Nuclear Physics B 846 [PM] (2011) 650–676 www.elsevier.com/locate/nuclphysb

Sharp existence and uniqueness theorems for non-Abelian multiple vortex solutions Chang-Shou Lin a , Yisong Yang b,∗ a Department of Mathematics, National Taiwan University, Taipei, Taiwan 10617, ROC b Department of Mathematics, Polytechnic Institute of New York University, Brooklyn, NY 11201, USA

Received 29 December 2010; accepted 19 January 2011 Available online 22 January 2011

Abstract Vortices in non-Abelian gauge field theory play essential roles in the mechanism of color confinement and are governed by systems of nonlinear elliptic equations of complicated structure. In this paper, we present a series of sharp existence and uniqueness theorems for multiple vortex solutions of the non-Abelian BPS equations over R2 and on a doubly periodic domain. Our methods are based on calculus of variations which may be used to analyze more extended problems. The necessary and sufficient conditions for the existence of a unique solution in the doubly periodic situation are expressed in terms of physical parameters involved explicitly. © 2011 Elsevier B.V. All rights reserved. Keywords: Non-Abelian gauge field theory; BPS vortices; Confinement; Higgs condensed solitons; Existence and uniqueness

1. Introduction Vortices have important applications in many fundamental areas of physics including superconductivity [1,15], particle physics [14], optics [5], and cosmology [29]. The first and also the best-known rigorous multiple vortex construction in gauge field theory is due to Taubes [15,26, 27] regarding the existence and uniqueness of static solutions of the Abelian Higgs model or the Ginzburg–Landau model [11] governed by the energy density

* Corresponding author.

E-mail address: [email protected] (Y. Yang). 0550-3213/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2011.01.019

C.-S. Lin, Y. Yang / Nuclear Physics B 846 [PM] (2011) 650–676

2 1 1 λ H = Fj2k + |Dj φ|2 + |φ|2 − 1 , 4 2 8

651

(1.1)

where Fj k = ∂j Ak − ∂k Aj is the magnetic field induced from the real-valued gauge potential field Aj , φ is the complex-valued scalar Higgs field, Dj φ = ∂j φ − iAj φ denotes the gaugecovariant derivative, λ > 0 is a coupling parameter, and summation convention is assumed over repeated indices j, k = 1, 2, which satisfy the Euler–Lagrange equations of (1.1) given by Dj Dj φ =

 λ 2 |φ| − 1 φ, 2

i ∂j Fj k = (φDk φ − φDk φ), 2

(1.2) (1.3)

known as the Ginzburg–Landau equations [11]. At the critical coupling λ = 1 bordering typeI and type-II superconductivity, Eqs. (1.2) and (1.3) over R2 are equivalent to the self-dual or BPS (after the pioneering explorations of Bogomol’nyi [6] and Prasad–Sommerfield [19]) vortex equations D1 φ + iD2 φ = 0,  1 F12 + |φ|2 − 1 = 0, 2

(1.4) (1.5)

