#  Relations among topological solitons

Muneto Nitta  Department of Physics, and Research and Education Center for
Natural Sciences, Keio University, Hiyoshi 4-1-1, Yokohama, Kanagawa 223-8521,
Japan

###### Abstract

We clarify relations among topological solitons in various dimensions: a
domain wall, non-Abelian vortex, magnetic monopole and Yang-Mills instanton,
together with a (non-Abelian) sine-Gordon soliton, baby Skyrmion (lump), and
Skyrmion. We construct a composite configuration consisting of a domain wall,
vortex, magnetic monopole and Yang-Mills instanton (wall-vortex-monopole-
instanton) using the effective theory technique or moduli approximation.
Removing some solitons from such a composite, we obtain all possible composite
solitons in the form of solitons within a soliton, including all the
previously known configurations, yielding relations among topological
solitons.

###### pacs:

##  I Introduction

Topological solitons and instantons are ubiquitous in nature: they appear and
play significant roles in quantum field theories Rajaraman (1987); Manton and
Sutcliffe (2004); Shnir (2005); Vachaspati (2010); Dunajski (2010); Weinberg
(2012); Shnir (2018), supersymmetric field theories Tong (2005); *Tong:2008qd;
Eto  _et al._ (2006a); Shifman and Yung (2007); *Shifman:2009zz, cosmology
Kibble (1976); *Kibble:1980mv; Vilenkin (1985); *Hindmarsh:1994re;
*Vachaspati:2015cma; Vilenkin and Shellard (2000), QCD Eto  _et al._ (2014a),
and various condensed matter systems Mermin (1979) such as helium-3
superfluids Volovik (2009), superfluids Svistunov  _et al._ (2015), Josephson
junctions in superconductors Ustinov (2015), nonlinear media Pismen (1999);
*Bunkov:2000, and Bose-Einstein condensates (BECs) of ultracold atomic gasses
Kawaguchi and Ueda (2012). Formations and dynamics of topological solitons
have been discussed in various contexts. Among them, topological defects are
ubiquitously formed during phase transitions (the Kibble-Zurek mechanism)
Kibble (1976); *Kibble:1980mv; Zurek (1985); *Zurek:1996sj, and thus have been
studied extensively in cosmology and condensed matter physics. Dynamics of
topological solitons are mostly studied numerically. When exact or approximate
soliton solutions are available, their low-energy dynamics can be analytically
described by the moduli approximation Manton (1982) (for solitons with
supersymmetry, see Ref. Eto  _et al._ (2006b)).

Topological solitons | Type | Codim | SSB | Homotopy | Example | Single soliton moduli
---|---|---|---|---|---|---
Yang-Mills instantons | Gauge | 4 | Non | $\pi_{3}(G)$ | $\pi_{3}[SU(N)]\simeq{\mathbb{Z}}$ | ${\mathbb{R}}^{4}\times{\mathbb{R}}^{+}\times\frac{SU(N)}{SU(N-2)\times U(1)}$
Magnetic monopoles | Defect | 3 | Gauge | $\pi_{2}(G/H)$ | $\pi_{2}\left[\frac{SU(N)}{U(1)^{N-1}}\right]\simeq{\mathbb{Z}}^{N-1}$ | ${\mathbb{R}}^{3}\times U(1)$
Vortices | Defect | 2 | Gauge | $\pi_{1}(G/H)$ | $\pi_{1}[U(N)]\simeq{\mathbb{Z}}$ | ${\mathbb{R}}^{2}\times{\mathbb{C}}P^{N-1}$
Kinks, Domain walls | Defect | 1 | Global | $\pi_{0}(G/H)$ | $\pi_{0}({\mathbb{Z}}_{N})\simeq{\mathbb{Z}}_{N}$ | ${\mathbb{R}}\times U(N)$
Skyrmions | Texture | 3 | Global | $\pi_{3}(G/H)$ | $\pi_{3}[SU(N)]\simeq{\mathbb{Z}}$ | ${\mathbb{R}}^{3}\times\frac{SU(N)}{SU(N-2)\times U(1)}$
Baby Skyrmions(Lumps) | Texture | 2 | Global | $\pi_{2}(G/H)$ | $\pi_{2}[{\mathbb{C}}P^{N-1}]\simeq{\mathbb{Z}}$ |  ${\mathbb{R}}^{2}\times U(1)$ $({\mathbb{R}}^{2}\times{\mathbb{C}}^{*})$
Sine-Gordon solitons | Texture | 1 | Global | $\pi_{1}(G/H)$ | $\pi_{1}[U(N)]\simeq{\mathbb{Z}}$ | ${\mathbb{R}}\times{\mathbb{C}}P^{N-1}$
Hopfions | Texture | 3 | Global | $\pi_{3}(S^{2})$ | $\pi_{3}(S^{2})\simeq{\mathbb{Z}}$ | ${\mathbb{R}}^{3}\times U(1)$
Table 1:  “Type” implies the types of topological solitons classified into
topological gauge structures, defects, or textures, and “Codim” implies
codimensions on which soliton configurations depend. Topological gauge
structures imply topological solitons in pure gauge theories in an
$n$-dimensional space ${\mathbb{R}}^{n}$, classified by a homotopy group
$\pi_{n-1}(G)$ of a map from the boundary $S^{n-1}$ of ${\mathbb{R}}^{n}$ to a
gauge group $G$. Topological defects arise in SSB $G\to H$ of either gauge or
global symmetry $G$, classified by a homotopy group $\pi_{n-1}(G/H)$ of a map
from the boundary $S^{n-1}$ to the OPM $G/H$. Topological textures are
classified by a homotopy group $\pi_{n}(G/H)$ of a map from the whole
$n$-dimensional space ${\mathbb{R}}^{n}$ to the OPM $G/H$ with one point
compactification of the base space ${\mathbb{R}}^{n}$ to $S^{n}$ (justified by
the assumption that the boundary is mapped to the same point). “Example” gives
typical examples, and “Single soliton moduli” denotes moduli space of a single
soliton in such examples.

Topological solitons can be classified into topological defects, topological
textures, and topological gauge structures(instantons). Topological defects
are formed due to the aforementioned Kibble-Zurek mechanism when spontaneous
symmetry breaking (SSB) $G\to H$ of gauge or global symmetry $G$ occurs.
Nontrivial homotopy groups $\pi_{n-1}(G/H)$ of the order parameter manifold
(OPM) $G/H$ admit the existence of topological defects of codimension $n$
(particle-like in an $n$ dimensional space ${\mathbb{R}}^{d}$), which can be
surrounded by $S^{n-1}$ at the boundary of ${\mathbb{R}}^{n}$. One of the
remarkable features of topological defects is the fact that the unbroken
symmetry $H$ is partially or fully recovered around their cores. On the other
hand, topological textures are classified by a homotopy group $\pi_{n}(G/H)$
as a map from the whole $n$-dimensional space ${\mathbb{R}}^{n}$ to the OPM
$G/H$ with one point compactification of the base space ${\mathbb{R}}^{n}$ to
$S^{n}$, which can be justified by the assumption that the boundary is mapped
to the same point. Gauge structures imply nontrivial pure gauge field
configurations without SSB classified by a homotopy group $\pi_{n-1}(G)$ of
the gauge group $G$ on an $n$-dimensional space ${\mathbb{R}}^{n}$. Taking
into account their dimensionality up to four space-time dimensions, there are
following eight topological solitons, as summarized in Table 1:

Yang-Mills instantons are solutions to self-dual Yang-Mills equations Belavin
_et al._ (1975) which are of codimension four (point-like in the Euclidean
space ${\mathbb{R}}^{4}$) and can be particle-like in $d=5+1$ dimensions. The
(framed) moduli space of a single $SU(N)$ instanton is
${\mathbb{R}}^{4}\times{\mathbb{R}}^{+}\times\frac{SU(N)}{SU(N-2)\times
U(1)}$. For multi-instanton configurations and their moduli space, the Atiyah-
Drinfeld-Hitchin-Manin (ADHM) construction is available Atiyah  _et al._
(1978); *Corrigan:1983sv. Instantons have important applications to determine
non-perturbative effects in quantum field theories in four space-time
dimensions, in particular, with supersymmetry Seiberg and Witten (1994a);
*Seiberg:1994aj; *Dorey:2002ik; *Nekrasov:2002qd.

‘t Hooft-Polyakov magnetic monopoles are particle-like topological defects in
$d=3+1$ ’t Hooft (1974); *Polyakov:1974ek (see Refs. Shnir (2005); Preskill
(1984); *Weinberg:2006rq; *Konishi:2007dn as a review) and instanton-like in
$d=2+1$. The simplest SSB $SU(2)\to U(1)$ admits a nontrivial second homotopy
group $\pi_{2}[SU(2)/U(1)]\simeq{\mathbb{Z}}$, supporting monopoles. This can
be generalized to an $SU(N)$ gauge group maximally broken by the adjoint Higgs
field as $SU(N)\to U(1)^{N-1}$, with
$\pi_{2}[SU(N)/U(1)^{N-1}]\simeq{\mathbb{Z}}^{N-1}$. In the Bogomol’nyi-
Prasad-Sommerfield (BPS) limit, analytic solutions are available Bogomolny
(1976); *Prasad:1975kr, for which the Nahm construction offers solutions of
multiple BPS monopoles and their moduli space Nahm (1980). The moduli spaces
of two monopoles Atiyah and Hitchin (1985) and well-separated multiple
monopoles Gibbons and Manton (1995) are available. There are several
alternative methods such as the Donaldson’s rational map Donaldson (1984).
When the $SU(N)$ gauge symmetry is not maximally broken by the adjoint Higgs
field, there remains a partially unbroken non-Abelian gauge symmetry in the
vacuum. In this case, monopoles are non-Abelian monopoles Goddard  _et al._
(1977); *Weinberg:1979zt; *Weinberg:1982ev; *Auzzi:2004if. In high-energy
phenomenology and cosmology, grand unified theories (GUTs) always admit GUT
monopoles (the monopole problem) because of $\pi_{2}[G/(SU(3)\times
SU(2)\times U(1))]\simeq{\mathbb{Z}}$ Dokos and Tomaras (1980);
*Lazarides:1980va; *Preskill:1979zi.

