## Abstract

Two sharp existence and uniqueness theorems are presented for solutions of multiple vortices arising in a six-dimensional brane-world supersymmetric gauge field theory under the general gauge symmetry group *G*=*U*(1)×*SU*(*N*) and with *N* Higgs scalar fields in the fundamental representation of *G*. Specifically, when the space of extra dimension is compact so that vortices are hosted in a 2-torus of volume |*Ω*|, the existence of a unique multiple vortex solution representing *n*_{1},…,*n*_{N}, respectively, prescribed vortices arising in the *N* species of the Higgs fields is established under the explicitly stated necessary and sufficient condition
where *e* and *g* are the *U*(1) electromagnetic and *SU*(*N*) chromatic coupling constants, *v* measures the energy scale of broken symmetry and is the total vortex number; when the space of extra dimension is the full plane, the existence and uniqueness of an arbitrarily prescribed *n*-vortex solution of finite energy is always ensured. These vortices are governed by a system of nonlinear elliptic equations, which may be reformulated to allow a variational structure. Proofs of existence are then developed using the methods of calculus of variations.

## 1. Introduction

The concept of solitons is important in quantum field theory. Solitons, realized as the static solutions of gauge field equations, categorized into domain walls, vortices, monopoles and instantons, and often of topological origins (Rajaraman 1982; Ryder 1996; Manton & Sutcliffe 2004), give rise to locally concentrated field configurations and are essential for the description of various fundamental interactions and phenomenologies. Vortices, arising in two spatial dimensions, were first discovered by Abrikosov (1957) in the form of a mixed state in a type-II superconductor in which the vortex-lines represent partial penetration of magnetic field into the superconductor as a consequence of partial destruction of superconductivity, in the context of the Ginzburg–Landau theory (cf. Ginzburg & Landau (1965)). In quantum field theory, Nielsen & Olesen (1973) showed that vortices arise in the Abelian Higgs model. These vortices may be used to model the so-called dual strings, better known as the Nambu–Goto strings (cf. Goto (1971)), which is a basic construct in string theory (Zwiebach 2004). As a consequence of the Julia–Zee theorem (Julia & Zee 1975; Spruck & Yang 2009), the vortex equations of the Abelian Higgs model in the static limit are exactly the Ginzburg–Landau equations. Although these equations are fundamentally important, fairly thorough understanding of their solutions has only been achieved in a few extreme situations where one assumes either that the magnetic field is absent (Neu 1990; E Weinan 1994; Bethuel *et al.* 1994; Lin 1996), that the solutions are radially symmetric ((Berger & Chen (1989); Chen *et al.* (2009) and B. J. Plohr 1980, unpublished thesis) or that a critical coupling is maintained so that the interaction between vortices vanishes (Jaffe & Taubes 1980; Wang & Yang 1992). In literature, this last situation is commonly referred to as the self-dual or BPS limit after the pioneering work of Bogomol'nyi (1976) and Prasad & Sommerfield (1975). In fact, in the study of non-Abelian gauge field equations, it is only in the BPS limit that tractable opportunities are available for gaining some good understanding of the solutions of various equations, owing to the difficulties associated with the presence of non-Abelian symmetry groups. The first array of existence results for non-Abelian vortices were obtained (Spruck & Yang 1992*a*,*b*) for the equations governing electroweak vortices formulated by Ambjorn & Olesen (1988, 1989*a*,*b*, 1990), which were later sharpened (Bartolucci & Tarantello 2002; Chen & Lin 2010). Although the BPS vortices are present only when the coupling constants satisfy a certain critical condition (in the Ginzburg–Landau theory, this is the interface between type-I and type-II superconductivity; in the Abelian Higgs model, this is when the masses of the gauge and Higgs bosons coincide), such solutions exhibit a full range of elegant and unambiguous features including exact topological characterization, quantization of flux and energy, and energy concentration, as determined experimentally. In recent years, the conceptual power of the BPS vortices in theoretical physics has been particularly explored in supersymmetric gauge field theory starting with the work of Seiberg & Witten (1994) in an attempt to use non-Abelian colour-charged monopoles and vortices to interpret quark confinement (Nambu 1974; 't Hooft 1974, 1978, 1981; Mandelstam 1975, 1980). For surveys on this exciting topic, see the works of Greensite (2011), Konishi (2009), Shifman & Unsal (2009) and Shifman & Yung (2007, 2009). Inspired by the importance of non-Abelian vortices in the linear confinement mechanism, through a so-called dual Meissner effect that extends the classical Meissner effect in superconductors, some systematic research has recently been carried out aimed at understanding the various BPS vortex equations obtained in (Eto *et al.* 2006*a*,*b*, 2009; Shifman & Yung 2004*a*,*b*, 2007, 2009; Gudnason *et al.* 2010) and a series of sharp existence theorems have been established (Lin & Yang 2011*a*,*b*; Lieb & Yang 2012). Unlike the BPS vortex equations in the electroweak theory (Ambjorn & Olesen 1988, 1989*a*,*b*, 1990; Spruck & Yang 1992*a*,*b*; Bartolucci & Tarantello 2002), the elegant structure of the BPS vortex equations in these supersymmetric models allow a complete understanding of their solutions, despite the apparent sophistication associated with various underlying non-Abelian gauge groups. The main contribution of the present paper is to prove two sharp existence theorems for the BPS vortex equations arising in the supersymmetric *U*(1)×*SU*(*N*) gauge theory discovered in the work of Eto *et al.* (2004) in the context of a six-dimensional brane-world scenario formalism (Hughes & Polchinski 1986; Hughes *et al.* 1986; Zwiebach 2004).