or their anti-self-dual counterpart via the correspondence (φ, Aj ) → (φ, −Aj ) so that the solutions are all characterized precisely by the zero distribution of the Higgs field φ [15,26,27] and the zeros of φ are the centers of magnetic vortices where superconductivity is absent. Solutions of such a characteristic are known as multiple vortices. Although λ = 1 corresponds to a non-physical, non-interacting, phase, the multiple vortex solutions represent co-existing normalsuperconducting states of a topological origin and theoretical significance. Recently, studies on vortices in non-Abelian gauge field theory have received considerable attention partly because of the interesting analogue [3,4,9,13,16,20–23,28] made between quark confinement and non-Abelian monopole confinement in the Higgs phase in the presence of nonAbelian vortices. These studies lead to the discovery of some very interesting new families of non-Abelian BPS vortex equations of various levels of challenges. In this paper, we shall concentrate on the non-Abelian BPS vortex equations derived in the recent studies of Auzzi, Bolognesi, Evslin, Konishi, and Yung [4], Gudnason, Jiang, and Konishi [12], Marshakov and Yung [18], and Shifman and Yung [22]. For the scalar quark Higgs field winding in the Abelian group U (1)2N where 2N is the number of quark flavors, we prove the existence and uniqueness of a multiple vortex solution realizing an arbitrarily prescribed vortex distribution over R2 , applying the variational method of Taubes [15,26,27] used for the Abelian Higgs model (Section 3). Using a constrained variational method, we will also establish the existence and uniqueness of a multiple vortex solution over a doubly periodic domain under a necessary and sufficient condition explicitly stated in terms of the coupling parameters, the total vortex number, and the size of the domain (Section 4). For the scalar quark field winding in the Abelian group SO(2N ), the system of equations becomes more complicated. Nevertheless, we will be able to obtain the same existence and uniqueness result over R2 , as in the case of U (1)2N . In this case, the existence problem over a doubly periodic domain becomes more involved. The necessary condition for existence is the same as in the previous U (1)2N situation but some additional sufficient conditions need to be imposed to ensure a successful execution of the same constrained variational method due to the restricted value of the optimal constant in the

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Trudinger–Moser inequality [2,10], similar to those arising in electroweak vortex solution studies [24,31]. However, we may apply the Leray–Schauder degree theory result obtained in [17], a homotopy flow, and a priori estimates to show that the necessary condition is also sufficient for the existence of a solution (Section 5). It is interesting that our methods may readily be extended to obtain variational constructions of multiple vortex solutions of the supersymmetric gauge field theory, formulated by Eto, Fujimori, Nagashima, Nitta, Ohashi, and Sakai in [8] and analyzed in [17] with non-variational methods, without any restriction (Section 6). Furthermore, our methods are shown to be equally effective in treating the existence and uniqueness problems for multiple vortex solutions induced from independently prescribed distributions of zeros of two complex scalar fields, instead of one (Section 6). The main results of the paper are summarized as Theorems 2.1, 5.1, and 6.2. 2. Non-Abelian BPS vortex equations Following Gudnason–Jiang–Konishi [12], the Lagrangian density of the non-Abelian Yang– Mills–Higgs model directly extending that of the Abelian Higgs model is taken to be 1 a aμν 1 0 0μν Fμν F − 2 Fμν F + (Dμ qf )† Dμ qf 2 4e 4g   v02 2 g 2  † a 2 e2  † 0 − qf t qf − √  − qf t qf  , 2 2 4N

L=−

(2.1)

for which the gauge group G is of the general form G = G × U (1) where G is a compact simple Lie group which may typically be chosen to be G = SO(2N ) or G = USp(2N ) (the unitary symplectic group). With the summation convention assumed over repeated indices, μ, ν = 0, 1, 2, 3 are time–space indices, a = 1, . . . , dim(G ) labels the generators of G , the index 0 indicates the Abelian (Maxwellian) gauge field, f = 1, . . . , Nflavor labels the matter flavors or “scalar quark” fields, qf , all are assumed to lie in the fundamental representation of G . The gauge fields, gaugecovariant derivatives, and field tensors are given by Aμ = A0μ + Aaμ t a ,

Dμ qf = ∂μ qf + iAμ qf ,

Fμν = ∂μ Aν − ∂ν Aμ + i[Aμ , Aν ], respectively, where the generators of  1  Tr t a t b = δ ab , 2

t0 = √

1 4N

G

(2.2) and U (1), i.e.,

12N ,

{t a }

and

t 0,

are normalized to satisfy (2.3)

with 1m denoting the m × m identity matrix. When the number of matter flavors is Nflavor = 2N , the scalar quark fields may be represented as a color-flavor mixed matrix q of size 2N × 2N . Restricting to static field configurations which are uniform with respect to the spatial coordinate x 3 , a Bogomol’nyi completion [6] may be performed to yield the BPS [6,19] vortex equations [4,7,12,18,22,23] D1 q + iD2 q = 0,  e2   †  0 Tr qq − v02 = 0, −√ F12 4N  T  g2  † a a t − qq − J † qq † J = 0, F12 4

