数理解析研究所講究録
第 1837 巻 2013 年 29-43

29Partial Differential Equations Arising from the
Chern-Simons Gauged $O(3)$ Sigma Model
Jongmin Han*

Abstract
In this paper, we survey recent progress for the study of partial differential equations arising from the self-dual Chern-Simons gauged $O(3)$ sigma model. We review
the classical $O(3)$ sigma model and its extension to gauge field models. Especially, theMaxwell gauged model and the Chern-Simons gauged models are described for broken
and unbroken symmetries. We derive the self-dual equations and their reduction to
the elliptic equations. We discuss recent progress for the existence of solutions for the
reduced equations.

1

Introduction

The classical $O(3)$ sigma model originates from the description of the planar ferromagnet.
This model allows an energy lower bound of Bogomol’nyi type which is saturated by thesolutions of the self-dual equations. All minimal energy solutions can be obtained bysolving these self-dual equations and they turn out to be meromorphic functions [3]. The
corresponding minimal energy is given by the degree of solutions.
One of the important property of these soliton solutions is the scale invariance. Dueto this conformal invariance, the size of these solitons can change arbitrarily during thetime evolution without costing any energy. As a result, this model becomes unsuitable asa model for particles.
There have been several results to break scale invariance of the model. The inclusionof a Skyrme term and a potential term leads to cs-called baby Skyrme model which have
soliton solutions of definite size [26, 27]. However, this model is nether integrable nor of
Bogomol’nyi type. Another trial is to add a potential term and prevent the solitons fromcollapsing.
suitable choice of the potential make the $mo$ del have self-dual structure andthe corresponding soliton solutions are called -lump [23, 24].
The third possibility of breaking the scale invariance of the sigma model is to introduce
a $U(1)$ gauge field. This was initiated by Schroers in [29], where he proposed a $U(1)$ gauged
model whose dynamics is governed by the Maxwell term. He introduce a new gauge
covariant derivative and derive the self-dual equations by adding a suitable potential. As
a consequence, this model possesses topological solitons which gives the lower bound of
$A$

$Q$

*Department of Mathematics,Kyung Hee University, Seoul, 130-701, Korea. email : jmhan@khu.ac.kr

30
energy. On the other hand, a different type of gauged $O(3)$ sigma model was studied in[1, 14, 21], where the Chem-Simons action is responsible for the dynamics. The additionof Chern-Simons action can give soliton solutions which have finite charge and angularmomentum besides broken scale invariance. The vacuum of the potential consists of twosymmetric phase which gives topological and nontopological solutions.
For both the Maxwell gauged sigma model and the Chem-Simons sigma gauged model,
one can estabhsh new Lagrangian by inserting a parameter which breaks the gauge symmetry. This approach was taken in [21, 30] and gives rich structures. As an vacuum manifold
it allows asymmetric phase as well as symmetric phase such that one may consider theplanar topological type solutions.
In this article, we review these models in detail and exhibit some mathematical problems related to them. One may refer to [33] for more general overview of the $O(3)$ models
in the physics literature.

Classical $O(3)$ Sigma Model

2

The Lagrangian of the classical planar ferromagnet model is given by$\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\cdot\partial^{\mu}\phi,$

where the spin vector
is given by(2.1)$\phi$:$\mathbb{R}^{2,1}arrow S^{2}$

describes a Heisenberg ferromagnet. The static energy$\mathcal{E}(\phi)=\int_{\mathbb{R}^{2}}\{(\partial_{1}\phi)^{2}+(\partial_{2}\phi)^{2}\}dx.$

Finite energy condition implies that goes to a constant unit vector at infinity which
is well. Hence, the Hopf degree
toa continuous map from
makes
group
homotopy
in
the
class
homotopy
represents
a
such
that
definedOne of the fundamental problem is to find a energy-minimizing configuration among eachtopological class$\phi$$S^{2}$$\phi$$S^{2}$$\deg\phi\in \mathbb{Z}$$\pi_{2}(S^{2})=\mathbb{Z}.$$\phi$$C_{N}=\{\phi:\mathbb{R}^{2}arrow S^{2}:\mathcal{E}(\phi)<\infty, \deg(\phi)=N\}, N\in \mathbb{Z}.$In the work of Belavin-Polyakov [3], it was shown that this static model has a self-dual. It isstructure as follows. We first recall the formula of degree for a function :densityofthecurrent
terms
degreeinconvenient to express the$\phi$$k_{\alpha}= \frac{1}{2}\epsilon_{\alpha\beta\gamma}\phi\cdot\partial^{\beta}\phi\cross\partial^{\gamma}\phi.$Then, the degree is given by$\deg\phi=\frac{1}{4\pi}\int_{\mathbb{R}^{2}}k_{0}dx=\frac{1}{4\pi}\int_{\mathbb{R}^{2}}\phi\cdot(\partial_{1}\phi\cross\partial_{2}\phi)dx,$$S^{2}arrow S^{2}$