An outline of the rest of the content of this article is as follows. In §2, we review the non-Abelian multiple vortex equations of Eto *et al.* (2004) and state our main existence theorems. In §3, we describe the nonlinear elliptic equation problem to be studied which is equivalent to the solution problem of the non-Abelian vortex equations. In §4, we consider multiple vortex solutions over a doubly periodic domain and identify a family of necessary conditions for the existence of such solutions. We then prove that these necessary conditions are also sufficient for existence. While doing this, we reveal various fine structures of the problem which will be seen to be useful for our later study of planar solutions. In §5, we prove the existence and uniqueness of a weak solution for the vortex equations over the full plane by the variational approach first developed for the scalar (Abelian) situation (Jaffe & Taubes 1980). In §6, we first establish pointwise decay properties of solutions near infinity. We then strengthen these results by obtaining some exponential decay estimates. In §7, we use the obtained exponential decay properties of the solutions to establish various anticipated flux quantization formulas.

## 2. Non-Abelian Bogomol'nyi–Prasad–Sommerfeld vortex equations and main results

In this section, we begin by a review of the non-Abelian vortex equations derived by Eto *et al.* (2004). For convenience, we use *x*^{1} and *x*^{2} to denote the coordinates of extra dimensions in their six-dimensional gauge field theory. Aimed at fixing notation here, we will be sketchy since details can be found in Eto *et al.* (2004).

We use {*t*^{a}} to denote the generators of *SU*(*N*) (*a*=1,…,*N*^{2}−1). Thus any element in the Lie algebra of *SU*(*N*), denoted by *su*(*N*), may be written as
2.1We use {*q*_{i}} (*i*=1,…,*N*) to denote *N* hypermultiplets in the fundamental representation of *SU*(*N*), i.e. each *q*_{i} is a -valued scalar Higgs field, which is such that each of its values is taken to be a column vector. With (*x*^{ℓ})=(*x*^{1},*x*^{2}), the gauge-covariant derivatives are given by
2.2where *A*_{ℓ} and lie in the Lie algebras of *U*(1) and *SU*(*N*), respectively, , so that the induced gauge field strength tensors are defined by
2.3where [·,·] denotes the matrix commutator.

With the earlier-mentioned notation, the non-Abelian vortex equations derived in Eto *et al.* (2004) (see also Auzzi *et al.* (2003); Hanany & Tong (2003)) are
2.4
2.5
and
2.6where *e*,*g*,*v*>0 are coupling parameters for which *e* represents the Abelian (electromagnetic) gauge coupling, *g* the non-Abelian (nuclear) gauge coupling, *v* the energy scale of the spontaneously broken ground state (vacuum), and dagger (^{†}) denotes the Hermitian conjugate.