(2.4) (2.5) (2.6)

C.-S. Lin, Y. Yang / Nuclear Physics B 846 [PM] (2011) 650–676

where J is the standard symplectic matrix   0 1N . J= −1N 0

653

(2.7)

In its general form, the system of the non-Abelian BPS vortex equations (2.4)–(2.6) appears hard to approach and an ansatz-based reduction may be made as a tool for further simplification. Here we concentrate on two ansätze presented in [12]. The first ansatz is based on letting the scalar quark fields wind in the Abelian subgroup U (1)N ⊂ G so that the Higgs-field matrix q becomes diagonal,   φ1N 0 , q= 0 ψ1N Aj = aj 12N + bj T , T = diag(1N , −1N ), where φ, ψ are two complex-valued scalar fields and aj , bj are two real-valued gauge potential vector fields. The BPS equations (2.4)–(2.6) take the form   (∂1 + i∂2 )φ = i [a1 + ia2 ] + [b1 + ib2 ] φ, (2.8)   (∂1 + i∂2 )ψ = i [a1 + ia2 ] − [b1 + ib2 ] ψ, (2.9)  α 2 a12 = ∂1 a2 − ∂2 a1 = − |φ| + |ψ|2 − γ , (2.10) 4  β b12 = ∂1 b2 − ∂2 b1 = − |φ|2 − |ψ|2 , (2.11) 4 where the positive parameters α, β, γ are given by v02 . (2.12) N Note that Eqs. (2.8)–(2.11) also appear in [4,18,22,23]. It will be illustrative to compare (2.8)–(2.11) with the Abelian Higgs BPS vortex equations (1.4)–(1.5) which may be rewritten as α = e2 ,

β = g2,

γ=

(2.13) (∂1 + i∂2 )φ = i(A1 + iA2 )φ,  1 2 (2.14) ∂1 A2 − ∂2 A1 = − |φ| − 1 . 2 Thus, as in the case of the Abelian Higgs BPS model [15,26,27], we are to look for n-vortex solutions so that the scalar field φ vanishes exactly at the prescribed zero set Z = {p1 , . . . , pn },

(2.15)

where possible repeated points are allowed to account for algebraic multiplicities of the zeros. A zero of multiplicity represents a vortex of local winding charge or vorticity . Solutions of (2.8)–(2.11) with the prescribed zero set (2.15) are to be obtained either over the full plane R2 satisfying the boundary condition [12]  γ |φ|, |ψ| → as |x| → ∞, (2.16) 2 or over a doubly periodic lattice cell domain Ω induced from the ’t Hooft type boundary condition [25,30]. Concerning these solutions, here is our existence theorem.

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Theorem 2.1. Consider the non-Abelian BPS vortex equations (2.8)–(2.11) defined over R2 or a doubly periodic cell domain Ω. (i) For any α, β, γ > 0 and any given zero-set data (2.15), these equations have a finite-energy unique smooth solution (φ, ψ, aj , bj ) over R2 so that (2.15) is the zero set of φ but ψ has no zero so that the boundary condition (2.16) is realized exponentially rapidly. (ii) When the equations are formulated over a doubly periodic domain Ω, for any given zero-set data (2.15), there is a smooth solution (φ, ψ, aj , bj ) with φ realizing these zeros if and only if the coupling parameters α, β, γ , the domain volume |Ω|, and the total vortex number n satisfy the condition   1 1 4πn + < γ |Ω|. (2.17) α β Furthermore, if there exists a solution, it is unique. We note that, in terms of the original parameters e, g, v0 , N , the condition (2.17) reads   v2 1 1 (2.18) 4πn 2 + 2 < 0 |Ω|. N e g Besides, in the special situation when α = β, it is consistent to impose |ψ| ≡ γ2 and aj ≡ bj (j = 1, 2) in (2.8)–(2.11). It is seen that, now, the system reduces to the form of (2.13)–(2.14) studied fully in [15,26,27] over R2 , and in [30] over a doubly periodic domain where the necessary and sufficient condition for the existence of an n-vortex solution reads 8πn < αγ |Ω|,

(2.19)

which is a specialization of the general condition (2.17). In the first part of this paper, we shall establish Theorem 2.1. To facilitate our computation, it will be convenient to adopt the derivatives 1 ∂ = (∂1 − i∂2 ), 2 and the notation a = a1 + ia2 ,

1 ∂ = (∂1 + i∂2 ), 2 b = b1 + ib2 .