31
By an elementary calculation,$\mathcal{E}(\phi)=\int_{\mathbb{R}^{2}}(|\partial_{1}\phi\pm\phi\cross\partial_{2}\phi|^{2}+|\partial_{2}\phi\mp\phi\cross\partial_{1}\phi|^{2})dx\pm\int_{\mathbb{R}^{2}}\phi\cdot(\partial_{1}\phi\cross\partial_{2}\phi)\geq 4\pi|\deg\phi|.$

The lower bound saturated if and only if(2.2)$\phi$

satisfying the following self-dual equations$\partial_{j}\phi=\mp\epsilon_{jk}(\phi\cross\partial_{k}\phi)$.

is a solution of (2.2) with upper signs, thenis a solution for lower signs. Hence,
we only consider upper signs in the following.
In order to simplify (2.2), we introduce the stereographic projection from the southpole $s=(0,0, -1)$ of
to obtain a map
defined byIf$-\phi$$\phi$$S^{2}$(2.3)$u:\mathbb{R}^{2}arrow \mathbb{R}^{2}$$u=(u_{1}, u_{2}) , u_{1}= \frac{\phi_{1}}{1+\phi_{3}}, u_{2}=\frac{\phi_{2}}{1+\phi_{3}}.$

Conversely, we have$\phi_{1}=\frac{2u_{1}}{1+|u|^{2}}, \phi_{2}=\frac{2u_{2}}{1+|u|^{2}}, \phi_{3}=\frac{1-|u|^{2}}{1+|u|^{2}}.$

Let $\mathcal{P}=\phi^{-1}(s)=\{p_{1}, \cdots,p_{N}\}$ . We note that$\lim_{xarrow p_{j}}|u(x)|^{2}=\lim_{xarrow p_{j}}\frac{1-\phi_{3}}{1+\phi_{3}}=\infty.$

We note that away from(2.4)

$p_{j},$$(2.2)$

becomes$\partial_{1}u_{1}=\partial_{2}u_{2}, \partial_{1}u_{2}=-\partial_{2}u_{1},$which implies that $u(z)=u_{1}(z)+iu_{2}(z)$ with $z=x_{1}+ix_{2}$ satisfies the Cauchy-Riemannequation. Let $\mathcal{Q}=\phi^{-1}(n)=\{q_{1}, \cdots, q_{N}\}$ . Then,(2.5)$u(z)=c \frac{(z-q_{1})\cdot.\cdot.\cdot.(z-q_{M})}{(z-p_{1})(z-p_{N})}$is a solution of (2.4).
We now calculate the degree of the solutions given by (2.5). It tums out that wehave an obstruction $M\leq N-1$ . The degree formula can be rewritten in terms of as
follows:$\phi$$u$$\deg\phi=\frac{i}{2\pi}\int_{\mathbb{R}^{2}}\frac{\partial_{1}u\partial_{2}\overline{u}-\overline{u}\partial_{2}u}{(1+|)^{2}}dx=\frac{1}{4\pi}\int_{\mathbb{R}^{2}}J_{12}dx,$where$J_{12}= \partial_{1}J_{2}-\partial_{2}J_{1}, J_{k}=\frac{i(u\partial_{k}\overline{u}-\overline{u}\partial_{k}u)}{(1+|u|^{2})}.$

Hence,$4 \pi\deg\phi=\lim_{rarrow\infty}\int_{|x|=r}J_{k}dx_{k}-\sum_{j=1}^{N}\lim_{rarrow 0}\int_{|x-p_{j}|=r}J_{k}dx_{k}.$32
We note that $J_{k}=(1+|u|^{2})^{-1}\cdot O(|x|^{2(M-N)-1})$ . Hence, if $N>M$ , the first term vanishes
identically. It is easy to see that each integral in the second term equals to $-4\pi$ . As aconsequence, we have $\deg\phi=N.$
In conclusion, the classical $O(3)$ sigma model has an energy lower bound of Bobomol’nyi type which is saturated by the solutions of the self-dual equations. All minimalenergy solutions can be obtained by solving this self-dual equations and they tum out
to be meromorphic functions having $N$ poles and $M$ zeros. The corresponding minimalenergy is given by the degree of solutions which equals $N$ . Moreover, we have a constraint$M\leq N-1.$

In the next two sections, we will consider gauge field model which is extensions of theclassical $O(3)$ sigma model in this section. We derive self-dual equations and find theminimal energy solutions as in this section.