Now, following Eto *et al.* (2004), we collectively rewrite the Higgs fields and gauge fields in the forms of *N*×*N* matrices,
2.7where *I*_{N} denotes the *N*×*N* identity matrix. Then the ansatz
2.8and
2.9where *ϕ*_{i} are complex-valued scalar fields and are real-valued vector fields, *i*=1,…,*N*, further reduces the equations (2.4)–(2.6) into (Eto *et al.* 2004):
2.10and
2.11for *i*=1,…,*N*.

In view of Jaffe & Taubes (1980), we see from (2.11) that the zeros of each *ϕ*_{i} are isolated with integer multiplicities. We may use *Z*(*ϕ*_{i}) to denote the set of zeros of *ϕ*_{i},
2.12so that the repetitions among the points *p*_{i,s} (*s*=1,…,*n*_{i}) take account of the multiplicities of these zeros. The first term on the right-hand side of (2.10) indicates that these zeros enhance the ‘vorticity’ field (*i*=1,…,*N*). However, the other terms on the right-hand side of (2.10) complicate the situation so that we are not able to assert that the maxima of are achieved at *Z*(*ϕ*_{i}) (*i*=1,…,*N*), which is what makes the problem interesting and challenging.

There are two partial differential equation problems to be studied.

The first one concerns the solutions of (2.10)–(2.11) with prescribed zero sets given in (2.12) over a bounded domain *Ω* in so that the field configurations are periodic modulo of the 't Hooft boundary condition ('t Hooft 1979; Wang & Yang 1992) at the boundary of *Ω*. The physical relevance of this problem is that such solutions may arise when the extra dimensions are compactified as two-dimensional tori. For this problem, here is our result.

### Theorem 2.1

*Consider the non-Abelian BPS vortex equations (2.10)–(2.11) for the field configurations* *(i=1,…,N) over a doubly periodic domain Ω with the given sets of zeros stated in (2.12) so that ϕ*_{i} *has exactly n*_{i} *zeros in Ω (i=1,…,N). Then a solution exists if and only if the conditions
*2.13*are fulfilled simultaneously, where* *is the total number of the N species of vortices. Furthermore, if a solution exists, it must be uniquely determined by the sets of zeros given in (2.12), up to gauge transformations.*

Although the condition (2.13) seems complicated, we may sum up the vortex numbers, *n*_{1},…,*n*_{N}, to get a simple consequence of the inequalities stated in (2.13), in the form
2.14It is interesting to notice that, now, the non-Abelian coupling constant *g* does not enter the condition (2.14) but the integer *N* which measures the ‘size’ of the non-Abelian symmetry.

The second problem concerns the solutions of (2.10)–(2.11) over the full plane . The form of the energy density described in Eto *et al.* (2004) and requirement of finite energy naturally impose the boundary condition
2.15for solutions. For this problem, here is our result.

### Theorem 2.2

*The non-Abelian BPS vortex equations (2.10)–(2.11) over the full plane* *described by the field configurations* *subject to the given sets of zeros stated in (2.12) and the boundary condition (2.15) always have a unique solution. Moreover, such a solution realizes the boundary condition (2.15) exponentially fast. More precisely, for an arbitrarily small number ε∈(0,1), there hold
*2.16*for |x| sufficiently large.*

We note that, intuitively, theorem 2.2 may be reinterpreted in view of theorem 2.1 in the context of vortices in a domain of infinite volume. In such a situation, the condition (2.13) of course becomes superfluous.

### Theorem 2.3

*In both situations stated as in theorems 2.1 and 2.2, the fluxes are quantized quantities given by the expressions
*2.17*which are seen to be determined by the algebraic numbers of zeros of the complex scalar fields ϕ*_{i}* (i=1,…,N), with the integrals evaluated over the doubly periodic domain Ω or* *, respectively.*

This result suggests that, although it is not clear whether the vorticity field indeed peaks at the zeros of the order parameter *ϕ*_{i}, the total number of zeros of *ϕ*_{i} determines the total vorticity (or flux) generated from the field in the full domain. For this reason, we may still regard the locations of the zeros of *ϕ*_{i} as the centres of vortices and refer to *n*_{i} as the *i*th vortex number, which extends the concept of vortices in the classical Abelian Higgs model (Jaffe & Taubes 1980). As a consequence, the integer *n* defined in (2.13) is well justified to be called the total vortex number of the solution.