Vortices or cosmic strings are topological defects of codimension two, which
are string-like in $d=3+1$, particle-like in $d=2+1$, and instanton-like in
$d=1+1$ (see Refs. Vilenkin and Shellard (2000); Manton and Sutcliffe (2004)
as a review). The simplest case is Abrikosov-Nielsen-Olesen (ANO) vortices
(magnetic flux tubes) Abrikosov (1957); *Nielsen:1973cs supported by
$\pi_{1}[U(1)]\simeq{\mathbb{Z}}$ when a $U(1)$ gauge symmetry is
spontaneously broken relevant for superconductors. Global analogues are global
vortices having logarithmically divergent tension in infinite space, examples
of which are given by axion strings Vilenkin and Everett (1982);
*Kawasaki:2013ae in cosmology, and superfluid vortices in superfluids or
atomic BECs in condensed matter physics. In an SSB of both local and global
symmetries, semilocal strings are known Vachaspati and Achucarro (1991);
*Achucarro:1999it. Some time ago, non-Abelian vortices containing non-Abelian
magnetic fluxes inside them were discovered Hanany and Tong (2003);
*Auzzi:2003fs, and since then they have been extensively studied in various
contexts (see Refs. Tong (2005); *Tong:2008qd; Eto  _et al._ (2006a); Shifman
and Yung (2007); *Shifman:2009zz as a review). The moduli space of a single
$U(N)$ vortex is ${\mathbb{C}}\times{\mathbb{C}}P^{N-1}$, and thus the low-
energy effective world-sheet theory is a two-dimensional ${\mathbb{C}}P^{N-1}$
model, explaining similarities between four dimensional gauge theories and
two-dimensional sigma models. Multiple vortex solutions and their moduli space
are available by the moduli matrix method Eto  _et al._ (2006c); *Eto:2006cx;
*Eto:2006db and half-ADHM construction Eto  _et al._ (2006a). The moduli space
metric can be calculated for well-separated vortices Fujimori  _et al._ (2010)
yielding the low-energy dynamics Eto  _et al._ (2011a), while coincident
vortices can be described group theoretically Eto  _et al._ (2010a). These
vortices can be generalized to non-Abelian semilocal strings Shifman and Yung
(2006); *Eto:2007yv and those of arbitrary gauge groups Eto  _et al._ (2008a);
*Eto:2009bg. In high energy phenomenology, similar non-Abelian vortices are
present in high density QCD (color superconductors) Balachandran  _et al._
(2006); *Nakano:2007dr; *Nakano:2008dc; *Eto:2009kg; *Eto:2009bh; *Eto:2009tr;
*Alford:2016dco (see Ref. Eto  _et al._ (2014a) as a review, and Refs. Alford
_et al._ (2019); *Chatterjee:2018nxe for recent developments) relevant for
neutron star cores and in two-Higgs doublet model (2HDM) Dvali and Senjanovic
(1993); *Eto:2018hhg; *Eto:2018tnk; *Eto:2021dca. On the other hand,
electroweak $Z$-strings in the Standard Model also have non-Abelian magnetic
fluxes Nambu (1977); *Vachaspati:1992fi but they are nontopological and
unstable.

Finally, the simplest topological defects are domain walls or kinks of
codimension one (see Refs. Manton and Sutcliffe (2004); Vachaspati (2010);
Shnir (2018) as a review). While the $\phi^{4}$ kink is the simplest example,
one of the more interesting examples is the ${\mathbb{C}}P^{1}$ domain wall
(or those in a $U(1)$ gauge theory with two complex Higgs scalar fields)
Abraham and Townsend (1992a); *Abraham:1992qv; *Arai:2002xa; *Arai:2003es.
These are relevant for ferromagnets and supersymmetric theories. These can be
generalized to ${\mathbb{C}}P^{N-1}$ domain walls (or those in a $U(1)$ gauge
theory with $N$ complex Higgs scalar fields) Gauntlett  _et al._ (2001a);
*Tong:2002hi. The moduli space of a single kink is ${\mathbb{R}}\times U(1)$
if masses are nondegenerate (for degenerate masses Eto  _et al._ (2008b)).
Further generalizations can be found as Grassmann domain walls (or a $U(N)$
gauge theory) Isozumi  _et al._ (2004a); *Isozumi:2004va; *Isozumi:2004vg;
*Eto:2004vy; Hanany and Tong (2006), those with generic $U(1)$ charges Eto
_et al._ (2005b), and those in nonlinear sigma models with the other target
spaces Arai  _et al._ (2009); *Arai:2011gg; *Lee:2017kaj; *Arai:2018tkf. Among
all the cases, a particularly important case in this paper is given by a non-
Abelian domain wall Shifman and Yung (2004a); Eto  _et al._ (2005c, 2008b)
that carries non-Abelian moduli ${\mathbb{R}}\times U(N)$, and thus its low-
energy effective theory is a $U(N)$ chiral Lagrangian (principal chiral
model), or the Skyrme model if a four derivative term is taken into account
Eto  _et al._ (2005c).

Apart from instantons and topological defects, another class of topological
solitons is given by topological textures such as Skyrmions. Skyrmions Skyrme
(1961); *Skyrme:1962vh are topological textures of the codimension three
characterized by $\pi_{3}[SU(2)]\simeq{\mathbb{Z}}$ in the chiral Lagrangian
of pions with a four derivative (Skyrme) term, which were proposed to describe
baryons Witten (1983) (see Refs. Manton and Sutcliffe (2004); Shnir (2018) as
a review). The moduli space of a single $SU(N)$ Skyrmion is
${\mathbb{R}}^{3}\times\frac{SU(N)}{SU(N-2)\times U(1)}$. While analytic
solutions are not available, there are some proposals to give approximate
configurations, such as the Atiyah-Manton construction based on Yang-Mills
instantons Atiyah and Manton (1989); *Atiyah:1992if and rational map ansatz
Houghton  _et al._ (1998); Ioannidou  _et al._ (1999).

Baby Skyrmions or planar Skyrmions are $2+1$ dimensional analogues of
Skyrmions characterized by the second homotopy group
$\pi_{2}(S^{2})\simeq{\mathbb{Z}}$, typically present in a $O(3)$ sigma model
with a potential term and a four derivative term Piette  _et al._ (1995a);
*Piette:1994ug (see Refs. Manton and Sutcliffe (2004); Shnir (2018) as a
review). They are string-like in $d=3+1$, particle-like in $d=2+1$ and
instanton-like in $d=1+1$. The case without a four-derivative term was known
earlier as lumps or sigma model instantons Polyakov and Belavin (1975). The
moduli space of a single lump (baby Skyrmion) is
${\mathbb{C}}\times{\mathbb{C}}^{*}$ (${\mathbb{C}}\times U(1)$). In condensed
matter physics, these solitons are simply called Skyrmions and are relevant in
various systems such as quantum Hall effects Ezawa (2013) and ferromagnets. In
particular there are great interests in chiral magnets because of experimental
finding of a Skyrmion lattice in chiral magnets Muhlbauer  _et al._ (2009);
*Yu:2010; *Han:2010by; *Lin:2014ada (see Refs. Barton-Singer  _et al._ (2020);
*Schroers:2019gbe; *Ross:2020hsw for analytic studies of Skyrmions and their
lattice in chiral magnets). The same Skyrmions can be present also in chiral
liquid crystals Foster  _et al._ (2019). The moduli space of a single Skyrmion
is only ${\mathbb{C}}$.

Finally, the lowest dimensional analogues of Skyrmions are sine-Gordon
solitons Perring and Skyrme (1962) (and double sine-Gordon solitons Condat
_et al._ (1983); *Gani:2017yla), characterized by the first homotopy group
$\pi_{1}[U(1)]\simeq{\mathbb{Z}}$ (see Refs. Manton and Sutcliffe (2004);
Shnir (2018) as a review). These solitons are of codimension one: planar in
$d=3+1$, string-like in $d=2+1$, and particle-like in $d=1+1$. They are
similar to domain walls but are distinct from them in the sense that the vacua
far from solitons are identical as the other Skyrmions, in contrast to domain
walls separating distinct vacua. In cosmology, a sine-Gordon soliton is
attached to an axion string (if the domain wall number is one) Vilenkin and
Everett (1982); *Kawasaki:2013ae. Double sine-Gordon solitons also appear in
various context such as 2HDM in high energy physics Eto  _et al._ (2018a);
*Eto:2018tnk. In condensed matter physics, (double) sine-Gordon solitons are
present in a Josephson junction (an insulator sandwiched by two
superconductors) Ustinov (1998, 2015). A lattice of (double) sine-Gordon
solitons is the ground state in chiral magnets (in a certain parameter
region), which is called a chiral soliton lattice Togawa  _et al._ (2012);
*togawa2016symmetry; *PhysRevB.97.184303; *PhysRevB.65.064433; *Ross:2020orc
(see Ref. ichiro Kishine and Ovchinnikov (2015) as a review). Recently, in
high energy physics, such chiral soliton lattices are found to be realized as
the ground states in QCD under rapid rotation Huang  _et al._ (2018);
*Nishimura:2020odq or in the strong magnetic field Brauner and Yamamoto
(2017). Among various extensions of sine-Gordon solitons, particularly
important ones in this study are non-Abelian sine-Gordon solitons supported by
$\pi_{1}[U(N)]\simeq{\mathbb{Z}}$ Nitta (2015a); Eto and Nitta (2015). In QCD,
a single non-Abelian sine-Gordon soliton can be bounded by global non-Abelian
strings Balachandran and Digal (2002a); *Nitta:2007dp; *Nakano:2007dq;
*Eto:2009wu; *Eto:2013bxa; Eto  _et al._ (2014a) and is stretched between two
chiral non-Abelian strings Eto and Nitta (2021). The moduli space of a single
$U(N)$ sine-Gordon soliton is ${\mathbb{R}}\times{\mathbb{C}}P^{N-1}$, and
thus the low energy theory is the ${\mathbb{C}}P^{N-1}$ model, as the case of
non-Abelian vortices. Such solitons can exist in a non-Abelian extension of a
Josephson junction (two color superconductors separated by an insulator or a
domain wall) Nitta (2015b, c). More interestingly, a chiral soliton lattice of
non-Abelian sine-Gordon solitons is found to be the ground state of QCD under
rapid rotation in a wide range of the parameter region Eto  _et al._ (2021b)
(instead of Abelian chiral soliton lattices Huang  _et al._ (2018);
*Nishimura:2020odq).

The last topological textures are Hopfions characterized by a Hopf map
$\pi_{3}(S^{2})\simeq{\mathbb{Z}}$ Faddeev and Niemi (1997) (see Refs. Radu
and Volkov (2008); Shnir (2018) as a review). Hopfions are present in an
$O(3)$ sigma model with a four derivative term, called the Faddeev-Skyrme
model. Typically these solitons are string-like and are linked Battye and
Sutcliffe (1998); *Battye:1998zn; *Kobayashi:2013xoa. In condensed matter
physics, Hopfions were suggested in 3He superfluids Volovik and V.P.Mineev
(1977) and superconductors Babaev  _et al._ (2002); *Rybakov:2018ktd, and have
been experimentally confirmed in a spinor BEC (but are unstable without a four
derivative term) Kawaguchi  _et al._ (2008); *Kawaguchi:2010mu; *Hall:2016;
*Ollikainen:2019dyh, liquid crystals Chen  _et al._ (2013); *Ackerman:2015;
*Ackerman:2017; *Ackerman:2017b; *Tai:2018; *Tai:2019, and magnetic materials
Kent  _et al._ (2021).