3

Maxwell Gauged $O(3)$ Sigma Model

3.1

Unbroken Model

as $|x|arrow\infty$ . Since
We recall that the finite energy condition for (2.1) implies that
$O(3)$
may assume that,wethe functional (2.1) is invariant under the group of rotation. This choice breaks the $O(3)$ symmetry down to $SO$ (2) $=U(1)$ , from which one
may consider a minimal potential hke $(1-n\cdot\phi)^{2}$ . In this case, we can consider the energyfunctional as$\phiarrow\phi_{\infty}$$\phi_{\infty}=n$$\mathcal{E}(\phi)=\int_{\mathbb{R}^{2}}\{(\partial_{1}\phi)^{2}+(\partial_{2}\phi)^{2}+(1-n\cdot\phi)^{2}\}dx.$

However, in this case there is no nontrivial critical points other than
We note that the vacuum state $\phi_{v}=n$ is invariant under the $SO$ (2) $=U(1)$ transfor$\phi\equiv n.$mation$(\phi_{1}, \phi_{2}, \phi_{3}) arrow (\phi_{1}\cos\theta-\phi_{2}\sin\theta, \phi_{1}\sin\theta+\phi_{2}\cos\theta, \phi_{3}) , \theta\in \mathbb{R}.$Hence, if we set $\psi=\phi_{1}+i\phi_{2}$ , then the Lagrangian(3.1)$\mathcal{L}=\frac{1}{2}\partial_{\mu}\psi\cdot\partial^{\mu}\overline{\psi}+\frac{1}{2}\partial_{\mu}\phi_{3}\cdot\partial^{\mu}\phi_{3}-\frac{1}{2}(1-n\cdot\phi)^{2}$is invariant under the global gauge transformation$\psiarrow e^{i\theta}\psi, \phi_{3}arrow\phi_{3}.$To enlarge this global symmetry into a local one, we need a gauge-covariant derivative. The componentwise expression is$D_{\mu}\psi=\partial_{\mu}\psi+iA_{\mu}\psi$$D_{\mu}\phi_{1}=\partial_{\mu}\phi_{1}-A_{\mu}\phi_{2}, D_{\mu}\phi_{2}=\partial_{\mu}\phi_{2}+A_{\mu}\phi_{1}, D_{\mu}\phi_{3}=\partial_{\mu}\phi_{3},$which is equal to$D_{\mu}\phi=\partial_{\mu}\phi+A_{\mu}(n\cross\phi)$.

33
We also introduce the Maxwell field
the following Maxwell gauged model(3.2)$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$. Then, we can extend (3.1) to$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}D_{\mu}\phi\cdot D^{\mu}\phi-\frac{1}{2}(1-n\cdot\phi)^{2}.$We define a new current density$j_{\alpha}= \frac{1}{2}\epsilon_{\alpha\beta\gamma}[\phi\cdot D^{\beta}\phi\cross D^{\gamma}\phi+F^{\beta\gamma}(1-n\cdot\phi)]=k_{\alpha}+\epsilon_{\alpha\beta\gamma}\partial^{\beta}[(1-n\cdot\phi)A^{\gamma}].$Then, the degree can be expressed as(3.3)$\deg\phi=\frac{1}{4\pi}\int_{\mathbb{R}^{2}}j_{0}dx=\frac{1}{4\pi}\int_{\mathbb{R}^{2}}[\phi\cdot D_{1}\phi\cross D_{2}\phi+F_{12}(1-n\cdot\phi)]dx.$

The static energy is given by$\mathcal{E}(\phi, A) = \int_{\mathbb{R}^{2}}\{(D_{1}\phi)^{2}+(D_{2}\phi)^{2}+F_{12}^{2}+(1-n\cdot\phi)^{2}\}dx$$= \pm 4\pi\deg\phi+\frac{1}{2}\int_{\mathbb{R}^{2}}\{(D_{1}\phi\pm\phi\cross D_{2}\phi)^{2}+(F_{12}\mp[1-n\cdot\phi])^{2}\}dx$$\geq 4\pi|\deg\phi|.$Hence, we get a energy lower bound $4\pi|\deg\phi|$ which is saturated by the following self-dual
equations: taking upper signs(3.4)$D_{1}\phi+\phi\cross D_{2}\phi = 0,$(3.5)$F_{12}-(1-n\cdot\phi) = 0.$

In terms of the new function$u$defined by (2.3), we can rewrite (3.4) and (3.5) as(3.6)$D_{1}u+iD_{2}u = 0,$(3.7)$F_{12}- \frac{2|u|^{2}}{1+|u|^{2}} = 0,$

where $D_{j}u=\partial_{j}u+iA_{j}u.$
It is obvious that $u(x)=0$ if and only if $x\in\phi^{-1}(n)$ . By virtue of the -Poincar\’eLemma (see [18]), the equation of (3.6) implies that is locally represented by a holomorphic factor up to a smooth function. Thus, the zeros of have integer multiplicities
and are realized by the points in. Similar arguments show that has poles at the
points in. Indeed, letting
and $\tilde{u}=1/u$ , we see that$\overline{\partial}$