With the quantized fluxes given in (2.17), the total non-Abelian vortex (string) tension, *T*_{NA}, may be computed (Eto *et al.* 2004) to assume the elegant exact value
2.18

The above-mentioned theorems will be established in the subsequent sections.

## 3. System of nonlinear elliptic equations

To proceed, we now adapt the complexified variables and derivatives defined by
3.1and convert the equations (2.10)–(2.11) into the following system:
3.2and
3.3away from the possible zeros of *ϕ*_{i}, for *i*=1,…,*N*.

Inserting (3.3) or into (3.2) (*i*=1,…,*N*) and using the relation , we arrive at the equations
3.4away from the zeros of *ϕ*_{i}, *i*=1,…,*N*. Thus, with the notation in (2.12) and the new variables
3.5we obtain the following system of nonlinear elliptic equations:
3.6governing the interaction of *N* species of vortices located at the prescribed set of points
3.7which is the set of zeros of the complex scalar fields *ϕ*_{1},…,*ϕ*_{N}.

We shall consider the solutions of the system (3.6) over a doubly periodic domain (a 2-torus) *Ω* realized by the 't Hooft periodic boundary condition ('t Hooft 1979; Wang & Yang 1992) and over the full plane . In the latter situation, we need to observe the boundary condition (2.15) at infinity. That is,
3.8

In §4, we shall first concentrate on the doubly periodic situation.

## 4. Necessary and sufficient condition for doubly periodic solutions

In this section, we study the equations (3.6) defined over a doubly periodic domain *Ω*. We conveniently rewrite these equations as
4.1where
4.2Let be a solution to
4.3(cf. Aubin (1982)). The substitutions
recast the equation (4.1) into
4.4

We use boldfaced letters to denote column vectors in . Thus, we set
and
and let *A*=(*a*_{ij})_{N×N} be the *N*×*N* matrix defined by (4.2). Then
where *a*=(1/*N*)(*e*^{2}/2−*g*^{2}/2*N*) and *b*=*g*^{2}/2*N*, which should not be confused with the group indices used in §2.

Now the equations (4.4) can be written in the vector form 4.5This system is challenging, and in order to tackle it, we seek a variational principle.

To find a variational principle, we need to use the property of the matrix *A*. It is easy to check that the matrix *A* is positive definite and its eigenvalues are
4.6Then, by the Cholesky decomposition theorem (Golub & Ortega 1992), we know that there is a unique upper triangular *N*×*N* matrix *T*=(*t*_{ij}) for which all the diagonal entries are positive, i.e. *t*_{ii}>0, *i*=1,…,*N*, such that
4.7

In fact, by direct computation, we have

Set **v**=(*v*_{1},…,*v*_{N})^{τ}, *L*=(*T*^{τ})^{−1}≡(*l*_{ij})_{N×N}. We introduce the new variable vector
4.8Then (4.5) takes the form
4.9

With the convention *α*_{N}=0, we may write (4.9) in the component form
4.10

It is easy to check that the above system of equations (4.10) are the Euler–Lagrange equations of the functional 4.11

Setting
4.12(note that these *q*_{i}'s should not be confused with the *N* scalar ‘quark’ fields in the fundamental representation of the gauge group denoted by the same notation in §2) we may integrate (4.10) to obtain
4.13Since *t*_{ii}>0 (*i*=1,…,*N*) and *α*_{i}>0 (*i*=1,…,*N*−1), the definition of *q*_{i} (*i*=1,…,*N*) given in (4.12) and the relation (4.13) lead to the necessary condition
4.14which appears to be complicated (or not sufficiently concrete to be fully appreciated).