(2.20)

(2.21)

As a consequence, away from Z, Eqs. (2.8) and (2.9) become i i ∂ ln φ = (a + b), ∂ ln ψ = (a − b), 2 2 which allow us to solve for a, b to get a = −i∂(ln φ + ln ψ),

b = −i∂(ln φ − ln ψ).

(2.22)

(2.23)

Using a12 = −i(∂a − ∂a),

b12 = −i(∂b − ∂b),

(2.10), (2.11), (2.23), and the fact that ∂∂ = ∂∂ = 14 , we have

(2.24)

C.-S. Lin, Y. Yang / Nuclear Physics B 846 [PM] (2011) 650–676

     ln |φ|2 + ln |ψ|2 = α |φ|2 + |ψ|2 − γ ,      ln |φ|2 − ln |ψ|2 = β |φ|2 − |ψ|2 ,

655

(2.25) (2.26)

away from Z. Now, setting u = ln |φ|2 , v = ln |ψ|2 , we arrive at the coupled system n    δps (x), (u + v) = α eu + ev − γ + 4π

(2.27)

s=1



(u − v) = β e − e u

v



+ 4π

n 

(2.28)

δps (x),

s=1

where we have included our consideration of the zero set Z of φ as given in (2.15). 3. Solution on full plane In this section, we prove the existence and uniqueness of the solution of the system of equations (2.27) and (2.28) over R2 satisfying the boundary condition     γ γ u → ln , v → ln as |x| → ∞. (3.1) 2 2 It will be convenient to work with the redefined variables and parameters γ γ γ , v → v + ln , α → α, 2 2 2 in this section. Thus (2.27) and (2.28) become u → u + ln

γ β → β, 2

n    δps (x), (u + v) = α eu + ev − 2 + 4π

(3.2)

(3.3)

s=1



(u − v) = β eu − e

 v

+ 4π

n 

(3.4)

δps (x),

s=1

and the boundary condition (3.1) is translated into u → 0,

v → 0 as |x| → ∞.

(3.5)

To proceed further, we introduce the background function [15] u0 (x) = −

n    ln 1 + μ|x − ps |−2 ,

μ > 0.

(3.6)

s=1

(Here and in the sequel, the parameter μ > 0 should not be confused with the spacetime index μ used in the context of various field equations.) Then we have u0 = −h + 4π

n 

δps (x),

h(x) = 4

s=1

Using the substitution u = u0 + w, we have

n  s=1

μ . (μ + |x − ps |2 )2

(3.7)

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  (w + v) = α eu0 +w + ev − 2 + h,   (w − v) = β eu0 +w − ev + h.

(3.8) (3.9)

Taking f = w + v,

g = w − v,

we change (3.8) and (3.9) into   1 1 f = α eu0 + 2 (f +g) + e 2 (f −g) − 2 + h,   1 1 g = β eu0 + 2 (f +g) − e 2 (f −g) + h.