As we have seen, various topological solitons appear in diverse subjects. It
is, however, usually the case that each soliton is studied individually, and a
unified understanding is yet to be clarified. The purpose of this paper is to
present connections among all kinds of topological solitons with a help of
composite solitons in the form of “solitons within a soliton,” where a
daughter soliton is trapped inside a mother soliton, as schematically shown in
Fig. 1.111 There are also other types of composite solitons: solitons ending
on a soliton or soliton junctions, such as a non-Abelian vortex ending on a
monopole Auzzi  _et al._ (2004b); *Eto:2006dx, electroweak $Z$-string ending
on a Nambu monopole Nambu (1977), axion domain wall(s) ending on an axion
string Vilenkin and Everett (1982); *Kawasaki:2013ae, and axial (chiral)
domain walls(s) ending on an axial vortex Balachandran and Digal (2002a);
*Nitta:2007dp; *Nakano:2007dq; *Eto:2009wu; *Eto:2013bxa; Eto  _et al._
(2014a, b); Eto and Nitta (2021). In addition, vortices ending on a domain
wall or stretched between domain walls, and domain wall junctions, which can
be BPS, are known. We do not consider these in this paper.  Thus, the mother
soliton should have larger world-volume dimensions (less codimensions) than
the daughter soliton (see Appendix A for the exceptional cases in which mother
and daughter solitons have the same dimensions where a single daughter soliton
must be split into fractional solitons). In general, the mother soliton
possesses moduli parameters or collective coordinates, that is, a set of
solutions with the same energy contains free parameters. Then, the low-energy
effective world-volume theories of the mother solitons can be constructed by
the moduli approximation Manton (1982) (with supersymmetry for BPS solitons
Eto  _et al._ (2006b)), where the moduli are promoted to fields on the world-
volume. It is usually a nonlinear sigma model whose target space is the moduli
space.

Figure 1: A relation of topological solitons. When a daughter “A” of
codimensions $n$, surrounded by $S^{n-1}$ in the bulk, is absorbed into a
mother “C” it becomes “B”. The effective theory of the moduli fields of C
admits a topological soliton B surrounded by $S^{m-1}$, which has the same
topological charge with A in the bulk. Thus, A and B can be identified.

Let the codimension of the daughter soliton be $n$ in the bulk so that the
daughter soliton is surrounded by $S^{n-1}$ as A in Fig. 1. When it is
absorbed into the mother soliton C, it can be expressed by a soliton (at B) in
the effective world-volume theory of the mother soliton 222 On the other hand,
a defect in the effective theory of the mother soliton represents a defect
ending on the mother defect. Examples are given by D-brane solitons Gauntlett
_et al._ (2001b); *Shifman:2002jm; *Isozumi:2004vg; Auzzi  _et al._ (2005a);
Eto  _et al._ (2009d) (see also Bowick  _et al._ (2003)). Higher dimensional
generalizations can be found in Ref. Gudnason and Nitta (2015a). We do not
consider these cases in this paper. . In this case, the daughter soliton
inside the mother soliton is surrounded by $S^{m-1}$, which is the
intersection of $S^{n-1}$ at B and the world-volume of the mother soliton,
where $m$ can be expressed by $n$ and the codimension $c$ of the mother
soliton as $m=n-c$. The topological charge of the daughter soliton is
characterized by $\pi_{n-1}$ in the bulk if it is a defect or gauge structure,
or by $\pi_{n}$ if it is a texture. It should be unchanged at A and B. In the
world-volume theory of the mother soliton, the topological charge is
characterized by $\pi_{m-1}$ if it is a defect, or by $\pi_{m}$ if it is a
texture.

All previously known examples are listed in the next section as a review. One
of typical examples is given by confined monopoles (the second in a summary in
the next section): In the Higgs phase, magnetic fluxes emanating from the
monopole are squeezed into flux tubes. If they are squeezed into two flux
tubes, stable configuration of a monopole attached by two vortices from the
both sides. In a certain situation, those vortices are non-Abelian vortices.
These two vortices can be regarded as a single non-Abelian vortex with non-
Abelian magnetic fluxes directed in two opposite directions. In this case, the
monopoles can be represented by a kink in the effective world-sheet theory of
a single non-Abelian vortex, which is the ${\mathbb{C}}P^{N-1}$ model Tong
(2004). This vortex monopole configuration has important applications to the
correspondence quantum effects in the two dimensional ${\mathbb{C}}P^{N-1}$
model and dimensional gauge theory Shifman and Yung (2004b); *Hanany:2004ea.

In this paper, as a configuration including all the composite solitons in the
form of solitons within a soliton summarized in the next section, we construct
a configuration consisting of four different solitons: a wall-vortex-monopole-
instanton schematically drawn in Fig. 2, which is the most general and unique
configuration. Relations among topological solitons deduced from this
configuration are summarized in Table 4. Further hidden relations which can be
obtained from Table 4 are summarized in Tables 4 and 4 (derivations are
explained in subsequent sections). We conjecture all possible relations among
topological solitons can be obtained from this configuration. In our setup,
Yang-Mills instantons, magnetic monopoles, non-Abelian vortices, and non-
Abelian domain walls are primary solitons — those who can live in the bulk.
The others, Skyrmions, baby Skyrmions (lumps), (non-Abelian) sine-Gordon
solitons, ${\mathbb{C}}P^{N-1}$ lumps, ${\mathbb{C}}P^{N-1}$ domain walls, and
$U(1)^{N-1}$ global vortices are all secondary solitons — those who can live
only as daughter solitons inside a mother soliton. All relations among the
secondary solitons can be deduced from Table 4 as discussed in subsequent
sections. Throughout the paper, we use the term “Yang-Mills instantons” as
codimensions four-objects in four Euclidean space in $d=4+1$ space-time.

Figure 2: A schematic picture of the configuration of the wall-vortex-
monopole-instanton in $d=4+1$. The codimensional direction $x^{1}$ of the wall
is not shown. The black box, blue sheet, green rod, and red circle denote the
non-Abelian domain wall, non-Abelian vortex, magnetic monopole, and Yang-Mills
instanton, respectively. YM, NA, SG and w.v. denote Yang-Mills, non-Abelian,
sine-Gordon and world-volume, respectively.

|  |  | Daughters |
---|---|---|---|---
Mothers | Moduli |  $U(N)$ NA vortex |  $\frac{SU(N)}{U(1)^{N-1}}$ monopole |  $SU(N)$ YM instanton
$U(N)$ NA wall | $U(N)$ |  $U(N)$ NA SG soliton |  $U(1)^{N-1}$ global vortex |  $SU(N)$ Skyrmion
$U(N)$ NA vortex | ${\mathbb{C}}P^{N-1}$ | – |  ${\mathbb{C}}P^{N-1}$ wall |  ${\mathbb{C}}P^{N-1}$ lump
$\frac{SU(N)}{U(1)^{N-1}}$ monopole | $U(1)$ | – | – | SG soliton

Table 2: Relations among all kinds of solitons. Mother solitons can host
daughter solitons inside them. The rightmost colum: An $SU(N)$ Yang-Mills
instanton becomes an $SU(N)$ Skyrmion, ${\mathbb{C}}P^{N-1}$ lump, and
($U(1)^{N-1}$ coupled) sine-Gordon soliton inside a non-Abelian domain wall,
non-Abelian vortex and monopole, respectively. The middle column: An $SU(N)$
monopole becomes $U(1)^{N-1}$ global vortex and ${\mathbb{C}}P^{N-1}$ kink
inside a non-Abelian domain wall and non-Abelian vortex, respectively. The
leftmost column: A non-Abelian vortex becomes a non-Abelian sine-Gordon
soliton inside a non-Abelian vortex.

|  | Daughters |
---|---|---|---
Mothers | Moduli |  $U(1)^{N-1}$ global vortex |  $SU(N)$ Skyrmion
$U(N)$ NA SG soliton | ${\mathbb{C}}P^{N-1}$ |  ${\mathbb{C}}P^{N-1}$ wall |  ${\mathbb{C}}P^{N-1}$ lump
$U(1)^{N-1}$ global vortex | $U(1)$ | – | SG soliton

Table 3: Hidden relations (1) among solitons deduced from Table 4.

|  | Daughter |
---|---|---|---
Mother | Moduli |  ${\mathbb{C}}P^{N-1}$ lump (baby skyrmion) |
${\mathbb{C}}P^{N-1}$ wall | $U(1)$ | SG soliton |

Table 4: Hidden relation (2) among solitons deduced from Table 4.

This paper is organized as follows. In Sec. II, we summarize all known
composite solitons in the form of solitons within a soliton. In Sec. III, we
present our model, $U(N)$ gauge theory coupled with Higgs scalar fields. In
Sec. IV, we construct the wall-vortex-monopole-instanton configuration in the
effective field theory technique. In Sec. V, we obtain various secondary
relations deduced from the configuration. Section VI is devoted to a summary
and discussion. In Appendices, we summarize solitons within a soliton which we
do not use in this paper. In Appendix A, we summarize composite solitons in
which mother and daughter solitons have the same dimensions. In Appendix B, we
summarize composite solitons in compactified spaces ${\mathbb{R}}^{d-1}\times
S^{1}$ with twisted boundary conditions.

##  II Solitons within a soliton: a review

There are already several known such composite solitons in the form of
solitons within a soliton in the literature, deducing relations among
topological solitons. In this section, let us summarize all known composite
solitons in the form of solitons within a soliton. In the first subsection, we
summarize composite solitons consisting of two different solitons, which we
call composite solitons of 2-generations. Then, in the second subsection, we
summarize composite solitons of 3-generations.

###  II.1 2-generations (mother and daughter)

We use a rule for names of composite solitons as follows – Mother daughter.

  1. 1.

Domain-wall baby-Skyrmions(lumps), or domain-wall vortices:

The simplest example of composite solitons is a domain-wall baby-Skyrmion.

    1. (a)

${\mathbb{C}}P^{1}$ lumps (baby Skyrmions, or semilocal vortices) are sine-
Gordon solitons inside a ${\mathbb{C}}P^{1}$ wall Nitta (2012);
*Kobayashi:2013ju (see also Kudryavtsev  _et al._ (1998); *Auzzi:2006ju;
*Jennings:2013aea; *Bychkov:2016cwc). Theoretically, this setup provides a
physical proof for the lower dimensional analogue of the Atiyah-Manton
construction Sutcliffe (1992); *Stratopoulos:1992hq. Physically, this
configuration is realized in two condensed matter systems. The first is
“domain-wall Skyrmions” in chiral magnets Cheng  _et al._ (2019);
*PhysRevB.102.094402; *Nagase:2020imn; *Yang:2021 (see also Kim and
Tserkovnyak (2017)). (In the condensed matter community, baby Skyrmions are
simply called Skyrmions.) See also Ref. Hongo  _et al._ (2020) for domain-wall
instantons in $d=1+1$ chiral magnets. The second is Josephson vortices (also
called fluxons) in a Josephson junction Ustinov (1998, 2015). A Josephson
junction can be regarded as a heavy tension limit of a domain wall in a $U(1)$
gauge theory coupled with two complex scalar fields Nitta (2012);
*Kobayashi:2013ju. The domain wall separates two vacua with only one of the
complex scalar fields developing a VEV, and thus the wall is an insulator
region of the Josephson junction. Then, vortices in the bulk are absorbed into
the wall to become Josephson vortices (fluxons), whose dynamics can be
described as sine-Gordon solitons in the domain wall effective theory.