In order to arrive at an explicit form of the necessary condition, recall the structure of the matrix *T* given in (4.7). Thus, with **p**=(*p*_{1},…,*p*_{N})^{τ} and **q**=(*q*_{1},…,*q*_{N})^{τ}, we can rewrite (4.13) in the matrix form
4.15With *L*=(*T*^{τ})^{−1}, we can solve (4.15) to get
4.16

On the other hand, because for any invertible *N*×*N* matrix *D* and the column vectors **X** and **Y** in satisfying **Y**^{τ}*D*^{−1}**X**≠1, the matrix *M*=*D*−**X****Y**^{τ} is invertible and
4.17Applying the formula (4.17) to the matrix
4.18we have
4.19Inserting (4.19) into (4.16), we obtain the solution
4.20where
4.21is the total vortex number. Substituting the values of the constants *a* and *b* in terms of the coupling constants *e* and *g* in (4.20), we have
4.22which leads us to the necessity of the condition (2.13).

Below, we shall show that, under the condition (2.13), the system (4.10) has a solution. We will use a direct minimization method as in Lieb & Yang (2012) Furthermore, in order to gain more insight into the technical structure of the problem, we shall also sketch a constrained minimization method to approach the problem.

We use *W*^{1,2}(*Ω*) to denote the usual Sobolev space of scalar-valued or vector-valued *Ω*-periodic *L*^{2}-functions whose derivatives are also in *L*^{2}(*Ω*). In the scalar case, we may decompose *W*^{1,2}(*Ω*) into so that any *f*∈*W*^{1,2}(*Ω*) can be expressed as
4.23

It is useful to recall the Moser–Trudinger inequality (Aubin 1982; Fontana 1993)
4.24With (4.24), it is clear that the functional *I* defined by (4.11) is a *C*^{1}-functional with respect to its argument (*v*_{1},…,*v*_{N})∈*W*^{1,2}(*Ω*), which is strictly convex and lower semi-continuous in terms of the weak topology of *W*^{1,2}(*Ω*).

We can suppress the functional *I* given in (4.11) into the form
4.25

Applying the Jensen inequality and using the fact that *t*_{ii}>0 (*i*=1,…,*N*) and *α*_{i}>0 (*i*=1,…,*N*−1), we have
4.26

Substituting (4.26) into (4.25), we have 4.27

In order to control the lower bound of (4.27), we recall the relation *T***q**=**p** with the quantities *q*_{i} (*i*=1,…,*N*) as defined in (4.22). That is,
4.28Therefore, we have
4.29Now set
4.30Then, we arrive at
4.31Thus, using the elementary inequality
4.32in (4.31), we obtain the coercive lower bound
4.33

It follows from (4.33) that *I*(**v**) is bounded from below and we may consider the following direct minimization problem:
4.34

Let be a minimizing sequence of (4.34). Because the function
4.35where *σ*,*q*>0 are constants, enjoys the property that as , we see from (4.31) that the sequences (*i*=1,…,*N*) are all bounded where is defined by (4.30) by setting and (*i*=1,…,*N*,*n*=1,2,…). Inverting the transformation (4.30), we see that the sequences (*i*=1,…,*N*) are also bounded. Without loss of generality, we may assume
4.36

In addition, using (4.33), we conclude that (*i*=1,…,*N*) are all bounded in *L*^{2}(*Ω*). Therefore, it follows from the Poincaré inequality that the sequences (*i*=1,…,*N*) are bounded in *W*^{1,2}(*Ω*). Without loss of generality, we may assume
4.37

Obviously, (*i*=1,…,*N*). Set (*i*=1,…,*N*). Then, (4.36) and (4.37) lead us to see that weakly in *W*^{1,2}(*Ω*) as (*i*=1,…,*N*). The weakly lower semi-continuity of *I* enables to conclude that solves (4.34), which is a critical point of *I* and a classical solution to the system (4.10) in view of the standard elliptic theory.

Because the matrix *A* is positive definite, it is easy to check that the functional *I* is strictly convex in *W*^{1,2}(*Ω*). So it has at most one critical point in *W*^{1,2}(*Ω*), which establishes the uniqueness of the solution to the equations (4.10).

