm{B}U(1)^{2})$.

For general $N$, we get

$H^{2}(SU(N)/U(1)^{N-1})=\mathbb{Z}^{\oplus(N-1)},\quad
H^{4}(SU(N)/U(1)^{N-1})=\mathbb{Z}^{\oplus\left({N(N-1)\over 2}-1\right)}.$
(102)

We note that $H^{2}(SU(N)/U(1)^{N-1})\simeq
H^{2}(\mathrm{B}U(1)^{N-1})\simeq\mathbb{Z}^{\oplus(N-1)}$. As we have seen in
the case $N=3$, however,

$\displaystyle H^{4}(SU(N)/U(1)^{N-1})$ $\displaystyle\simeq$ $\displaystyle
H^{4}(\mathrm{B}U(1)^{N-1})/d_{4}^{0,3}(H^{3}(SU(N)))$ (103)
$\displaystyle\simeq$ $\displaystyle\mathbb{Z}^{\oplus{N(N-1)\over
2}}/d_{4}^{0,3}(\mathbb{Z})$ $\displaystyle\simeq$
$\displaystyle\mathbb{Z}^{\oplus\left({N(N-1)\over 2}-1\right)}.$

Thus, we can express these topological terms as $f_{i}\wedge f_{j}$, but there
is a constraint compared with the $U(1)^{N-1}$ gauge theory. We have computed
the explicit relation among them in Appendix A.

##  Appendix C  Realization of topological terms by a free massive fermion

In this section, we present an explicit realization of the Hopf term (15) as a
partition function of a massive free fermion. Let us consider a $(2+1)$d
Euclidean relativistic Dirac fermion coupled with spin fields $\bm{n}$ which
take values on sphere $S^{2}$,

$\displaystyle{\cal
L}=\mathrm{i}\bar{\psi}(\not{D}+m\bm{n}\cdot\bm{\sigma})\psi,\quad\bm{n}^{2}=1.$
(104)

Here, $\not{D}=\gamma_{\mu}(\partial_{\mu}-\mathrm{i}A_{\mu})$ is a covariant
derivative with $A_{\mu}$ a spinc gauge field, $\gamma_{\mu}$’s are the 2 by 2
gamma matrices, and $\bm{\sigma}$ are Pauli matrices for the internal spin
space. It was shown that the large-scale dynamics of the spin fields $\bm{n}$
compared to the inverse of the mass, $1/m>0$, is described by the nonlinear
sigma model with the Hopf term Abanov:1999qz ,

$\displaystyle\exp\left({-S_{\rm eff}}\right)=\int
D\bar{\psi}D\psi\,\exp\left({-\int d^{3}x{\cal L}}\right),$ (105)

where151515 Strictly speaking, the imaginary part of (107) is defined as
$\displaystyle\frac{i}{4\pi}\int_{W_{4}}(\mathrm{d}\tilde{a}+2\mathrm{d}\tilde{A})\wedge\mathrm{d}\tilde{a}$
(106) with $\tilde{a}$ and $\tilde{A}$ extensions of $U(1)$ and spinc fields,
respectively, to a spinc bounding manifold $W_{4}$.

$\displaystyle S_{\rm eff}=\frac{m}{8\pi}\int
d^{3}x(\partial_{\mu}\bm{n})^{2}+\frac{\mathrm{i}}{4\pi}\int
a\mathrm{d}a+\frac{\mathrm{i}}{2\pi}\int A\mathrm{d}a.$ (107)

The imaginary part of the effective action $S_{\rm eff}$ is the Chern-Simons
term corresponding to the spinc bordism group $\tilde{\Omega}^{\rm
spin^{c}}_{4}(\mathbb{C}P^{1})=\mathbb{Z}$, and it describes two topological
responses: a skyrmion of the spin fields $\bm{n}$ has a unit $U(1)$ charge,
and the adiabatic $2\pi$ rotation of the skyrmion texture gives rise to the
Berry phase $(-1)$ on the ground state wave function. By restricting the
background spinc fields $A$ to a spin one, the third term of (107) vanishes
and only the $\mathbb{Z}_{2}$-quantized Hopf term of (107) remains, which is
nothing but the Hopf term (15) with $k=1$.