$u$

$u$

$\phi^{-1}(n)$$\phi^{-1}(s)$$\psi$$\tilde{D}_{j}’=\partial_{j}-iA_{j}$$\tilde{D}_{1}’\tilde{u}=-\tilde{u}^{2}\tilde{D}_{1}u=i\tilde{u}^{2}\tilde{D}_{2}u=-i\overline{D}_{2}’\tilde{u},$which leads us by the -Poincar\’e Lemma thatlet$\overline{\partial}$(3.8)$\tilde{u}$has zeros of integer multiplicities. Now$\{\begin{array}{l}v = |u|^{2},\mathcal{Q} = \phi^{-1}(n)=\{q_{1}, q_{2}, \cdots, q_{M}\},\mathcal{P} = \phi^{-1}(s)=\{p_{1},p_{2}, \cdots, p_{N}\}.\end{array}$

34
Each point$p_{j}$

or$q_{j}$

is counted as its multiplicity. Then, the equation (3.7) reduces to(3.9)$\Delta v=\frac{4e^{v}}{1+e^{v}}-4\pi\sum_{j=1}^{N}\delta_{p_{j}}+4\pi\sum_{j=1}^{M}\delta_{q_{j}}.$

The finite energy condition gives the boundary condition $\phiarrow n$ as $|x|arrow\infty$ , which isequivalent to $u(x)arrow 0$ and hence $v(x)arrow-\infty$ as
We are interested in the solutions of (3.9) having finite magnetic flux which is given$|x|arrow\infty.$

$v$

by

(3.10)$\Phi=\int_{\mathbb{R}^{2}}F_{12}dx=\int_{R^{2}}\frac{2e^{v}}{1+e^{v}}dx<\infty.$

The corresponding energy is also finite and the solution of (3.9) satisfying (3.10) is referred
as a finite-energy solution. If we set$v_{0}(x)=- \sum_{j=1}^{N}\ln|x-p_{j}|^{2}+\sum_{k=1}^{M}\ln|x-q_{k}|^{2},$then $w=v-v_{0}$ satisfies$\Delta w=\frac{4e^{uo+w}}{1+e^{uo+w}}\equiv F\in L^{\infty}(\mathbb{R}^{2})\cap L^{1}(\mathbb{R}^{2})$.

From a standard argument it follows that$\lim_{|x|arrow\infty}\frac{w(x)}{\ln|x|}=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}F(x)dx\equiv\alpha>0.$

As a consequence,(3.11)$v$

enjoys the following decays:$v(x)=-\beta\ln|x|+o(\ln|x|)$as$|x|arrow\infty,$

where $\beta=2N-2M-\alpha<2(N-M)$ . By the condition (3.10), we also see thatIn the sequel, if is a finite-energy solution of (3.9), then must satisfy (3.11) with$\beta\geq 2.$

$v$

$v$

(3.12)$\beta\in[2,2N-2M)$

and$N-M\geq 2.$

In this point of view, the main question about the existence of finite-energy solutions is
to verify whether the condition (3.12) is also sufficient. Concerning this problem, it wasproved in [31] that if $N-M\geq 2$ , then for each $\beta\in(2,4)$ there exists a unique finite-energy
solution of (3.9) satisfying that$v$

(3.13)$v(x)=-\beta\ln|x|+O(1)$More general result is the following.

as$|x|arrow\infty.$

35
Theorem 3.1 ([15]). Suppose that $N-M\geq 2.$(a) If is a finite-energy solution of (3.9) satisfying (3.11) for some $\beta\in(2,2N-2M)$ ,then satisfies (3.13).
(b) Conversely, for any $\beta\in(2,2N-2M)$ , there exists a unique finite-energy solutionof (3.9) satisfying (3.13). Moreover, we have
$u$

$u$

$v$

(3.14)$\int_{\mathbb{R}^{2}}\frac{4e^{v}}{1+e^{v}}dx=2\pi(2N-2M-\beta)$.

The critical case is that $\beta=2$ . The decay rate of the solution tums out to be different
from (3.13). We have the following partial result for the existence of solutions.

Theorem 3.2 ([15]). Let $N\geq 2$ be an integer and $M=0$ . Then, for $\beta=2,$ $(3.9)$
possesses a unique finite-energy solution satisfying (3.14). The solution satisfies the
$u$

$v$

following asymptotic behavior(3.15)$u(x)=-2\ln|x|-2$ ln ln $|x|+O(1)$as$|x|arrow\infty.$

It is still open to find solutions of (3.9) for the general case $M\neq 0$ when $\beta=2$ . Once
we find a solution of (3.9), we can find the corresponding solution of (3.4) and (3.5) byusing the relations ofand. So, the physical quantities of solutions satisfy thefollowing properties.
$v$