Below, we will develop our methods further by presenting a constrained minimization approach to the problem. For convenience, we rewrite the constraints (4.12) collectively as
4.38Recall that the values of *q*_{1},…,*q*_{N} are given by (4.22) which are obtained by solving the system of equations (4.13), or
4.39

We consider the multi-constrained minimization problem 4.40

Suppose that (4.40) allows a solution, say **v**=(*v*_{1},…,*v*_{N}). Then there are numbers (the Lagrange multipliers) in , say *ξ*_{1},…,*ξ*_{N}, such that, for *i*=1,…,*N*,
4.41where *w*_{1},…,*w*_{N} are test functions. Letting *w*_{1}=⋯=*w*_{N}=1 in the above equations and applying (4.39), we arrive at
4.42Consequently, *ξ*_{1}=⋯=*ξ*_{N}=0 so that (4.41) is exactly the weak form of the system (4.10). In other words, the Lagrange multipliers disappear automatically and a solution of (4.40) solves (4.10). Hence, it suffices to find a solution to (4.40).

In order to approach (4.40), we use the notation (4.23) to rewrite the constraints (4.38) as
4.43which may be resolved to yield
4.44where and *l*_{ij} are the entries of the lower triangular matrix *L*=(*T*^{τ})^{−1} with *T*=(*t*_{ij}).

To proceed, we use the constraints (4.38) to rewrite the action functional (4.11) or (4.25) as
4.45where we have inserted (4.44). However, using the relation **q**=*L*^{τ}**p**, we can rewrite (4.45) as
4.46where *C*_{0} is a constant depending only on *L*, **p** and **q**. By virtue of the Jensen inequality, we have
4.47Using the condition *q*_{1},…,*q*_{N}>0 and the lower bound (4.47) in (4.46), we obtain the following coercive inequality:
4.48

Now the proof of solvability of (4.40) follows from a standard argument.

In fact, let be a minimizing sequence of (4.40). In view of (4.48) and the Poincaré inequality, we see that is bounded in *W*^{1,2}(*Ω*). Without loss of generality, we may assume that converges weakly in *W*^{1,2}(*Ω*) to an element . The compact embedding
4.49implies that in *L*^{p}(*Ω*) (*p*≥1) as . In particular, (*i*=1,…,*N*). In view of (4.24) and (4.49), we see that the functionals defined by the right-hand sides of (4.44) are continuous in (*i*=1,…,*N*) with respect to the weak topology of *W*^{1,2}(*Ω*). Therefore, some (*i*=1,…,*N*) as , where is given in (4.44). In other words, satisfies the constraints (4.38) and solves the constrained minimization problem (4.40).

## 5. Solution on full plane

In this section, we prove the existence and uniqueness of the solution to equations (4.1)–(4.2) over satisfying the natural boundary condition (*i*=1,…,*N*) at infinity as given in (3.8). Under the translation (*i*=1,…,*N*) and the rescaling of the coefficient matrix, *v*^{2}*a*_{ij}↦*a*_{ij} (*i*,*j*=1,…,*N*), the equations (4.1) become
5.1where
5.2subject to the boundary condition
5.3

As in Jaffe & Taubes (1980) and Yang (2000, 2001), we introduce the background function
5.4Then we have
5.5Let , *i*=1,…,*N*. Then the equations (5.1) become
5.6

Set
and
Thus, the equations (5.6) can be written in the vector form
5.7where *A*=(*a*_{ij})_{N×N}.

As before we use the transformation (4.8) to change (5.7) into
5.8for *i*=1,…,*N*. It is direct to check that (5.8) are the variational equations of the energy functional
5.9where
5.10which may also be rewritten as
5.11where (·,·)_{2} and ∥·∥_{2} denote the inner product and norm of , respectively.

It is clear that the functional *I* is a *C*^{1}-functional with respect to **v** and its Fréchet derivative satisfies
5.12

Set
5.13**w**=(*w*_{1},…,*w*_{N})^{τ} and **g**=(*g*_{1},…,*g*_{N})^{τ}. Then **w**=*T*^{τ}**v** or **v**=*L***w**. Thus,
5.14where **H**=(*H*_{1},…,*H*_{N})^{τ}=*L*^{τ}*L***h**=*A*^{−1}**h**. Inserting (5.13) and (5.14) into (5.12), we obtain
5.15

To estimate the right-hand side of (5.15), we consider the quantity
5.16where *w*,*u*_{0},*H* stand for one of the functions *w*_{i}, , *H*_{i}, for *i*=1,…,*N*, respectively.