A corresponding lattice model of (107) is the following imaginary-time-
dependent Hamiltonian of fermions on a closed two-dimensional square lattice
Qi:2008ew

$\displaystyle\hat{H}(\tau)$
$\displaystyle=\sum_{{\bm{x}}\in\mathbb{Z}^{2}}\Bigg{[}\left(\hat{\psi}^{\dagger}_{{\bm{x}}+\hat{x}}\frac{-\sigma_{z}\tau_{z}+\mathrm{i}\tau_{x}}{2}\hat{\psi}_{\bm{x}}+\hat{\psi}^{\dagger}_{{\bm{x}}+\hat{y}}\frac{-\sigma_{z}\tau_{z}+\mathrm{i}\tau_{y}}{2}\hat{\psi}_{\bm{x}}+{\rm
h.c.}\right)$
$\displaystyle\qquad\qquad+\hat{\psi}^{\dagger}_{\bm{x}}(m\bm{n}({\bm{x}},\tau)\cdot\bm{\sigma}\tau_{z}+\lambda\sigma_{z}\tau_{z})\hat{\psi}_{\bm{x}}\Bigg{]},$
(108)

where $\hat{\psi}^{\dagger}_{\bm{x}}/\hat{\psi}_{\bm{x}}$ are complex fermion
creation/annihilation operators, $\sigma_{\mu},\tau_{\mu}$ are Pauli matrices,
$\bm{n}(\bm{x},\tau)$ are spin fields with $\bm{n}^{2}=1$, and $m>0$ and
$\lambda$ are parameters controlling whether the skyrmion becomes a fermion or
not. For a uniform configuration of the spin fields
$\bm{n}(\bm{x},\tau)\equiv\underline{\bm{n}}$, this Hamiltonian can be written
as $\hat{H}=\sum_{\bm{k}}\psi^{\dagger}_{\bm{k}}{\cal
H}({\bm{k}},\underline{\bm{n}})\psi_{\bm{k}}$ with a one-particle Hamiltonian

$\displaystyle{\cal H}({\bm{k}},\underline{\bm{n}})=\sin k_{x}\tau_{x}+\sin
k_{y}\tau_{y}+(m\underline{\bm{n}}\cdot\bm{\sigma}+(\lambda-\cos k_{x}-\cos
k_{y})\sigma_{z})\tau_{z}$ (109)

in the Bloch-momentum space. The energy eigenvalues of ${\cal
H}({\bm{k}},\underline{\bm{n}})$ are given by

$\displaystyle\varepsilon({\bm{k}})^{2}=\sin^{2}k_{x}+\sin^{2}k_{y}+m^{2}\underline{n}_{1}^{2}+m^{2}\underline{n}_{2}^{2}+(m\underline{n}_{3}+\lambda-\cos
k_{x}-\cos k_{y})^{2},$ (110)

which has a finite energy gap unless $m=|\lambda|,|\lambda+2|,|\lambda-2|$ is
satisfied. For $\lambda=2$ the low-energy Hamiltonian around the Gamma point
${\bm{k}}\sim\bm{0}$ reduces to the Dirac model (104), thus it is expected
that the ground state with a skyrmion indicates desired topological responses
for an appropriate parameter region. Hereafter, we set $m=1$ and focus on
$\lambda=0,\pm 2,\pm 4$ as representatives of different invertible phases.

A skyrmion texture in a finite square lattice with sites
$\\{1,\dots,L_{x}\\}\times\\{1,\dots,L_{y}\\}$ in the periodic boundary
conditions is introduced, for example, by setting the unit vector $\bm{n}$ to

$\displaystyle n_{1}(x,y,\tau=0)=\sin\left(\frac{2\pi
x}{L_{x}}-\pi\right)\cos\left(\frac{\pi y}{L_{y}}-\frac{\pi}{2}\right),$ (111)
$\displaystyle n_{2}(x,y,\tau=0)=\sin\left(\frac{2\pi
x}{L_{x}}-\pi\right)\cos^{2}\left(\frac{\pi x}{L_{y}}-\frac{\pi}{2}\right),$
(112) $\displaystyle n_{3}(x,y,\tau=0)=1-2\cos^{2}\left(\frac{\pi
x}{L_{x}}-\frac{\pi}{2}\right)\cos^{2}\left(\frac{\pi
y}{L_{y}}-\frac{\pi}{2}\right).$ (113)