$v,$

Theorem 3.3. Let$(\phi_{\beta}, A_{\beta})$

we have(3.16)$(\phi, A)$$u$of (3.4) and (3.5) corresponding to $\beta\in(2,2N-2M)$ . Then,energy :$\mathcal{E}(\phi_{\beta}, A_{\beta})=\deg\phi_{\beta}=4\pi N,$

magnetic flux :$\Phi(\phi_{\beta}, A_{\beta})=\int_{\mathbb{R}^{2}}\frac{2e^{v}}{1+e^{v}}dx=\pi(2N-2M-\beta)$,$|D\phi_{\beta}|^{2}=O(|x|^{-\beta})$,decays :$(\phi_{\beta})_{1}^{2},$$(\phi_{\beta})_{2}^{2},$$(F_{12})_{\beta}=1-(\phi_{\beta})_{3}^{2},$where $\beta=2(N-M)-2N\alpha.$

3.2

Broken Model

We consider a more general situation:(3.17)$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}D_{\mu}\phi\cdot D^{\mu}\phi-\frac{1}{2}(\tau+n\cdot\phi)^{2},$where is a real number in [-1, 1]. We note that if $|\tau|=1,then$ the vacuum state is given
by two symmetric phases $\phi_{v}=\pm n$ which is invariant under $U(1)$ transformation. On theother hand, if $|\tau|<1$ , then we have broken phases $n\cdot\phi_{v}=1.$
We define a new current density$\tau$$j_{\alpha}= \frac{1}{2}\epsilon_{\alpha\beta\gamma}[\phi\cdot D^{\beta}\phi\cross D^{\gamma}\phi-F^{\beta\gamma}(\tau+n\cdot\phi)]=k_{\alpha}-\epsilon_{\alpha\beta\gamma}\partial^{\beta}[(\tau+n\cdot\phi)A^{\gamma}].$

36
The corresponding conserved topological charge is defined by(3.18)$T= \frac{1}{4\pi}\int_{\mathbb{R}^{2}}j_{0}dx=\frac{1}{4\pi}\int_{\mathbb{R}^{2}}[\phi\cdot D_{1}\phi\cross D_{2}\phi-F_{12}(\tau+n\cdot\phi)]dx.$

Comparing this formula with (3.3), we can see that if $\tau=-1$ , then $T$ is the degree of
However, may not be an integer in general. The static energy is given by$\phi.$$T$$\mathcal{E}(\phi, A) = \int_{\mathbb{R}^{2}}\{(D_{1}\phi)^{2}+(D_{2}\phi)^{2}+F_{12}^{2}-(\tau+n\cdot\phi)^{2}\}dx$$= \pm 4\pi T+\frac{1}{2}\int_{\mathbb{R}^{2}}\{(D_{1}\phi\pm\phi\cross D_{2}\phi)^{2}-(F_{12}\pm[\tau+n\cdot\phi])^{2}\}dx$$\geq 4\pi|T|.$

Hence, we get a energy lower bound $4\pi|T|$ which is saturated by the following self-dual
equations: taking upper signs(3.19)$D_{1}\phi+\phi\cross D_{2}\phi = 0,$(3.20)$F_{12}+(\tau+n\cdot\phi) = 0.$

In terms of the new function$u$defined by (2.3), we can rewrite (3.4) and (3.5) as(3.21)$D_{1}u+iD_{2}u = 0,$(3.22)$F_{12}- \frac{(1-\tau)|u|^{2}-(1+\tau)}{1+|u|^{2}} = 0,$

where $D_{j}u=\partial_{j}u+iA_{j}u$ . Using the notations of (3.8) and proceeding as before, we change(3.7) int
$0$(3.23)$\Delta v=\frac{2(1-\tau)e^{v}-2(1+\tau)}{1+e^{v}}-4\pi\sum_{j=1}^{N}\delta_{p_{j}}+4\pi\sum_{j=1}^{M}\delta_{q_{j}}.$It is not difficult to see that if is a solution of (3.23) for $\tau\in(-1,0]$ , then $-v$ is also
. Therefore, from now onwith the change of the roles of ’s anda solution for$(3.23)$$-1\leq\tau\leq0$
to the unbroken case$\tau=-1,$
corresponds. Whenwe may assume that
(3.9). So, we consider only the broken case $|\tau|<1$ in the following. The finite energy
as $|x|arrow\infty$ , which is equivalent tocondition gives the boundary condition$v$$-\tau$$p_{j}$$q_{j}’ s$$\phi_{3}arrow-\tau$$|u(x)|^{2}arrow(1+\tau)/(1-\tau)$(3.24). Letting$w=v+\ln a, a=(1-\tau)/(1+\tau)\geq 1,$

we have(3.25)$\Delta w=\frac{4a(e^{w}-1)}{(1+a)(e^{w}+a)}-4\pi\sum_{j=1}^{N}\delta_{p_{j}}+4\pi\sum_{j=1}^{M}\delta_{q_{j}},$$warrow 0$

as $|x|arrow\infty.$

Concerning this equation, we have the following existence result.