As in Jaffe & Taubes (1980), we decompose *w* into its positive and negative parts, *w*=*w*_{+}−*w*_{−} with and . Then *M*(*w*)=*M*(*w*_{+})+*M*(*w*_{−}). Using the inequality *e*^{t}−1≥*t* (), we have *e*^{u0+w}−1≥*w*+*u*_{0}, which leads to
5.17where we have used the fact that .

On the other hand, using the inequality 1−*e*^{−t}≥*t*/(1+*t*) (*t*≥0), we can estimate *M*(−*w*_{−}) from below as follows:
5.18

From the definition of the function *H*, we may choose *μ*>0 large enough so that (say). Note also that . Thus, we have
5.19Summarizing these facts, we see that (5.18) enjoys the lower bound
5.20where and in the sequel, *C* denotes a generic but irrelevant positive constant.

From (5.17) and (5.20), we have 5.21

Using (5.21) in (5.15), we arrive at 5.22

Moreover, since the matrix *T* is invertible and *v* and *w* are related through **w**=*T*^{τ}**v**, we can find a positive constant *C*_{0} such that
5.23

Inserting (5.23) into (5.22), we get 5.24

We now recall the standard Gagliardo–Nirenberg–Sobolev inequality (cf. Ladyzhenskaya & Uraltseva (1968); Ladyzhenskaya (1969)) 5.25

Consequently, we have 5.26As a result of (5.26), we have 5.27

From (5.24), (5.27) and the relation between *v* and *w*, we may conclude with the coercive lower bound
5.28where *C*_{1},*C*_{2} are some constants. In view of the estimate (5.28), the existence of a critical point of the functional *I* in the space follows in a standard way.

In fact, from (5.28), we may choose *R*>0 large enough such that
5.29Consider the optimization problem
5.30Let {**v**^{(n)}} be a minimizing sequence of (5.30). Without loss of generality, we may assume that this sequence is also weakly convergent. Let **v** be its weak limit. Thus, using the fact that the functional *I* is weakly lower semi-continuous, we have *I*(**v**)≤*η*. Of course, because norm is also weakly lower semi-continuous. Hence, *I*(**v**)=*η* and **v** solves (5.30). We show next that **v** is a critical point of the functional *I*. In fact, we only need to show that **v** is an interior point, or For suppose otherwise that . Then, in view of (5.29), we have
5.31Therefore, when *t*>0 is sufficiently small, we see by virtue of (5.31) that *I*(**v**−*t***v**)<*I*(**v**)=*η*. However, because , we arrive at a contradiction to the definition of **v** or (5.30). Thus, *v* is a critical point of *I*.

Finally, the strict convexity of *I* says that *I* can only have at most one critical point; so we have the conclusion that *I* has exactly one critical point in . Of course, this critical point is a solution of (5.1), which must be smooth by virtue of the standard elliptic regularity theory.

The asymptotic behaviour of the solution will be studied in §6.

## 6. Asymptotic behaviour of planar solution

Let **v**=(*v*_{1},…,*v*_{N}) denote the solution of (5.1) obtained in §5. Here, we aim to establish the pointwise decay properties for **v**. Our tools are based on elliptic *L*^{p}-estimates and the maximum principle.

To proceed, we first recall the following embedding inequality (Ladyzhenskaya & Uraltseva 1968; Gilbarg & Trudinger 1977; Jaffe & Taubes 1980) 6.1From (6.1), we may infer for . To see this fact, we use a MacLaurin series expansion to get 6.2By virtue of (6.1) and (6.2), we obtain 6.3which confirms our claim that because it is easily shown that the series on the right-hand side of (6.3) is convergent.

Now consider (5.8). Because (*i*=1,…,*N*) and the right-hand side of (5.8) may be rewritten as
6.4we see that the right-hand side of each of the equations in (5.8) belongs to . Thus, we may resort to the standard elliptic *L*^{2}-estimates to deduce that (*i*=1,…,*N*). In particular, as because we are in two dimensions.