(3.10)

(3.11) (3.12)

(Here and in the sequel, the function f should not be confused with the flavor index f and the function g with the coupling constant g used in the context of various gauge field equations.) It is clear that (3.11) and (3.12) are the Euler–Lagrange equations of the action functional

  1 1 1 1 I (f, g) = dx |∇f |2 + |∇g|2 + 2 eu0 + 2 (f +g) − eu0 − (f + g) 2α 2β 2 R2

    1 1 g f + . + 2 e 2 (f −g) − 1 − (f − g) + h 2 α β

(3.13)

It is clear that the functional I is a C 1 -functional for f, g ∈ W 1,2 (R2 ) and its Fréchet derivative satisfies

 1  1 1 DI (f, g)(f, g) = dx |∇f |2 + |∇g|2 + eu0 e 2 (f +g) − 1 (f + g) α β R2

    1   f g + . + eu0 − 1 (f + g) + e 2 (f −g) − 1 (f − g) + h α β

(3.14)

On the other hand, (3.10) gives us |∇f |2 + |∇g|2 = 2|∇w|2 + 2|∇v|2 .

(3.15)

As a consequence of (3.14) and (3.15), with σ = max{α, β}, we obtain

2 |∇w|2 + |∇v|2 dx DI (f, g)(f, g) − σ 2

R2

      h 1 h 1 1 1 dx w eu0 +w − 1 + + + v ev − 1 + − 2 α β 2 α β

R2

≡ 2M1 (w) + 2M2 (v).

(3.16)

As in [15], we decompose v and w into their positive and negative parts, w = w+ − w− and v = v+ − v− , where q+ = max{q, 0} and q− = − min{q, 0} for q ∈ R. Using the elementary inequality et − 1  t, we have

t ∈ R,

(3.17)

C.-S. Lin, Y. Yang / Nuclear Physics B 846 [PM] (2011) 650–676

eu0 +w − 1 + which leads to

    h 1 h 1 1 1 +  u0 + w + + , 2 α β 2 α β





M1 (w+ ) 

2 w+ dx + R2

1  2

2 w+ dx R2

(3.18)

   h 1 1 w+ u0 + + dx 2 α β

R2



657

1 − 2



  h 1 1 2 dx. + u0 + 2 α β

(3.19)

R2

On the other hand, using the inequality t , t  0, 1 − e−t  1+t we have     1 h 1 + − eu0 −w− w− 1 − 2 α β       1 h 1 = w− 1 − + + eu0 1 − e−w− − eu0 2 α β     w− 1 h 1  w− 1 − − eu 0 + + eu 0 2 α β 1 + w−       2 w− h 1 1 w− 1 h 1 u0 = + + + . 1− 1−e − 1 + w− 2 α β 1 + w− 2 α β In view of (3.7), we see that we may choose μ > 0 large enough so that   1 1 h(x) + < 1, x ∈ R2 . α β On the other hand, since 1 − eu0 and h both lie in L2 (R2 ), we have    2 w− w−  h 1 1  u0 − dx + C(ε), 1 − e + dx  ε   1 + w− 2 α β 1 + w− R2

(3.20)

(3.21)

(3.22)

(3.23)

R2

where ε > 0 may be chosen to be arbitrarily small. Combining (3.21)–(3.23), we obtain 2 w− 1 dx − C1 (ε), M1 (−w− )  4 1 + w−

(3.24)

R2

provided that ε < 1/4. From (3.19) and (3.24), we get w2 1 M1 (w)  dx − C, 4 1 + |w|

(3.25)

R2

where and in the sequel we use C to denote an irrelevant generic positive constant. Similar estimates may be made for M2 (v). Thus, (3.16) gives us  

2 1 v2 w2 2 2 DI (f, g)(f, g) − |∇w| + |∇v| dx  + dx − C. (3.26) σ 2 1 + |w| 1 + |v| R2

R2

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We now recall the standard Sobolev inequality   v 4 dx  2 v 2 dx |∇v|2 dx, v ∈ W 1,2 R2 . R2

R2

(3.27)

R2

Consequently, we have 2   2 v dx = R2

R2



2 R2

 |v|  1 + |v| |v| dx 1 + |v| v2 dx (1 + |v|)2



4 R2

1  2

(1 + |v|)2 v dx

 v 2 + v 4 dx

R2

dx

  v 2 dx 1 + |∇v|2 dx

R2

2 2





v2









+C 1+

R2

R2

As a result of (3.28), we have 1   2 2 2 v dx  C 1 + |∇v| dx + R2

2

R2

R2

R2

v2 dx (1 + |v|)2

4

 +

4  |∇v| dx 2

. (3.28)

R2

 v2 dx . (1 + |v|)2

Applying (3.29) in (3.26), we arrive at   DI (f, g)(f, g)  C1 w W 1,2 (R2 ) + v W 1,2 (R2 ) − C2 .