37
Theorem 3.4 ([32]). Suppose that $|\tau|<1.$(a) For any $N,$ $M>0$ , there exists a unique solution of (3.25) decaying exponentiallyat the infinity.
(b) The corresponding solutionof (3.19) and (3.20) enjoys the following:$w$$(\phi, A)$energy :$\mathcal{E}(\phi, A)=\frac{1}{2}(1-\tau)(N-M)$

magnetic flux :$\Phi(\phi, A)=2\pi(N-M)$ .

(3.26),

Proof. Since the proof of the part (b) was not explicitly given in [32], we provide it here.
It is easy to see that $\Phi(\phi, A)=2\pi(N-M)$ . It follows from (3.3) and (3.18) that
$T= \deg\phi-\frac{1}{4\pi}\int_{\mathbb{R}^{2}}H_{12},$where$H_{12}=\partial_{1}H_{2}-\partial_{2}H_{1}$and$H_{k}=A_{k}( \tau+n\cdot\phi)=A_{k}\frac{(1+\tau)-(1-\tau)|u|^{2}}{1+|u|^{2}}, k=1,2.$

If we set

$J_{12}=\partial_{1}J_{2}-\partial_{2}J_{1}$and$G_{12}=\partial_{1}G_{2}-\partial_{2}G_{1}$, where with $k=1,2,$$J_{k}= \frac{iu\overline{\tilde{D}_{k}u}-i\overline{u}\tilde{D}_{k}u}{1+|u|^{2}}=\frac{\epsilon_{kl}\partial_{l}v}{1+e^{v}},$$G_{k}=- \frac{2|u|^{2}A_{k}}{1+|u|^{2}}=\frac{(1+\tau)e^{v}-(1-\tau)}{2(1+e^{v})}\epsilon_{kl}\partial_{l}(v-\sum_{j=1}^{N}\ln|x-p_{j}|^{2}+\sum_{j}^{M}|x-q_{j}|^{2})$,then we obtain from [33] that

(3.27)$\{\begin{array}{l}T=\frac{1}{4\pi}\int_{\mathbb{R}^{2}}J_{12}-\frac{1+\tau}{4\pi}\Phi(\phi, A) ,\deg\phi=T+\frac{1}{4\pi}\int_{\mathbb{R}^{2}}H_{12}.\end{array}$Here, we used the identity $H_{k}-G_{k}=(1+\tau)A_{k}$ . Since
nentially to zero at infinity, we obtain that

(3.28)$v$

and its derivative decay expo-$\int_{\mathbb{R}^{2}}J_{12}dx=4\pi(N-M) , \int_{\mathbb{R}^{2}}H_{12}dx=0.$

As a consequence, we have
$T= \frac{1}{2}(1-\tau)(N-M)$ .

The proof is finished.

$\square$38
It is worth while to mention the difference between Theorem 3.3 and Theorem 3.4. In
the broken model (3.17), the magnetic flux and the energy are quantized by (3.26). Thisis one of the features of classical $U(1)$ gauged planar vortices in $(2+1)$ (for example theAbelian Higgs model [18] : they either have quantized magnetic flux in which case they
are topologically stable, or have arbitrary flux in which case they are not topologicallystable. So the quantization of the magnetic flux is responsible for the topological stabilityof soliton solutions. However, in the unbroken model (3.2), the topological stability ofsoliton solutions is independent of the magnetic flux. The solitons can carry arbitrary
magnetic flux in a certain range, but yet be topologically stable such that the energy isquantized by the degree of the scalar field. See (3.16).
$)$

4

Chern-Simons Gauged $O(3)$ Sigma ModelIn this section, we consider the model in the previous section when the action is govemedby the Chem-Simons force instead of the Maxwell force. We consider both the broken andthe unbroken models. The Lagrangian is given by$\mathcal{L}=-\frac{1}{4}\epsilon^{\mu\nu\rho}F_{\mu\nu}A_{\rho}+\frac{1}{2}D_{\mu}\phi\cdot D^{\mu}\phi+\frac{1}{2\kappa^{2}}(\tau+n\cdot\phi)^{2}(n\cross\phi)^{2}.$Here, the constant $\kappa>0$ is the Chem-Simons coupling constant representing the strengthof the Chem-Simons action. While $\tau\in[-1,1]$ determines the vacuum manifold of thepotential. If $|\tau|=1$ , we have only the symmetric vacua $\phi_{v}=\pm n$ which is fixed under the$U(1)$ gauge group. So, the case $|\tau|=1$ gives the unbroken model. If $|\tau|<1$ , we have notwhich is invariant
only the symmetric vacuum but also a asymmetric vacuum
but not fixed under the $U(1)$ gauge group. Hence, we have the broken model for $|\tau|<1.$
The Gauss law equation yields$n\cdot\phi_{v}=\tau$$\kappa F_{12}-2A_{0}(\tau+(n\cdot\phi)^{2})=0.$