We can establish similar decay properties for |∇*v*_{i}|(*i*=1,…,*N*). To this end, we rewrite the right-hand sides of (5.8) as
6.5All these belong to the space for any *p*>2 due to the embedding and the definition of . Therefore, elliptic *L*^{p}-estimates enable us to conclude that (*i*=1,…,*N*;*p*>2). In particular, we have as , *i*=1,…,*N*.

To obtain suitable exponential decay estimates for the solution, it suffices to consider (5.1) outside the disc , where 6.6

For convenience, we write (5.1) in in the form 6.7

With the vectors **u**=(*u*_{1},…,*u*_{N})^{τ} and **V**=(*e*^{u1}−*u*_{1}−1,…,*e*^{uN}−*u*_{N}−1)^{τ}, we may rewrite (6.7) as
6.8

Let *O* be an *N*×*N* orthogonal matrix so that
6.9where *λ*_{1},*λ*_{2},…,*λ*_{N} are as given in (4.6). Thus, in terms of the new variable vector
6.10Substituting (6.10) into (6.7) and using (6.9) and the behaviour of as , we arrive at
6.11where as . Consequently, for any *ε*∈(0,1), we can find a suitably large *R*_{ε}>*R* such that
6.12Thus, using a suitable comparison function, the property **U**=0 at infinity, and the maximum principle, we can obtain a constant *C*(*ε*)>0 to achieve
6.13

We next derive some exponential decay estimates for |∇*u*_{i}| (*i*=1,…,*N*). For given ℓ=1,2, we differentiate (6.7) to obtain
6.14Set **v**=(∂_{ℓ}*u*_{1},…,∂_{ℓ}*u*_{N})^{τ} and *E*(*x*)=diag{*e*^{u1(x)},…,*e*^{uN(x)}}. Then the system (6.14) becomes
6.15where *I*_{N} is the *N*×*N* unit matrix. Consequently, we have
6.16where *a*(*x*)→0 as . Hence, as before, we obtain the estimate
6.17

## 7. Consequences of asymptotic estimates

For the function *h*_{i} given in (5.5), we can directly compute to get
7.1Moreover, for the solution (*U*_{1},…,*U*_{N}) of the system (5.6), we have where is defined by (5.4) (*i*=1,…,*N*) and (*u*_{1},…,*u*_{N}) is the unique solution of (5.1) subject to the boundary condition (5.3) whose derivatives are seen to vanish at infinity exponentially fast. Thus, we infer that |∇*U*_{1}|,…,|∇*U*_{N}| all vanish at infinity at least as fast as |*x*|^{−3}. Consequently, we have
7.2Thus, integrating (5.6) and applying (7.1) and (7.2), we obtain the quantized integrals
7.3

Now recall the relation between the functions *u*_{1},…,*u*_{N} and the Higgs scalar fields *ϕ*_{1},…,*ϕ*_{N} described in §2. Because
7.4we see that
7.5if |*x*| is large. Using (7.5) in (2.10), we conclude that the curvatures (*i*=1,…,*N*) vanish at infinity exponentially fast at the same rate. Furthermore, since the Higgs fields *ϕ*_{1},…,*ϕ*_{N} and gauge fields may be constructed from *u*_{1},…,*u*_{N} following the expressions (Yang 2001)
7.6and
7.7*i*=1,…,*N*, we see that the covariant derivatives obey
7.8when |*x*| is sufficiently large. Hence, inserting the value *λ*_{N}=*g*^{2}/2*N*, we obtain all the decay estimates stated in Theorem 2.2.

Finally, applying (7.3) and (7.4) in (2.10) and noting (5.2), we obtain the quantized flux formulas stated in theorem 2.3 in the full plane case.

In the situation of a doubly periodic domain, the same flux quantization conclusion follows simply from integrating (4.1) and no further consideration is needed.

## Acknowledgements

S.C.'s research was partially supported by Henan Basic Science and Frontier Technology Program Funds under grant 112300410054.

- Received March 11, 2012.
- Accepted May 29, 2012.

- This journal is © 2012 The Royal Society