(3.29)

(3.30)

In view of (3.10), we have the identity f 2 + g 2 = 2w 2 + 2v 2 . Thus, using (3.15) and (3.31) in (3.30), we conclude with the coercive lower bound   DI (f, g)(f, g)  C1 f W 1,2 (R2 ) + g W 1,2 (R2 ) − C2 .

(3.31)

(3.32)

With (3.32), we can now show that the existence of a critical point of the action functional (3.13) follows by using a standard argument as in [15,32]. In fact, from (3.32), we can choose R > 0 large enough such that   

(3.33) inf DI (f, g)(f, g)  f, g ∈ W 1,2 R2 , f W 1,2 (R2 ) + g W 1,2 (R2 ) = R  1 (say). Now consider the minimization problem 

η = min I (f, g)  f W 1,2 (R2 ) + g W 1,2 (R2 )  R .

(3.34)

Let {(fk , gk )} be a minimizing sequence of (3.34). Without loss of generality, we may assume that {(fk , gk )} weakly converges to an element (f, g) in W 1,2 (R2 ). The weakly lower semicontinuity of I implies that (f, g) solves (3.34). To show that (f, g) is a critical point of I , it suffices to see that it is an interior point. That is, f W 1,2 (R2 ) + g W 1,2 (R2 ) < R.

(3.35)

C.-S. Lin, Y. Yang / Nuclear Physics B 846 [PM] (2011) 650–676

659

Suppose otherwise that f W 1,2 (R2 ) + g W 1,2 (R2 ) = R. Then for t ∈ (0, 1) the point (1 − t)(f, g) is interior which gives us   I (1 − t)f, (1 − t)g  η = I (f, g). (3.36) On the other hand, we have

  I ((1 − t)(f, g)) − I (f, g) d  lim = I (1 − t)(f, g)  t→0 t dt t=0 = −DI (f, g)(f, g)  −1.

Consequently, if t > 0 is sufficiently small, (3.37) leads to   I (1 − t)(f, g) < I (f, g) = η,

(3.37)

(3.38)

which contradicts (3.36). Therefore, the existence of a critical point of I follows. Note that the part in the integrand of I which does not involve the derivatives of f and g may be rewritten as      1  1 f g Q(f, g) = 2 eu0 + 2 (f +g) − eu0 + 2 e 2 (f −g) − 1 − 2f + h + , (3.39) α β whose Hessian is easily checked to be positive definite. Thus, the functional I is strictly convex. As a consequence, I can have at most one critical point (f, g) in the space W 1,2 (R2 ). Using the methods in [15,32], it can be shown that the right-hand sides of (3.11) and (3.12) belong to L2 (R2 ). We may now apply the standard elliptic theory to (3.11) and (3.12) to infer f, g ∈ W 2,2 (R2 ). In particular, f and g approach zero as |x| → ∞, which renders the validity of the boundary condition (3.5). Linearizing (3.3) and (3.4) around u = 0 and v = 0, in a neighborhood of infinity of R2 , we have (u + v) ≈ α(u + v),

(u − v) ≈ β(u − v),

(3.40)

which indicate that u and v vanish exponentially fast at infinity. Inserting this information into (3.8) and (3.9), we see that the associated functions w and v and the right-hand sides of (3.8) and (3.9) all lie in L2 (R2 ). Consequently, the pair f and g defined by (3.10) yields a W 2,2 (R2 )solution of (3.11) and (3.12), which must be the unique critical point of the functional I produced earlier. In conclusion, we can state Theorem 3.1. For any distribution of the points p1 , . . . , pn ∈ R2 , the system of nonlinear elliptic equations (3.3) and (3.4) subject to the boundary condition (3.5) has a unique solution. Furthermore, the solution realizes the boundary condition (3.5) exponentially fast. This theorem establishes the part (i) stated in Theorem 2.1. 4. Solution on doubly periodic domain Next, we consider the compact case setting in which the vortices lie in a doubly periodic domain Ω (say). Let u0 be a doubly periodic function over Ω which solves the equation