    2. (b)

${\mathbb{C}}P^{N-1}$ lumps are $U(1)^{N-1}$ coupled sine-Gordon solitons
inside ${\mathbb{C}}P^{N-1}$ walls Fujimori  _et al._ (2016).

The aforementioned Josephson junction of two superconductors can be
generalized to multi-layered $N$ parallel Josephson junctions. This case can
be described by $N-1$ parallel domain walls in the ${\mathbb{C}}P^{N-1}$ model
Fujimori  _et al._ (2016). A single Josephson vortex (fluxson) can be
described in each wall, while the interaction between Josephson vortices in
different domain walls can be also studied Fujimori  _et al._ (2016).

    3. (c)

Non-Abelian vortices are non-Abelian sine-Gordon solitons inside a non-Abelian
domain wall Nitta (2015b). A non-Abelian domain wall Shifman and Yung (2004a);
Eto  _et al._ (2005c, 2008b) can describe a non-Abelian Josephson junction (of
two color superconductors) once a Josephson term is introduced. The effective
theory of the non-Abelian domain wall is then a $U(N)$ chiral Lagrangian with
a pion mass or non-Abelian sine-Gordon model Nitta (2015b, c). Then, non-
Abelian vortices Hanany and Tong (2003); *Auzzi:2003fs absorbed into the
domain wall become non-Abelian Josephson vortices described as a non-Abelian
sine-Gordon soliton Nitta (2015a); Eto and Nitta (2015).

Similar (but different) configurations that cannot be regarded as solitons
within a soliton can be found.333 Similar configurations can be found as axion
domain wall(s) ending on an axion string Vilenkin and Everett (1982);
*Kawasaki:2013ae, axial (chiral) domain wall(s) ending on an axial (chiral)
vortex in QCD Balachandran and Digal (2002b, a); *Eto:2013bxa; Eto  _et al._
(2014a), and wall-vortex composites in GUT Kibble  _et al._ (1982);
*Everett:1982nm, 2HDMs Dvali and Senjanovic (1993); *Eto:2018hhg;
*Eto:2018tnk; *Eto:2021dca, the Gerogi-Machacek model Chatterjee  _et al._
(2018), and supersymmetric theories Ritz  _et al._ (2004); *Bolognesi:2007bh.
These cases are not solitons within a soliton except for some special cases in
which a vortex is attached by two domain walls of the same tension. The wall
decays quantum mechanically Preskill and Vilenkin (1993).

  2. 2.

Vortex monopoles:

The second simplest example of composite solitons is a vortex monopole.

    1. (a)

Monopoles are ${\mathbb{C}}P^{N-1}$ kinks inside a non-Abelian vortex
(confined monopoles) Tong (2004). This vortex monopole configuration (as well
as vortex instanton configuration given below) has important applications to
the correspondence of quantum effects in the two dimensional
${\mathbb{C}}P^{N-1}$ model and four dimensional gauge theory Shifman and Yung
(2004b); *Hanany:2004ea.

    2. (b)

Non-Abelian monopoles on multiple non-Abelian vortices can be also realized
Nitta and Vinci (2011). Mathematically, this provides a physical realization
of the Donaldson’s rational map Donaldson (1984).

    3. (c)

This correspondence can be generalized to $SO,USp$ groups Eto  _et al._
(2011b).

These can be obtained from vortex instantons given below by the Scherk-Schwarz
dimensional reduction Scherk and Schwarz (1979) or an $S^{1}$ compactification
with a twisted boundary condition Eto  _et al._ (2005d). There are similar
(but different) configurations that cannot be regarded as soliton within a
soliton.444 As similar configurations, there are vortices ending on monopoles
in QCD Nambu (1974); *Mandelstam:1974vf; *Mandelstam:1974pi, GUTs Langacker
and Pi (1980), supersymmetric QCD Auzzi  _et al._ (2004b); *Auzzi:2004yg;
*Eto:2006dx; *Cipriani:2011xp; *Chatterjee:2014rqa, and high density QCD
Gorsky  _et al._ (2011); *Eto:2011mk. A Nambu monopole in the Standard Model
Nambu (1977); Achucarro and Vachaspati (2000); Eto  _et al._ (2013) is
attached by a single $Z$-string, similar to the above. On the other hand, a
Nambu monopole in 2HDMs Eto  _et al._ (2020a); *Eto:2020hjb; *Eto:2020opf is
attached by two non-Abelian $Z$-strings of the same tension, similar to vortex
monopoles in the topic of this paper. However, it cannot be regarded as a kink
on a single $Z$-string since magnetic fluxes spread out spherically from the
monopole. See further Refs. Hindmarsh and Kibble (1985); *Kibble:2015twa;
*Kleihaus:2003nj; *Kleihaus:2003xz; *Ng:2008mp for similar objects. These
cases are not solitons within a soliton except for some special cases in which
a monopole is attached by vortex strings of the same tension.

  3. 3.

Domain-wall monopoles:

Monopoles are $U(1)^{N-1}$ global vortices inside a non-Abelian domain wall
Nitta (2015c). We call them Josephson monopoles. These can be obtained from
domain-wall instantons given below by the Scherk-Schwarz dimensional reduction
Scherk and Schwarz (1979) or an $S^{1}$ compactification with a twisted
boundary condition.

  4. 4.

Vortex instantons:

Yang-Mills instantons are ${\mathbb{C}}P^{N-1}$ lumps inside a non-Abelian
vortex Hanany and Tong (2004); Eto  _et al._ (2005d); Fujimori  _et al._
(2008). Fractional instantons and calorons are transformed to vortex monopoles
with the twisted boundary condition Eto  _et al._ (2005d). This composite
soliton can be applied for correspondence between the vortex counting and
instanton counting Fujimori  _et al._ (2015); *Fujimori:2012ab.

  5. 5.

Domain-wall instantons:

Yang-Mills instantons are Skyrmions inside a non-Abelian domain wall Eto  _et
al._ (2005c) (see also Nitta (2014, 2015c)). We also call these instantons
Josephson instantons. Originally, the term “domain-wall Skyrmion” was used for
this configuration, but in this paper we call it a “domain-wall instanton”
because of the rule of names (mother-daughter). This configuration provides a
proof of the Atiyah-Manton construction of Skyrmions Atiyah and Manton (1989);
*Atiyah:1992if. The situation is also similar to holographic QCD Hata  _et
al._ (2007).

  6. 6.

Vortex Skyrmions:

Skyrmions are sine-Gordon solitons inside a global vortex in a Skyrme model
with a twisted mass term Gudnason and Nitta (2016). Similar configurations can
be found in a BEC-Skyrme model (Skyrme model with a BEC motivated potential
term) Gudnason and Nitta (2014a); *Gudnason:2014gla.

  7. 7.

Domain-wall Skyrmions and sine-Gordon Skyrmions :

The final examples are domain-wall Skyrmions and sine-Gordon Skyrmions.

    1. (a)

Skyrmions are baby Skyrmions (lumps) inside a “non-Abelian” $S^{2}$ domain
wall Nitta (2013a); *Gudnason:2014nba (see also Refs. Kudryavtsev  _et al._
(2000); *Gudnason:2013qba). This configuration can be generalized to higher
dimensions (called matryoshka Skyrmions) Nitta (2013b). Similar configurations
can be found in a BEC-Skyrme model Gudnason and Nitta (2018). This
configuration provides a translational (or Donaldson’s type) rational map
ansatz of $SU(2)$ Skyrmions Houghton  _et al._ (1998). Physically, domain wall
Skyrmions were discussed in a chiral soliton lattice in QCD Chen  _et al._
(2022).

    2. (b)

Skyrmions are ${\mathbb{C}}P^{N-1}$ lumps inside a non-Abelian sine-Gordon
soliton Eto and Nitta (2015). This configuration provides a translational (or
Donaldson’s type) rational map ansatz of $SU(N)$ Skyrmions Ioannidou  _et al._
(1999).

  8. 8.

Lump-string Hopfions:

Hopfions are sine-Gordon solitons inside a ${\mathbb{C}}P^{1}$ lump-string.

These are all known examples of composite solitons consisting of two
generations. Some cases are BPS states preserving a fraction of
supersymmetries if one embeds the theories to supersymmetric theories Eto  _et
al._ (2006e).

###  II.2 3-generations (grandmother, mother and daughter)

In addition, there are a few examples of three generations, that is, composite
solitons of three different dimensions. We use a rule for names of composite
solitons as grandmother-mother-daughter.

  1. 1.

Vortex-monopole-instanton:

Yang-Mills instantons inside a monopole inside a non-Abelian vortex Nitta
(2013c) (see also Refs. Nitta (2013d, 2014)). Instantons are sine-Gordon
solitons inside the monopole and lumps inside the vortex at the same time,
while the monopole is a kink inside the vortex.

  2. 2.

Wall-vortex-instanton:

Yang-Mills instantons inside a vortex inside a non-Abelian domain wall Nitta
(2015c). Instantons are lumps inside a vortex and Skyrmions inside the domain
wall at the same time, while the vortex is a non-Abelian sine-Gordon soliton
inside the domain wall.

  3. 3.

Wall-monopole-instanton:

Yang-Mills instantons inside a monopole inside a non-Abelian domain wall Nitta
(2015c). Instantons are sine-Gordon solitons inside a monopole and Skyrmions
inside the domain wall at the same time, while the monopole is a kink inside
the vortex. This configuration can be obtained as the Scherk-Schwarz
dimensional reduction Scherk and Schwarz (1979) of the wall-vortex-instanton.

As already mentioned, in the following sections, we construct a configuration
of four generations as a configuration including all the aforementioned
composite solitons in the form of solitons within a soliton of two or three
generations: a wall-vortex-monopole-instanton schematically drawn in Fig. 2.

##  III The model: non-Abelian gauge theory in the Higgs phase

The theory that we consider is a $U(N)$ gauge theory coupled with Higgs scalar
fields in the Higgs phase in $d=4+1$ dimensions with the following matter
contents: a $U(N)$ gauge field $A_{\mu}(x)$, two $N$ by $N$ charged complex
scalar fields $H(x)=(H_{1}(x),H_{2}(x))$, and a real neutral adjoint $N$ by
$N$ scalar field $\Sigma(x)$. The $U(N)$ gauge (color) symmetry acts on fields
as

$\displaystyle\,A_{\mu}\to gA_{\mu}g^{-1}+ig\partial_{\mu}g^{-1},\quad H\to
gH,\quad\Sigma\to g\Sigma g^{-1},\quad\,g\in U(N)_{\rm C}.$ (1)

The Lagrangian is given as follows:

$\displaystyle{\cal L}\,=\,-\frac{1}{4g^{2}}{\rm
tr}\,F_{\mu\nu}F^{\mu\nu}+\frac{1}{g^{2}}{\rm tr}\,(D_{\mu}\Sigma)^{2}\,+{\rm
tr}\,|D_{\mu}H|^{2}+{\cal L}_{J}-V$ (2)

where $V$ is the potential term

$\displaystyle V\,=\,{g^{2}\over 4}{\rm tr}\,(HH^{\dagger}-v^{2}{\bf
1}_{N})^{2}\,+{\rm tr}\,|\Sigma H-HM|^{2},$ (3)

and $D_{\mu}$ is the covariant derivative, given by
$D_{\mu}H=\partial_{\mu}H-iA_{\mu}H$ and
$D_{\mu}\Sigma=\partial_{\mu}\Sigma-i[A_{\mu},\Sigma]$, $g$ is the gauge
coupling constant that we take common for the $U(1)$ and $SU(N)$ factors of
$U(N)$, $v$ is a real constant representing the vacuum expectation value of
$H$, and $M$ is a $2N$ by $2N$ mass matrix for $H$ given below. Apart from the
Josephson term ${\cal L}_{J}$, the model is a truncation of the bosonic part
of ${\cal N}=2$ supersymmetric theory (with eight supercharges) in $d=4+1$ Eto
_et al._ (2006a).