This construction is based on the smash product $S^{1}\wedge S^{1}\cong S^{2}$
Higashikawa:2018qsh . We numerically find that the number of electrons of the
ground state $\ket{\Omega}$ (equivalently, the number of negative eigenvalues
of one-particle Hamiltonian of (107)) in the presence of a skyrmion is

$\displaystyle\Delta
N:=\braket{\Omega}{\hat{N}}{\Omega}-2L_{x}L_{y}=\left\\{\begin{array}[]{ll}-1&(\lambda=\pm
2),\\\ 2&(\lambda=0),\\\ 0&(\lambda=\pm 4),\end{array}\right.$ (117)

for sufficiently large $L_{x}$ and $L_{y}$. Here, $2L_{x}L_{y}$ is the number
of electrons of the ground state without skyrmions. Thus, we confirm that a
skyrmion behaves as an electron for $1<|\lambda|<3$. We also confirm that
$\Delta N$ coincides with the second Chern number
$-\frac{1}{8\pi^{2}}\int_{T^{2}\times S^{2}}\mathrm{tr}{\cal
F}({\bm{k}},\bm{n})^{2}$ of the semiclassical Hamiltonian ${\cal
H}({\bm{k}},\bm{n})$ over the Bloch-momentum space $T^{2}$ and the spin space
$S^{2}$ with the identity map $\bm{n}:S^{2}\to S^{2}$.

Next, we evaluate the imaginary-time evolution of the Hamiltonian with the
$2\pi$ rotation of skyrmion, which can be introduced by hand as

$\displaystyle\tilde{n}_{1}(x,y,\tau)+\mathrm{i}\tilde{n}_{2}(x,y,\tau)=\mathrm{e}^{2\pi\mathrm{i}\tau/T}(n_{1}(x,y,\tau=0)+\mathrm{i}n_{2}(x,y,\tau=0)),$
$\displaystyle\tilde{n}_{3}(x,y,\tau)=n_{3}(x,y,\tau=0).$ (118)

For the adiabatic limit $T\to\infty$, the ground state $\ket{\Omega}$ stays in
the instantaneous ground state $\ket{\Omega(\tau)}$, which is defined by
$\hat{H}(\tau)\ket{\Omega(\tau)}=E_{\Omega}(\tau)\ket{\Omega(\tau)}$, and
acquires the Berry phase as in

$\displaystyle
P\mathrm{e}^{-\int_{0}^{T}\hat{H}(\tau)\mathrm{d}\tau}\ket{\Omega}\sim\mathrm{e}^{-\int_{0}^{T}E_{\Omega}(\tau^{\prime})\mathrm{d}\tau^{\prime}}\mathrm{e}^{-\oint\braket{\Omega(\tau)}{\mathrm{d}\Omega(\tau)}}\ket{\Omega}.$
(119)

The Berry phase
$\mathrm{e}^{-\oint\braket{\Omega(\tau)}{\mathrm{d}\Omega(\tau)}}$ can be
computed as the product of fermion ground-state overlaps

$\displaystyle\braket{\Omega(\tau+\delta\tau)}{\Omega(\tau)}=\det[\Psi(\tau+\delta\tau)^{\dagger}\Psi(\tau)].$
(120)

Here, $\Psi(\tau)=(\phi_{1}(\tau),\dots,\phi_{N}(\tau))$ is the
$4L_{x}L_{y}\times N$ matrix composed of one-particle eigenvectors
$\phi_{j}(\tau),j=1,\dots,N$, which are eigenvectors of the one-particle
Hamiltonian defined in (108) having negative energies. We then numerically
find that

$\displaystyle\mathrm{e}^{-\oint\braket{\Omega}{\mathrm{d}\Omega}}=\left\\{\begin{array}[]{ll}-1&(\lambda=\pm
2),\\\ 1&(\lambda=0,\pm 4),\end{array}\right.$ (123)

as expected. The skyrmion with an odd $U(1)$ charge has an odd half-integer
spin.

It is not evident from the point of view of the lattice model that the Berry
phase of the Hamiltonian is always quantized to $\pm 1$. We leave this issue
to future studies.

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