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u0 = 4π

n 

δps (x) −

s=1

4πn . |Ω|

(4.1)

Using the new variable w so that u = u0 + w, we can modify (2.27) and (2.28) into  4πn  , (w + v) = α eu0 +w + ev − γ + |Ω|  4πn  (w − v) = β eu0 +w − ev + . |Ω| To proceed further, we take 1 1 g = (w − v). f = (w + v), 2 2 Then the governing system of equations becomes  2πn α  u0 +f +g + ef −g − γ + e , 2 |Ω|  2πn β g = eu0 +f +g − ef −g + . 2 |Ω|

f =

It is clear that these equations are the Euler–Lagrange equations of the action functional

1 1 |∇f |2 + |∇g|2 + eu0 +f +g + ef −g I (f, g) = α β Ω   4πn 4πn + −γ f + g dx. α|Ω| β|Ω|

(4.2) (4.3)

(4.4)

4πn > γ . Hence we It is easily seen that the functional (4.4) is not bounded from below when α|Ω| cannot carry out a direct minimization procedure for the problem. In fact, we need to consider a constrained minimization procedure instead. Integrating (4.2) and (4.3), we obtain 4πn eu0 +f +g dx + ef −g dx = γ |Ω| − , (4.5) α Ω Ω 4πn eu0 +f +g dx − ef −g dx = − . (4.6) β Ω

Ω

Or equivalently,    1 1 1 eu0 +f +g dx = γ |Ω| − 4πn + ≡ η1 , 2 α β Ω   1 4πn 4πn ef −g dx = γ |Ω| + − ≡ η2 . 2 β α

(4.7)

(4.8)

Ω

Of course, the conditions (4.7) and (4.8) imply that the existence of an n-vortex solution requires that η1 > 0 and η2 > 0, which is simply

C.-S. Lin, Y. Yang / Nuclear Physics B 846 [PM] (2011) 650–676

η1 > 0,

661

(4.9)

since η1 > 0 contains η2 > 0. We can prove that (4.9) is in fact sufficient for existence as well. Theorem 4.1. The system of the non-Abelian vortex equations (4.2) and (4.3) has a solution if and only if (4.9) holds or   1 1 4πn + < γ |Ω|. (4.10) α β Furthermore, if a solution exists, it must be unique, which can be constructed through solving a multiply constrained minimization problem. This theorem establishes the part (ii) stated in Theorem 2.1. We use W 1,2 (Ω) to denote the usual Sobolev space of doubly periodic functions over the cell domain Ω. The proof of Theorem 4.1 is based on the solution of the constrained minimization problem 

min I (f, g)  f, g ∈ W 1,2 (Ω) and satisfy the constraints (4.7) and (4.8) . (4.11) We first show that the Lagrange multipliers do not rise as a problem. To this end, let (f, g) be a solution of (4.11). Then there are numbers λ1 , λ2 ∈ R such that  

4πn ˜ 2 ∇f · ∇ f˜ + eu0 +f +g + ef −g − γ + f dx α α|Ω| Ω u0 +f +g ˜ f dx + λ2 ef −g f˜ dx, ∀f˜, (4.12) = λ1 e

Ω

Ω

Ω

  4πn 2 u0 +f +g f −g −e + ∇g · ∇ g˜ + e g˜ dx β β|Ω| = λ1 eu0 +f +g g˜ dx − λ2 ef −g g˜ dx, ∀g. ˜



Ω

(4.13)