In the massless case $M=0$, the flavor symmetry is the maximum $SU(2N)$. This
is explicitly broken by the mass matrix $M$ that we take

$\displaystyle M={\rm diag.}(m{\bf 1}_{N}+\Delta M,-m{\bf 1}_{N}-\Delta
M),\quad\Delta M={\rm diag.}(m_{1},m_{2},\cdots,m_{N})$ (4)

with a real mass $m$ and real mass shifts $m_{a}$ much smaller than $m$:
$m_{a}\ll m$. For $m\neq 0$ with $\Delta M=0$, the flavor symmetry is
$SU(N)_{\rm L}\times SU(N)_{\rm R}\times U(1)_{\rm L-R}$, given by

$\displaystyle H_{1}\to H_{1}U_{\rm L}e^{+i\alpha},\quad H_{2}\to H_{2}U_{\rm
R}e^{-i\alpha},\quad U_{\rm L,R}\in SU(N)_{\rm L,R},\quad e^{i\alpha}\in
U(1)_{\rm L-R},$ (5)

while for $\Delta M\neq 0$ with non-degenerate mass perturbation $m_{a}\neq
m_{b}$ for $a\neq b$, the flavor symmetry is further explicitly broken to
$U(1)^{N-1}_{\rm L}\times U(1)^{N-1}_{\rm R}\times U(1)_{\rm L-R}$. In this
paper, we consider the non-degenerate masses. Without loss of generality, we
can assume $m_{r}>m_{r+1}$.

In the Lagrangian in Eq. (2), ${\cal L}_{J}$ consists of scalar couplings that
we call the Josephson interactions

$\displaystyle{\cal L}_{J}\,=\,{\cal L}_{J,1}+{\cal L}_{J,2},$ (6)
$\displaystyle{\cal L}_{J,1}\,=\,-\gamma{\rm
tr}\,(H_{1}^{\dagger}H_{2}+H_{2}^{\dagger}H_{1})\,\,$ (7) $\displaystyle{\cal
L}_{J,2}=-\sum_{r=1}^{N-1}{\beta_{r}^{2}\over v^{2}}[{\rm
tr}\,(H_{1}X_{r}H_{2}^{\dagger})+{\rm tr}\,(H_{2}X_{r}H_{1}^{\dagger})]$ (8)

where $X_{r}=X_{r}^{\dagger}$ ($r=1,\cdots,N-1$) are elements of $SU(N)$
algebra into which $\sigma_{1}$ is embedded as a diagonal submatrix. Here,
${\cal L}_{J,1}$ gives a Josephson interaction of two color superconductors
separated by a domain wall, given below.

The vacuum structures of the model are as follows. In the massless case $m=0$,
$\Delta M=0$, $\gamma=0$, $\beta_{r}=0$, the vacuum can be taken without the
lost of generality as

$\displaystyle\,H\,=\,\left(v{\bf 1}_{N},{\bf
0}_{N}\right),\,\quad\quad\Sigma={\bf 0}_{N}\,\,$ (9)

by using the $SU(2N)$ flavor symmetry. The unbroken symmetry is $SU(N)_{\rm
C+L}\times SU(N)_{\rm R}\times U(1)$, in which the factor $SU(N)_{\rm C+L}$ is
the color-flavor locked (global) symmetry. The moduli space of vacua is the
complex Grassmann manifold Higashijima and Nitta (2000)

$\displaystyle\,\,Gr_{2N,N}\,\simeq\,{SU(2N)\over SU(N)\times SU(N)\times
U(1)}.\,$ (10)

In the massive case, $m\neq 0$ but still $\Delta M=0$, the above vacua are
split into the following two disjoint vacua

$\displaystyle H\,=\,\left(v{\bf 1}_{N},{\bf 0}_{N}\right),\quad\Sigma=+m{\bf
1}_{N}:\quad SU(N)_{\rm C+L},\,$ $\displaystyle H\,=\,\left({\bf 0}_{N},v{\bf
1}_{N}\right),\quad\Sigma=-m{\bf 1}_{N}:\quad SU(N)_{\rm C+R}\,$ (11)

with the unbroken color-flavor locked (global) symmetries $g=U_{\rm L}$ and
$g=U_{\rm R}$, respectively. These vacua are color-flavor locked vacua that
can be interpreted as non-Abelian color superconductors. With the non-
degenerate mass deformation $\Delta M\neq 0$, each vacuum in Eq. (11) is
shifted to

$\displaystyle H=\left(v^{\prime}{\bf 1}_{N},{\bf
0}_{N}\right),\quad\Sigma=+m{\bf 1}_{N}+\Delta M:\quad U(1)^{N-1}_{\rm C+L},$
$\displaystyle H=\left({\bf 0}_{N},v^{\prime}{\bf
1}_{N}\right),\quad\Sigma=-m{\bf 1}_{N}-\Delta M:\quad U(1)^{N-1}_{\rm C+R},$
(12)

where $v^{\prime}$ is shifted from $v$.

In the following sections, we often work in the strong coupling (nonlinear
sigma model) limit $g\to\infty$ for explicit calculations. In this limit, we
have the constraints

$\displaystyle HH^{\dagger}=v^{2}{\bf
1}_{N},\quad\Sigma=v^{-2}HMH^{\dagger},\quad A_{\mu}={i\over
2}v^{-2}[H\partial_{\mu}H^{\dagger}-(\partial_{\mu}H)H^{\dagger}],\,$ (13)

and the model is reduced to the Grassmann sigma model with the target space
given in Eq. (10) together with a potential term, known as the massive
(twisted-mass deformed) Grassmann sigma model Arai  _et al._ (2005). In this
limit, vortices reduce to Grassmann sigma model lumps.

In this paper, we consider the following hierarchical symmetry breakings:

$\displaystyle m\quad\gg\quad\gamma\quad\gg\quad\Delta
m_{r}\quad\gg\quad\beta_{r}$ (14) wall vortex monopole instanton

with $\Delta m_{r}\equiv m_{r}-m_{r+1}$ ($r=1,2,\cdots,N-1$). The second line
denotes topological solitons that form when turning on the corresponding
parameters. In the following, we turn on these parameters from the left to
right gradually.

##  IV Wall-Vortex-Monopole-Instanton

In this section, we construct the wall-vortex-monopole-instanton configuration
in the moduli approximation, by gradually turning of hierarchical parameters
in Eq. (14). In each subsection we construct the wall, vortex, monopole, and
instanton with turning on $m$, $\gamma$, $\Delta m_{r}$, and $\beta_{r}$ in
Eq. (14), respectively.

###  IV.1 The first hierarchy: non-Abelian domain wall

We first turn on the mass $m$ in the hierarchy in Eq. (14). In the sigma model
limit, a non-Abelian domain wall solution interpolating between the two vacua
in Eq. (11) perpendicular to the coordinate $x^{1}$ can be given by Isozumi
_et al._ (2004a); Shifman and Yung (2004a); Eto  _et al._ (2005c, 2008b)

$\displaystyle H=\,H_{\rm wall}(x^{1})\,={v\over\sqrt{1+e^{\mp
2m(x^{1}-X^{1})}}}\left({\bf 1}_{N},e^{\mp m(x^{1}-X^{1})}U\right),$ (15)

with $\Sigma$ and $A_{1}$ in Eq. (13). The width of the wall is $m^{-1}$.
Here, $X^{1}$ is the position (translational modulus) of the domain wall in
the coordinate $x^{1}$ and $U$ contains group-valued moduli $U\in U(N)$:

$\displaystyle\,(X^{1},U)\in{\cal M}_{\rm wall}\,\simeq\,{\mathbb{R}}\times
U(N).\,$ (16)

The effective theory of the non-Abelian domain wall can be constructed by
using the moduli approximation Manton (1982); Eto  _et al._ (2006b); First, we
promote the moduli parameters $X^{1}$ and $U$ to moduli fields $X^{1}(x^{i})$
and $U(x^{i})$, respectively ($i=0,2,3,4$) on the world-volume of the domain
wall, and then perform integration over the codimension $x^{1}$. We thus
obtain the effective theory given by Shifman and Yung (2004a); Eto  _et al._
(2005c, 2008b):

$\displaystyle\,{\cal L}_{\rm wall}$ $\displaystyle=$ $\displaystyle\int
dx^{1}{\cal L}(H=H_{\rm wall}(x^{1};X^{1}(x^{i}),U(x^{i})))$ (17)
$\displaystyle=$ $\displaystyle{v^{2}\over
2m}\partial_{i}X^{1}\partial^{i}X^{1}-f_{\pi}^{2}{\rm
tr}\,\left(U^{\dagger}\partial_{i}UU^{\dagger}\partial^{i}U\right),\,\quad
f_{\pi}^{2}\equiv{v^{2}\over 4m},$

which is a $U(N)$ chiral Lagrangian, or principal chiral model. If we
calculate the next leading order of the derivative expansion of the effective
theory, we would obtain the Skyrme term Eto  _et al._ (2005c).

###  IV.2 The second hierarchy: non-Abelian vortex trapped inside the wall

Now we turn on the Josephson interaction $\gamma$ as the second largest
parameter in the hierarchy in Eq. (14), so that the domain wall becomes the
Josephson junction. Let us evaluate the effect of $\gamma$ perturbatively in
the domain wall effective theory in Eq. (17) provided that $\gamma$ is small
enough not to affect the domain wall solution at the leading order. We thus
find the pion mass term Nitta (2015b)

$\displaystyle\Delta{\cal L}_{{\rm wall},J}$ $\displaystyle=$
$\displaystyle\int dx^{1}{\cal L}_{J,1}(H=H_{\rm
wall}(x^{1};X^{1}(x^{i}),U(x^{i})))$ (18) $\displaystyle=$
$\displaystyle-m^{\prime 2}({\rm tr}\,U+{\rm tr}\,U^{\dagger}),\quad m^{\prime
2}\equiv{\pi\gamma\over 2m}.$

This potential term lifts the $U(N)$ vacuum manifold, leaving the unique
vacuum $U={\bf 1}_{N}$ as the case of the usual chiral Lagrangian.