Then, by means of the Gauss law equation, the static energy is given by$\mathcal{E}(\phi)$$=$$\int_{\mathbb{R}^{2}}\frac{\kappa^{2}F_{12}^{2}}{(n\cross\phi)^{2}}+(D_{1}\phi)^{2}+(D_{2}\phi)^{2}-\frac{1}{\kappa^{2}}(\tau+n\cdot\phi)^{2}(n\cross\phi)$$= \pm 4\pi T+\int_{\mathbb{R}^{2}}\{(\frac{\kappa F_{12}}{|n\cross\phi|}\pm\frac{1}{\kappa}(\tau+n\cdot\phi)|n\cross\phi|)^{2}+(D_{1}\phi\pm\phi\cross D_{2}\phi)^{2}\}dx$$\geq 4\pi|T|,$

where the equality holds if and only if$\phi$

satisfies the self-dual equations$D_{1}\phi\pm\phi\cross D_{2}\phi = 0,$(4.1)$F_{12} \pm\frac{2}{\kappa^{2}}(\tau+n\cdot\phi)(n\cross\phi)^{2} = 0.$

We take the upper signs of (4.1).

39
In terms of the new function$u$defined by (2.3), we can rewrite (4.1) as(4.2)$D_{1}u+iD_{2}u = 0,$(4.3)$F_{12}- \frac{4|u|^{2}\{(1-\tau)|u|^{2}-(1+\tau)\}}{(1+|u|^{2})^{3}} = 0,$

where $D_{j}u=\partial_{j}u+iA_{j}u$ . Under the setting of (3.8) and proceeding as before, we change(4.2) and (4.3) into
(4.4)$\Delta v+\frac{1}{\epsilon^{2}}f(u, \tau)=-4\pi\sum_{j=1}^{N}\delta_{p_{j}}+4\pi\sum_{j=1}^{M}\delta_{q_{j}},$where$\epsilon^{2}=8/\kappa^{2}$and$f(v, \tau)=\frac{e^{v}\{(1-\tau)-(1+\tau)e^{v}\}}{(1+e^{v})^{3}}.$We have three types of boundary condition from the finite energy condition:
topological $BC$ :
(4.5)

$\{$

$v(x) arrow\ln\frac{1-\tau}{1+\tau}$for$|\tau|<1,$

nontopological $BC$ of type I : $v(x)arrow-\infty,$
nontopological $BC$ of type II : $v(x)arrow+\infty.$

For the topological solitons, we have the following result.

Theorem 4.1. (a) [32] There exists a solution of (4.4) which decays exponentially to zeroat infinity. The corresponding energy and the magnetic flux are quantized.
(b) [13] If is small enough, the solution is unique.
$\epsilon$

The uniqueness of solutions for generalis still an open problem. In the following,
we focus on the nontopological solutions. In particular, let us consider the case when there
is only one vortex or antivortex point. In this case, we want to obtain radial solutions. To
see this, let $v(r;\tau, s)$ denote the solution of the following initial value problem:$\epsilon>0$(4.6)$\{\begin{array}{ll}v"+\frac{1}{r}v’+f(v, \tau)=0 for r>0,v(r;\tau, s)=2N\ln r+s+o(1) as rarrow 0.\end{array}$Here,
always denotes. Since $f(-v, -\tau)=-f(v, \tau),$ $w=-v$ satisfies (4.4)
with and $2N$ replaced $by-\tau and-2N$ . So without loss of generality, we might assume
that $N\geq 0$ . Define
by$v’$$\frac{dv}{dr}(r;\tau, s)$$\tau$$\beta(\tau, s)$$\beta(\tau, s)=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}f(v(r;\tau, s),$ $\tau)dx=\int_{0}^{\infty}f(v(r;\tau, s), \tau)$ $dr$.

Then, we have
(4.7)$v(r;\tau, s)=[2N-\beta(\tau, s)]\ln r+I_{N,\beta,\tau}+O(r^{2+2N-\beta})$for large$r.$

For the unbroken case $|\tau|=1$ , we have a result for the existence of solutions from [1] for
certain range of . This result is extended in [9] as follows.
$\beta$40
Theorem 4.2 ([9]). (a) Let $\tau=-1$ . If $N\geq 0$ and $\beta>4N+4$ , then there exists ais the corresponding
nontopological solutionof (4.6) which satisfies (4.7). If
solution of (4.1), then$(\phi_{\beta}, A_{\beta})$$u_{\beta}$(4.8)$\mathcal{E}(\phi_{\beta}, A_{\beta})=\Phi(\phi_{\beta}, A_{\beta})=2\pi\beta.$(b) If $N\geq 1$ and $\beta\in(0,2N-1], then (4.6)$ possesses a unique nontopological solutionof type such that satisfies (4.7) for $\beta\in(0,2N-1)$ , and$\Pi$$u_{\beta}$(4.9)$u_{\beta}$$u_{\beta}(r)=\ln r+$ln ln $r+O(1)$near$\infty$for$\beta=2N-1.$(c) Let $\tau=1$ . If $N\geq 0$ and $\beta<\min\{0,4N-4\}$ , then (4.6) admits a nontopologicalsolutionof type which satisfies (4.7). There is no nontopological solution of typeto the equation (4.6).
(d) In both cases $(b)$ and $(d)$ , we have