When the non-Abelian vortex is placed parallel to the non-Abelian Josephson
junction (domain wall), it is absorbed into the junction to minimize the total
energy. The resulting configuration can be described as a non-Abelian sine-
Gordon soliton in the $U(N)$ chiral Lagrangian in Eq. (17) with the mass term
in Eq. (18). A non-Abelian sine-Gordon soliton (perpendicular to the $x^{2}$
coordinate) is given by Nitta (2015a); Eto and Nitta (2015):

$\displaystyle\,U\,=\,U_{\rm vortex}(x^{2})=V{\rm
diag}\,(u(x^{2}),1,\cdots,1)V^{\dagger}={\bf 1}_{N}+(u-1)\phi\phi^{\dagger}$
$\displaystyle\,u(x^{2})\,=\,\exp i\theta_{\rm
SG}(x^{2}-X^{2})=\exp\left(4i\,\arctan\exp[m^{\prime\prime}(x^{2}-X^{2})]\right),\,$
(19)

with $m^{\prime\prime 2}=\frac{m^{\prime
2}}{f_{\pi}^{2}}=\frac{2\pi\gamma}{v^{2}}$. The width of the soliton is
$m^{\prime\prime-1}\sim v/\sqrt{\gamma}$, and the tension of the soliton is
$T_{\rm SG}=8m^{\prime\prime}$. Here $X^{2}$ is translational modulus in the
coordinate $x^{2}$, and $V$ are orientational moduli taking a value in
${\mathbb{C}}P^{N-1}\simeq{SU(N)\over SU(N-1)\times U(1)}$, or more precisely
$\phi\in{\mathbb{C}}^{N}$ ($\phi^{\dagger}\phi=1$) are homogeneous coordinates
of ${\mathbb{C}}P^{N-1}$. Therefore, the moduli of the non-Abelian sine-Gordon
soliton are

$\displaystyle{\cal M}_{\rm
vortex}\simeq{\mathbb{R}}\times{\mathbb{C}}P^{N-1}$ (20)

that coincide with the moduli of the non-Abelian vortex in the bulk, except
for one translation modulus fixed to be the position $X^{1}$ of the wall. It
was also shown in Ref. Nitta (2015b) from the flux matching that this is
precisely a non-Abelian vortex.

The effective theory of the sine-Gordon soliton with the world-volume
$x^{\alpha}$ ($\alpha=0,3,4,5$) can be also obtained by the moduli
approximation Eto and Nitta (2015):

$\displaystyle{\cal L}_{\rm vortex}$ $\displaystyle=$ $\displaystyle\int
dx^{2}{\cal L}_{\rm wall}(U=U_{\rm
vortex}(x^{2};X^{2}(x^{\alpha}),\phi(x^{\alpha})))$ (21) $\displaystyle=$
$\displaystyle
C_{X}\partial_{\alpha}X^{2}\partial^{\alpha}X^{2}+C_{\phi}\left[\partial_{\alpha}\phi^{\dagger}\partial^{\alpha}\phi+(\phi^{\dagger}\partial_{\alpha}\phi)(\phi^{\dagger}\partial^{\alpha}\phi)\right]$

with the constants $C_{X}=\frac{f_{\pi}^{2}T_{\rm
SG}}{2}=\sqrt{2\pi}\frac{v{\sqrt{\gamma}}}{m}$ and
$C_{\phi}=\frac{f_{\pi}^{2}T_{\rm SG}}{m^{\prime\prime
2}}=\sqrt{2\over\pi}{v^{3}\over m\sqrt{\gamma}}$. This is the
${\mathbb{C}}P^{N-1}$ model written in terms of the homogeneous coordinates
$\phi$. The four derivative correction to the vortex effective theory was
obtained for a non-Abelian vortex in the bulk Eto  _et al._ (2012), but we do
not need it in our study.

###  IV.3 The third hierarchy: monopole trapped inside the vortex inside the
wall

Now let us turn on $\Delta m_{r}$ (in $\Delta M$) in the hierarchy in Eq.
(14). First, $\Delta M$ induces a twisted mass in the $U(N)$ chiral Lagrangian
as the domain wall effective theory in Eq. (17). This potential can be
obtained by the Scherk-Schwarz dimensional reduction Nitta (2015c):

$\displaystyle V_{\rm wall}={v^{2}\over 4m}{\rm tr}\,([\Delta
M,U]^{\dagger}[\Delta M,U]),$ (22)

implying that diagonal $U$ has lower energy. The sine-Gordon soliton (vortex
in the bulk) in Eq. (19) should be either of $N$ diagonal embedding.

In fact, the vortex effective theory in Eq. (21) is also modified by the
twisted mass $\Delta M$ as Nitta (2015c):

$\displaystyle V_{\rm vortex}$ $\displaystyle=$ $\displaystyle
C_{\phi}\left[(\phi^{\dagger}\Delta M\phi)^{2}-\phi^{\dagger}(\Delta
M)^{2}\phi\right].$ (23)

The Lagrangian in Eq. (21) with this potential term is known as the massive
${\mathbb{C}}P^{N-1}$ model. For non-degenerate mass deformation $\Delta M$,
this potential admits $N$ discrete vacua

$\displaystyle\phi_{a}^{T}=v(0,\cdots,0,1,0,\cdots),\quad a=1,\cdots,N$ (24)

where only the $a$-component is nonzero. These correspond to embedding of $u$
into diagonal elements in the solution $U$ in Eq. (19).

For later use, it is often enough to consider a ${\mathbb{C}}P^{1}$
submanifold. The vortex effective theory of the $r$-th ${\mathbb{C}}P^{1}$
submanifold parametrized by a homogeneous coordinate $u$ in
$\phi=\frac{1}{\sqrt{1+|u|^{2}}}(0,\cdots,0,1,u,0\cdots)$, where only $r$-th
and $(r+1)$-th components are nonzero, can be written as

$\displaystyle{\cal L}_{\rm
vortex,{\mathbb{C}}P^{1}}=C_{X}\partial_{\alpha}X^{2}\partial^{\alpha}X^{2}+C_{\phi}\left[{\partial_{\alpha}u^{*}\partial^{\alpha}u-\delta
m_{r}^{2}|u|^{2}\over(1+|u|^{2})^{2}}\right].$ (25)

with $\delta m_{r}\equiv m_{r+1}-m_{r}$. The vacua in the vortex theory, $u=0$
(the north pole) and $u=\infty$ (south pole) of the target space
${\mathbb{C}}P^{1}$, correspond to embedding of $u$ to the upper-left and
lower-right elements of the non-Abelian sine-Gordon solution $U$ in Eq. (19),
respectively.

The Lagrangian (23) admits $N-1$ multi-kink solutions Gauntlett  _et al._
(2001a); *Tong:2002hi; Isozumi  _et al._ (2004a); *Isozumi:2004va;
*Isozumi:2004vg; *Eto:2004vy, where the constituent kink connecting the $r$-th
and $(r+1)$-th vacua has the mass $E_{{\rm kink},r}$ ($r=1,\cdots,N-1$) which
is proportional to the mass of a monopole $E_{{\rm monopole},r}$ Tong (2004);
Shifman and Yung (2004b); *Hanany:2004ea; Nitta and Vinci (2011):

$\displaystyle E_{{\rm kink},r}=C_{\phi}\delta m_{r},\quad E_{{\rm
monopole},r}={4\pi\over g^{2}}\delta m_{r}.$ (26)

When the vortex goes to the bulk outside the wall where the vortex becomes
BPS, the Kähler moduli $C_{\phi}$ in the vortex effective theory in Eq. (23)
is replaced by ${4\pi\over g^{2}}$. Accordingly, the energy of kinks on the
vortex becomes $E_{{\rm kink},r}=E_{{\rm monopole},r}={4\pi\over g^{2}}\delta
m_{r}$. Then, one can confirm the kinks on the vortex represent the monopoles,
and from the BPS properties, they carry correct monopole charges. When the
vortex gets back into the wall, the energy becomes the first in Eq. (26)
different from the monopole energy in the bulk [the second in Eq. (26)], but
still carrying the same monopole charges.

A single monopole solution can be constructed by restricting ourselves to the
$r$-th ${\mathbb{C}}P^{1}$ submanifold in Eq. (25). It is a domain wall
interpolating the two vacua $u=0$ and $u=\infty$ Abraham and Townsend (1992a)
in the vortex effective theory (25). We place it in the $x^{3}$-coordinate as

$\displaystyle u=u_{\rm monopole}(x^{3})=e^{\mp\delta
m_{r}(x^{3}-X^{3})+i\varphi},$ (27)

where $\mp$ represents a monopole and an anti-monopole with the width
$1/\delta m_{r}$. Here, $X^{3}$ and $\varphi$ are moduli parameters
representing the position in the $x^{3}$-coordinate and $U(1)$ phase of the
(anti-)monopole. The moduli of the monopole is then

$\displaystyle{\cal M}_{\rm monopole}\simeq{\mathbb{R}}\times U(1)$ (28)

coinciding with the monopole moduli except for two translational moduli fixed
to the positions $X^{1}$ and $X^{2}$ of the domain wall and the vortex.

Let us construct the effective theory of the single monopole-string by
promoting the moduli $X^{3}$ and $\varphi$ to fields $X^{3}(x^{m})$ and
$\varphi(x^{m})$ ($m=0,4$) Manton (1982); Eto  _et al._ (2006b) on the
monopole string Nitta (2013c):

$\displaystyle{\cal L}_{r-{\rm th\;monopole}}$ $\displaystyle=$
$\displaystyle\int dx^{3}{\cal L}_{{\rm vortex},{\mathbb{C}}P^{1}}(u=u_{\rm
monopole}(x^{3};X^{3}(x^{m}),\varphi(x^{m})))$ (29) $\displaystyle=$
$\displaystyle{C_{\phi}\over 2\delta
m_{r}}[(\partial_{m}X^{3})^{2}+(\partial_{m}\varphi)^{2}]$

which is a free theory, a sigma model with the target space
${\mathbb{R}}\times U(1)$.

###  IV.4 The fourth hierarchy: instanton inside the monopole inside the
vortex inside the wall

Finally, we turn on the smallest parameters $\beta_{r}$ in the hierarchy in
Eq. (14). The term proportional to the parameter $\beta_{r}$ perturbatively
induces the following deformation term to the effective Lagrangian on the non-
Abelian domain wall in Eq. (17):

$\displaystyle\Delta{\cal L}_{{\rm wall},\beta}$ $\displaystyle=$
$\displaystyle\int dx^{1}{\cal L}_{J,2}(H=H_{\rm
wall}(x^{1};X^{1}(x^{i}),U(x^{i})))$ (30) $\displaystyle=$
$\displaystyle\sum_{r}\frac{\pi\beta_{r}^{2}}{2m^{\prime\prime}}{\rm
tr}\,[X_{r}(U+U^{\dagger})].$

Next, this term induces the following deformation term to the effective
Lagrangian of the non-Abelian vortex:

$\displaystyle\Delta{\cal L}_{{\rm vortex},\beta}$ $\displaystyle=$
$\displaystyle\int dx^{2}\Delta{\cal L}_{{\rm wall},\beta}(U=U_{\rm
vortex}(x^{2};X^{2}(x^{\alpha}),\phi(x^{\alpha})))$ (31) $\displaystyle=$
$\displaystyle-\sum_{r}\frac{4\pi\beta_{r}^{2}}{m^{\prime\prime}{}^{2}}(\phi^{\dagger}X_{r}\phi).$

This is known as a momentum map (or a D-term in supersymmetric theories). It
reduces in the $r$-th ${\mathbb{C}}P^{1}$ submanifold to

$\displaystyle\Delta{\cal L}_{{\rm
vortex},\beta}=-\frac{4\pi\beta_{r}^{2}}{m^{\prime\prime}{}^{2}}\frac{1}{1+|u|^{2}}\left(\begin{array}[]{cc}1&u^{*}\end{array}\right)\sigma_{1}\left(\begin{array}[]{c}1\cr
u\end{array}\right)=-\frac{4\pi\beta_{r}^{2}}{m^{\prime\prime}{}^{2}}\frac{u+u^{*}}{1+|u|^{2}}.$
(35)

Finally, this deformation induces the deformation term in the monopole
effective action. By considering a single monopole-string, the effective
Lagrangian is given by

$\displaystyle\Delta{\cal L}_{r{\rm-th\;monopole},\beta}$ $\displaystyle=$
$\displaystyle\int dx^{3}\Delta{\cal L}_{{\rm vortex},\beta}(u=u_{\rm
monopole}(x^{3};X^{3}(x^{m}),\varphi(x^{m})))$ $\displaystyle=$
$\displaystyle-C_{r}\cos\varphi\equiv-V_{\rm monopole}$ $\displaystyle C_{r}$
$\displaystyle\equiv$ $\displaystyle\frac{\pi^{2}\beta_{r}^{2}}{2\delta
m_{r}m^{\prime\prime}{}^{2}}=\frac{\pi\beta_{r}^{2}v^{2}}{4\delta
m_{r}\gamma}.$ (36)

We thus arrive at the monopole effective Lagrangian summarized as

$\displaystyle{\cal L}_{r{\rm-th\;monopole},\beta}$ $\displaystyle=$
$\displaystyle{C_{\phi}\over 2\delta
m_{r}}[(\partial_{m}X^{3})^{2}+(\partial_{m}\varphi)^{2}-D_{r}\cos\varphi],$
$\displaystyle D_{r}$ $\displaystyle\equiv$ $\displaystyle\frac{2\delta
m_{r}C_{r}}{C_{\phi}}={\pi^{3/2}\over 2\sqrt{2}}{m\beta_{r}^{2}\over
v\sqrt{\gamma}}.$ (37)

This is nothing but the sine-Gordon model.

We can consider multi-monopoles (multi-walls in the ${\mathbb{C}}P^{N-1}$
model). In that case, we will obtain a $U(1)^{N-1}$ coupled sine-Gordon model
Fujimori  _et al._ (2016) in which the interaction of sine-Gordon solitons on
different domain walls was studied. In our purpose, a single monopole is
enough.

The monopole effective action in Eq. (37) admits sine-Gordon solitons. As the
simplest, a single soliton solution is given by

$\displaystyle\varphi=\varphi_{\rm
instanton}(x^{4})=4\arctan\exp{\sqrt{D_{r}\over 2}(x^{4}-X^{4})}+\pi.\ \quad$
(38)

The width of the soliton is $\Delta x^{4}\sim
1/\sqrt{D_{r}}\sim\sqrt{v}\gamma^{1/4}/\sqrt{m}\beta_{r}$. This carries a lump
charge in the ${\mathbb{C}}P^{1}$ model as the vortex effective action Nitta
(2012); *Kobayashi:2013ju :

$\displaystyle T_{\rm lump}$ $\displaystyle\equiv$ $\displaystyle\int
d^{2}x{i(\partial_{i}u^{*}\partial_{j}u-\partial_{j}u^{*}\partial_{i}u)\over(1+|u|^{2})^{2}}=2\pi
k$ (39)

with $k=1\in\pi_{2}({\mathbb{C}}P^{1})$, which is also the topological charge
$\pi_{2}({\mathbb{C}}P^{N-1})$ of the whole ${\mathbb{C}}P^{N-1}$. The lump
charge in the vortex induces the instanton charge in the bulk, and thus lumps
in the vortex effective action correspond to Yang-Mills instantons in the bulk
Hanany and Tong (2004); Eto  _et al._ (2005d).

Combining all discussions together, we finally reach the configuration in Fig.
2.

##  V Relations among all topological solitons

###  V.1 Subconfigurations

Once we obtain the composite state of the wall-vortex-monopole-instanton, we
can obtain various configurations consisting of a fewer number of solitons by
removing some solitons from it.

Taking limits to send various parameters to zero, we obtain subconfigurations
as schematically shown in Fig. 3. We start from (c) wall-vortex-monopole-
instanton. We first remove one soliton from (c). By taking the limit $m\to 0$,
$\gamma\to 0$, or $\delta m\to 0$, we can remove the domain wall, vortex, or
monopole to reach (f) the vortex-monopole-instanton, (b) wall-monopole-
instanton, and (d) wall-vortex-instanton, respectively. Next, we remove two
solitons from (c). By taking the limit $\delta m,\gamma\to 0$, $m,\delta m\to
0$, or $m,\gamma\to 0$, we to reach (a) the wall-instanton, (g) vortex-
instanton, or (e) monopole-instanton, respectively. Only (e) is unstable as it
is, but it can be stable if we make a closed loop Nitta (2014). We also can
remove instantons from each configuration in the limit $\beta_{a}\to 0$.

Figure 3: Removing constituent solitons. Starting from (c) the wall-vortex-
monopole-instanton, we can remove the domain wall, vortex, or monopole in the
limits $m\to 0$, $\gamma\to 0$ or $\delta m\to 0$, respectively to reach (f)
the vortex-monopole-instanton, (b) wall-monopole-instanton or (d) wall-vortex-
instanton, respectively. We can remove two solitons in the limits $\delta
m,\gamma\to 0$, $m,\delta m\to 0$ or $m,\gamma\to 0$ to reach (a) the wall-
instanton, (g) vortex-instanton or (e) monopole-instanton, respectively. We
also can remove instantons from each configuration in the limit $\beta_{a}\to
0$.

###  V.2 Hidden relations

All possible relations among topological solitons (domain walls, vortices,
monopoles and Yang-Mills instantons) obtained thus far are summarized in Table
4. Mother solitons can host solitons with less world-volume dimensions or
equivalently of more codimensions as daughters. The first row of the table
shows that a $U(N)$ domain wall can host a $U(N)$ non-Abelian vortex,
$SU(N)/U(1)^{N-1}$ monopole and $SU(N)$ instanton, and in the domain wall
effective theory, which is the $U(N)$ chiral Lagrangian, these solitons are
realized as a $U(N)$ non-Abelian sine-Gordon soliton, $U(1)^{N-1}$ global
vortex and $SU(N)$ Skyrmion, respectively. In the second row, a vortex can
host a monopole and instanton as a ${\mathbb{C}}P^{N-1}$ kink and
${\mathbb{C}}P^{N-1}$ lump, respectively in the vortex world-volume theory,
that is the ${\mathbb{C}}P^{N-1}$ model. Finally in the third row, a magnetic
monopole-string can host an $SU(N)$ Yang-Mills instanton as a sine-Gordon
soliton in the monopole world-volume effective theory which is the sine-Gordon
model.

Table 4 contains further hidden relations among topological solitons.

|
---|---
(a) | (b)
Figure 4: Hidden relations. (a) Within a domain wall, an $SU(N)$ Skyrmion at A
turns to a ${\mathbb{C}}P^{N-1}$ lump at B inside a $U(N)$ non-Abelian sine-
Gordon soliton (that is a non-Abelian vortex in the bulk point of view), and
to a sine-Gordon soliton at C in a $U(1)^{N-1}$ global vortex (that is a
monopole in the bulk point of view). Within a non-Abelian vortex, a
${\mathbb{C}}P^{N-1}$ lump at B becomes a sine-Gordon soliton at C inside a
${\mathbb{C}}P^{N-1}$ kink. (b) A $U(1)^{N-1}$ global vortex (that is a
monopole in the bulk point of view) becomes a ${\mathbb{C}}P^{N-1}$ kink
inside a $U(N)$ non-Abelian sine-Gordon soliton.

If one places an instanton inside a domain wall as A in Fig. 4(a), it is a
Skyrmion in the domain wall effective theory as denoted above. Then, if this
Skyrmion moves into B on a non-Abelian vortex, it becomes a
${\mathbb{C}}P^{N-1}$ lump in the vortex effective theory. This relation
implies that a $U(N)$ non-Abelian sine-Gordon in the $U(N)$ chiral Lagrangian
can host an $SU(N)$ Skyrmion as a ${\mathbb{C}}P^{N-1}$ lump in its world-
volume effective theory which is the ${\mathbb{C}}P^{N-1}$ model Eto and Nitta
(2015). This corresponds to the first row and second column in Table 4.

If the Skyrmion at A moves to C on a monopole Fig. 4(a) directly without
passing through a vortex, it becomes a sine-Gordon soliton. This implies that
a $U(1)^{N-1}$ global vortex in the $U(N)$ chiral Lagrangian can host an
$SU(N)$ Skyrmion as a sine-Gordon soliton in its world-volume effective theory
which is a sine-Gordon model. This corresponds to the second row and second
column in Table 4. The case of $N=2$ was constructed in Ref. Gudnason and
Nitta (2016). This relation for general $N$ is not studied yet.

We also can consider a monopole as a daughter soliton instead of the
instanton. As in Fig. 4(b), if a monopole inside a domain wall is trapped into
a non-Abelian vortex inside the domain wall as indicated by the arrow in the
figure, it becomes a ${\mathbb{C}}P^{N-1}$ kink. This implies that a $U(N)$
non-Abelian sine-Gordon in the $U(N)$ chiral Lagrangian can host a
$U(1)^{N-1}$ global vortex (which is a monopole in the bulk point of view) as
a ${\mathbb{C}}P^{N-1}$ kink in its world-volume effective theory which is the
${\mathbb{C}}P^{N-1}$ model. This corresponds to the first row and first
column in Table 4. This relation is not studied yet.

Finally, going back to Fig. 4(a), if the instanton in the bulk (that is a
${\mathbb{C}}P^{N-1}$ lump (or baby Skyrmion) inside the non-Abelian vortex)
at B moves to C on a monopole in the bulk (that is a ${\mathbb{C}}P^{N-1}$
kink in the non-Abelian vortex), it becomes a sine-Gordon soliton, see Table
4. This relation implies that a ${\mathbb{C}}P^{N-1}$ domain wall in the
massive ${\mathbb{C}}P^{N-1}$ model can host a ${\mathbb{C}}P^{N-1}$ lump as a
sine-Gordon soliton Nitta (2012); *Kobayashi:2013ju; Fujimori  _et al._
(2016). In the literature, this relation is known as a domain wall (baby)
Skyrmion in field theory, attracting a lot of attention recently in condensed
matter physics because of experimental confirmations in chiral magnets Cheng
_et al._ (2019); *PhysRevB.102.094402; *Nagase:2020imn; *Yang:2021 (see also
Kim and Tserkovnyak (2017)).