$I$

$\Pi$

$u\beta$(4.10)$\mathcal{E}(\phi_{\beta}, A_{\beta})=4\pi N, \Phi(\phi_{\beta}, A_{\beta})=\pi\beta.$

For the unbroken case $|\tau|<1$ , we have the following result.
Theorem 4.3 ([12]). Suppose $N$ is a nonnegative integer and $\tau\in(-1,1)$ is given.
(a) There exists a unique $s_{*}=s_{*}(\tau)$ such that $\beta(\tau, s_{*})=2N,$ $i.e$ . (4.6) has a uniquetopological solution $v(r;\tau, s)$ .
(b) $\beta(\tau, s)>0$ if and only if $s<s_{*}$ . In this case, $v(r;\tau, s)$ is a nontopological solutionof type I. Moreover, $\beta(\tau, \cdot):(-\infty, s_{*})arrow(4N+4, \infty)$ is strictly increasing in and$s$satisfies$\lim_{sarrow-\infty}\beta(\tau, s)=4N+4, \lim_{sarrow s}.\beta(\tau, s)=+\infty.$(c) For $\beta<\min\{O, 4N-4\}$ , there exists an unique $s\in(s_{*}, \infty)$ such that $\beta(\tau, s)=\beta,$
and $v(r;\tau, s)$ is a nontopolo9ical solution of type$\Pi.$(d) Let $(\phi, A)$ be the corresponding solution of (4.1). The energy
the topological charge $4\pi T(\phi, A)$ , is given by$\mathcal{E}(\phi_{*}, A_{*}) = 2\pi N(1-\tau)$$\mathcal{E}(\phi, A)$, which equals,(4.11)$\mathcal{E}(\phi_{(\beta)}, A_{(\beta)})$(e) The degree of$\phi$$=$$\{\begin{array}{ll}\pi(1-\tau)\beta, \beta>4N+4,4\pi N-\pi\beta(1+\tau) , \beta<\min\{0,4N-4\}.\end{array}$is given by$\deg\phi_{*} = \frac{1}{2}(1-\tau)N,$(4.12)$\deg\phi_{(\beta)}$$=$$\{\begin{array}{ll}0, \beta>4N+4,N, \beta<\min\{0,4N-4\}.\end{array}$41(f) The total magnetic flux$\Phi$and the total charge $Q=-\kappa\Phi$ are given by$\Phi(\phi_{*}, A_{*}) = 2\pi N,$(4.13)$\Phi(\phi_{(\beta)}, A_{(\beta)}) = \pi\beta.$

Radially symmetric solutions may be used to the study of non-radial solutions. In
particular, the linearized operator at each radial solution plays a crucial role in constructing
blow-up solutions. Indeed, using Theorem 4.2 and Theorem 4.3, one can show that thelinearized operator $\triangle+f’(u, \tau)$ is an isomorphism on suitable function spaces. Then,following the argument of [7, 22], we can construct bubbling solutions [10, 11].
Now let us investigate the properties of solutions for broken and unbroken models. In
the broken case $|\tau|=1$ , the vacuum of the potential consists of two symmetric phase which
gives nontopological solutions of type I and II. The type II solution is topologically stable
and the degree is an integer. The energy is quantized but the magnetic flux is fractional.
See Theorem 4.2 $(b)-(d)$ . For the nontopological solution of type I, the energy and themagnetic flux are fractional. So it looks like the classical planar nontopological solutions[4]. See Theorem 4.2 (a).
In the broken model $|\tau|<1$ , we have two symmetric phases and one asymmetric phase,depending on the parameter . Two symmetric phases gives nontopological solutions oftype I and II. The asymmetric phase yields a topological solution. The nontopologicalsolutions have fractional magnetic flux and energy. The type II solution has integer degree
but the type II solution has fractional degree. In the Schroers’s model, the energy is equal$\tau$

to the degree of the solution. However, in the Chern-Simons model they are different for
type II solutions. In contrast, the topological solutions have quantized magnetic flux andenergy and look like the classical planar Chern-Simons system [19, 20]. In this case, thedegree is fractional. See Theorem 4.3 $(d)-(f)$ .
We close this section with two remarks. First, we can consider sigma models under thet’Hooft type periodic condition. This leads us to consider (4.4) on a flat torus. One canfind some results in [6, 11, 22, 25] for this direction. Second, one may consider the partial
differential equations arising from the self-dual Maxwell-Chern-Simons gauged $O(3)$ sigma
model [21]. This $mo$ del unifies the Maxwell $O(3)$ sigma model and the Chern-Simons $O(3)$sigma model. Recent progress for this model can be found in [5, 8, 16, 17, 28].

Acknowledgements
This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science andTechnology (2009-0086382).

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