##  VI Summary and Discussion

By using the effective field theory technique or the moduli approximation, we
have constructed the wall-vortex-monopole-instanton configuration in Fig. 2 as
the most general composite soliton in the form of solitons within a soliton.
It reduces to all the previously known composite solitons, providing relations
among various topological solitons in various dimensions summarized in Tables
4, 4 and 4. We have conjectured that this exhausts all possible relations
among topological solitons.

We have considered non-degenerate masses $m_{a}\neq m_{b}$ if $a\neq b$,
giving rise to Abelian domain walls in the ${\mathbb{C}}P^{N-1}$ model on a
non-Abelian vortex as well as ‘t Hooft-Polyakov monopoles. If we consider
partially degenerate masses, the ${\mathbb{C}}P^{N-1}$ domain walls Eto  _et
al._ (2008b) and monopoles on the vortex Nitta and Vinci (2011) are non-
Abelian. In this case, instantons as sine-Gordon solitons on the monopole-
string may be non-Abelian sine-Gordon solitons.

Among all major topological solitons in Table 1, we have not considered
Hopfions. Hopfions inside a non-Abelian vortex was studied before Nitta
(2013d), but it is unclear to what it corresponds in the bulk.

As one of the most important applications, one can consider quantum effects,
or non-perturbative effects in different dimensions. Instanton corrections to
monopoles correspond to vortex corrections to kinks through instanton vortex
configurations. This gives a correspondence between a four dimensional $U(N)$
gauge theory and a two dimensional ${\mathbb{C}}P^{N-1}$ model. Hopefully,
this relation can be generalized to other mother-daughter relations completed
in this paper.

Electrically charged solitons or dyonic extensions can be also studied. As for
dyonic extensions of individual solitons, dyonic instantons Lambert and Tong
(1999), dyons Julia and Zee (1975), dyonic vortices Collie (2009);
*Eto:2014gya, Q-lumps Leese (1991); *Abraham:1991ki and Q-kinks Abraham and
Townsend (1992a); *Abraham:1992qv are known. Dyonic extensions of composite
solitons are not well studied except for few examples: domain wall networks
and strings stretched between walls Eto  _et al._ (2007d, 2006e).

We have not studied fermions in this paper. When fermions are coupled to the
system, topological solitons are often accompanied by fermion zero modes, such
as on a kink Jackiw and Rebbi (1976) and a vortex Jackiw and Rossi (1981). The
existence of such fermions is ensured by the index theorem. For instance, the
index theorem and fermion zero modes were studied for non-Abelian vortices in
supersymmetric $U(N)$ gauge theory Hanany and Tong (2003) and in dense QCD
Yasui  _et al._ (2010); *Fujiwara:2011za. However, they have not been well
understood for composite solitons such as solitons within a soliton, and thus
remain as one of future problems.

Probably, a more elegant framework to systematically understand composite
solitons is offered by higher-form symmetry and higher group Gaiotto  _et al._
(2015). In fact, mathematical structures of the axion electrodynamics
admitting composite solitons were recently clarified in terms of higher-form
symmetries and higher groups Hidaka  _et al._ (2020a); *Hidaka:2020izy;
*Hidaka:2021mml; *Hidaka:2021kkf (see also Refs. Hidaka  _et al._ (2019);
*Hidaka:2019mfm). Mathematical structures behind the configuration in the
present study are yet to be clarified.

Finally, we hope that our most general configuration itself has a direct
application in condensed matter physics. See Refs. Volovik (2020);
*Volovik:2020zqc for composite solitons in helium-3 superfluids.

## Acknowledgements

The author thanks Minoru Eto and Sven Bjarke Gudnason for collaborations in
various works. This work is supported in part by the JSPS Grant-in-Aid for
Scientific Research (KAKENHI Grant No. JP18H01217).

##  Appendix A Mother and daughter of the same dimensions

As composite solitons in the form of solitons within a soliton, mother and
daughter solitons can have the same dimensions in the following cases. In
these cases, a single daughter soliton must be split into a set of fractional
solitons.

  1. 1.

Domain-wall sine-Gordon solitons:

The double sine-Gordon model admits a false (metastable) vacuum in addition to
the true vacuum in a certain parameter region Condat  _et al._ (1983);
*Gani:2017yla (see also Refs. Eto  _et al._ (2018a, b); Ross  _et al._
(2021b)). In this case, a single soliton profile has double peaks which can be
identified with a molecule of two constituents. Each constituent is an
(unstable) domain wall or anti-domain wall separating the true and false
vacua, carrying half sine-Gordon soliton winding. They are linearly confined
by the false vacuum energy present between them. In the triple sine-Gordon
model, a single sine-Gordon soliton is split into three constituents with 1/3
sine-Gordon topological charges Eto  _et al._ (2014a).

  2. 2.

Vortex baby-Skyrmions (or vortex lump):

In the baby Skyrme model ($O(3)$ or ${\mathbb{C}}P^{1}$ model) with the easy
plane potential, a single baby Skyrmion is split into a pair of global vortex
and anti-global vortex with half baby-Skyrmion topological charges,
constituting a half baby-Skyrmion molecule Jaykka and Speight (2010);
*Kobayashi:2013aja; *Kobayashi:2013wra; *Leask:2021hzm. Each half baby-
Skyrmion (lump) can be interpreted as an Ising spin inside a global
(anti-)vortex. Half baby-Skyrmions (lumps) are sometimes called merons. For a
${\mathbb{C}}P^{N-1}$ model with a generalization of the easy plane potential,
a single baby Skyrmion is split into a set of $N$ $U(1)^{N-1}$ global-vortex
Skyrmions with $1/N$ lump ($\pi_{2}$) topological charges Akagi  _et al._
(2021).

A $U(1)$ gauged ${\mathbb{C}}P^{1}$ model (without a Skyrme term) or
$U(1)^{N-1}$ gauged ${\mathbb{C}}P^{N-1}$ model are also known, for which
vortices are local, and configurations are BPS, and thus can be embedded into
a supersymmetric theory Schroers (1995); *Schroers:1996zy; *Baptista:2004rk;
*Nitta:2011um. This can be applied to the stabilization of semilocal strings
Eto  _et al._ (2016). See Ref. Samoilenka and Shnir (2018) for the case with a
Skyrme term.

  3. 3.

Monopole Skyrmions:

In a Skyrme model with the potential term admitting $S^{2}$ vacua, a single
$SU(2)$ Skyrmion is split into a pair of a global monopole and an anti-global
monopole with half-Skyrmion topological charges, constituting a half Skyrmion
molecule Gudnason and Nitta (2015c). Each half Skyrmion can be interpreted as
an Ising spin inside a global (anti-)monopole.

An $SU(2)$ gauged version is also known, for which monopoles are local (‘t
Hooft-Polyakov type) Arthur and Tchrakian (1996); *Kleihaus:1999ea;
*Grigoriev:2002qc. It is also called a Skyrmed monopole.

##  Appendix B Solitons on compactified spaces with twisted boundary
conditions

Topological solitons on compactified spaces ${\mathbb{R}}^{d-1}\times S^{1}$
with twisted boundary conditions along $S^{1}$ have half topological charges
together with topological charges in ${\mathbb{R}}^{d-1}$. These solitons are
also composite solitons in the form of solitons within a soliton.

  1. 1.

Monopole instantons:

Yang-Mills instantons on ${\mathbb{R}}^{3}\times S^{1}$ are called calorons.
The situation with twisted boundary condition is known as calorons with a
nontrivial holonomy along $S^{1}$ Lee (1998); *Lee:1998bb; *Kraan:1998pm;
*Kraan:1998sn. A single $SU(2)$ instanton can be decomposed into a pair of a
monopole string and an anti-monopole string winding along $S^{1}$ with half
instanton charges. A bion is a pair of a monopole-string with $+1/2$ instanton
charge and an anti-monopole string with $-1/2$ instanton charge, and has been
considered for applications to confinement and mass gap Unsal (2009);
*Anber:2011de; *Argyres:2012ka. Similarly, an $SU(N)$ instanton is decomposed
into $N$ monopole strings with $1/N$ instanton charges.

  2. 2.

Vortex Skyrmions:

Skyrmions on ${\mathbb{R}}^{2}\times S^{1}$ with twisted boundary conditions
are considered with Harland and Ward (2008a) and without Nitta (2015d) the
Skyrme term. A single $SU(2)$ Skyrmion can be decomposed into a pair of a
global vortex string and an anti-global vortex string winding along $S^{1}$
with half Skyreme ($\pi_{3}$) charges. A bion is a pair of a global vortex-
string with $+1/2$ Skyrme charge and an anti-global vortex string with $-1/2$
Skyrme charge Nitta (2015d). Likewise, a single $SU(N)$ Skyrmion is decomposed
into $N$ $U(1)^{N-1}$ global-vortex Skyrmions with $1/N$ Skyrme charges Nitta
(2015d). Applications of bions in this case are yet to be clarified. This
relation can be embedded inside a non-Abelian domain wall, reproducing the
aforementioned monopole instantons Nitta (2015c).

  3. 3.

Domain-wall lumps:

    1. (a)

${\mathbb{C}}P^{N-1}$ lumps on ${\mathbb{R}}^{1}\times S^{1}$ with twisted
boundary conditions were first found in Ref. Eto  _et al._ (2005d) (see also
Refs. Bruckmann (2008); *Brendel:2009mp; *Harland:2009mf; *Bruckmann:2014sla).
See Ref. Harland and Ward (2008b) for baby Skyrmions for the case with the
Skyrme term. A single ${\mathbb{C}}P^{1}$ lump can be decomposed into a pair
of a domain wall and an anti-domain wall winding along $S^{1}$ with half lump
($\pi_{2}$) charges. Similarly a single ${\mathbb{C}}P^{N-1}$ lump can be
decomposed into a set of $N$ domain walls with $1/N$ lump ($\pi_{2}$) charges.
A bion is a pair of a domain wall with $+1/2$ lump charge and an anti-domain
wall with $-1/2$ lump charge, and similar for the ${\mathbb{C}}P^{N-1}$ model
Dunne and Unsal (2012); *Dunne:2012zk; *Misumi:2014jua; *Misumi:2014rsa;
*Fujimori:2018kqp, which has been extensively applied to resurgence theory of
the ${\mathbb{C}}P^{N-1}$ model. This relation can be embedded inside a non-
Abelian vortex reproducing the aforementioned monopole instantons Eto  _et
al._ (2005d). Recently, Monte Carlo simulations have been performed Misumi
_et al._ (2019); *Fujimori:2020zka in which fractional lumps and bions have
been observed.

    2. (b)

Grassmann lumps on ${\mathbb{R}}^{1}\times S^{1}$ with twisted boundary
conditions were first found in Refs. Eto  _et al._ (2006f); *Eto:2007aw. Bions
and application to the resurgence theory can be found in Refs. Misumi  _et
al._ (2015b); *Dunne:2015ywa.

  4. 4.

Lump-string Hopfions: Hopfions on ${\mathbb{R}}^{2}\times S^{1}$ were studied
in Ref. Kobayashi and Nitta (2013c).

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