Ω

Setting f˜ ≡ 1 and g˜ ≡ 1 in (4.12) and (4.13), respectively, and using the constraints (4.7) and (4.8), we obtain λ1 = λ2 = 0. Using this result in (4.12) and (4.13), we see that the pair (f, g) solves (4.2) and (4.3) as expected. We next show that the constrained minimization problem (4.11) has a solution under the condition (4.10). Substituting the constraints (4.7) and (4.8) in (4.4), we can rewrite (4.4) as  

1 4πn 4πn 1 2 2 I (f, g) = (4.14) |∇f | + |∇g| + −γ f + g dx + η1 + η2 . α β α|Ω| β|Ω| Ω

Setting f =f +f,

g = g  + g,

f , g ∈ R, Ω

f  dx =

Ω

g  dx = 0,

(4.15)

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we can derive from (4.7) and (4.8) the expressions     1 1 1   u0 +f  +g  e dx − ln ef −g dx , f = ln(η1 η2 ) − ln 2 2 2 

g=

1 η1 ln 2 η2

Ω

 −

1 ln 2





(4.16)

Ω

  1     eu0 +f +g dx + ln ef −g dx . 2

Ω

(4.17)

Ω

Inserting (4.16) and (4.17) into (4.14), we have  

4πn 1   2 1   2 4πn I (f, g) = ∇f + ∇g − γ |Ω| f + g + η1 + η2 dx + α β α β Ω



=

    1   2 1   2     eu0 +f +g dx + η2 ln ef −g dx ∇f + ∇g dx + η1 ln α β

Ω

Ω



   1 4πn 2πn η1 + η1 + η2 + − γ |Ω| ln(η1 η2 ) + ln . 2 α β η2

Ω

On the other hand, Jensen’s inequality implies that    1   1 u dx u0 +f  +g    u0 + f + g dx = |Ω|e |Ω| Ω 0 , e dx  |Ω| exp |Ω| Ω

(4.18)

(4.19)

Ω



e

f  −g 

Ω

     1  f − g dx = |Ω|. dx  |Ω| exp |Ω|

(4.20)

Ω

Inserting these into (4.18), we arrive at the coercive lower estimate

1   2 1   2 I (f, g)  ∇f + ∇g dx − C, α β

(4.21)

Ω

where C > 0 is an irrelevant constant. From (4.21), we know that the existence of solution of (4.11) follows. In fact, let {(fk , gk )} be a minimizing sequence of the problem (4.11) and set 1 1 fk = fk dx, gk = gk dx. (4.22) |Ω| |Ω| Ω

fk

Ω

Then, with = fk − f k and = gk − g k . We have f k = 0 and g k = 0. In view of (4.21), and applying the Poincaré inequality, we see that {(fk , gk )} is bounded in W 1,2 (Ω). Without loss of generality, we may assume that {(fk , gk )} converges weakly in W 1,2 (Ω) to an element (f  , g  ) (say). The compact embedding W 1,2 (Ω) → Lp (Ω),

gk

p  1,

then implies (fk , gk ) → (f  , g  ) in Lp (Ω) (p  1) as k → ∞. In particular, f  = g  = 0.

(4.23)

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Furthermore, recall the Trudinger–Moser inequality   1 eF dx  C0 exp |∇F |2 dx , F ∈ W 1,2 (Ω), F = 0. 16π Ω

663

(4.24)

Ω

In view of (4.23) and (4.24), we see that the functionals defined by the right-hand sides of (4.16) and (4.17) are continuous in f  , g  with respect to the weak topology of W 1,2 (Ω). Therefore, f k → f , g k → g as k → ∞, as given in (4.16) and (4.17). In other words, (f, g) = (f + f  , g + g  ) satisfies the constraints (4.7) and (4.8) and solves the constrained minimization problem (4.11). Since the functional (4.4) is convex, it can at most have one critical point. Thus the uniqueness of a solution of the system of equations (4.2) and (4.3) follows. 5. Vortices in the SO(2N) model In this section, we show that our method may be applied to establish the same existence and uniqueness theorem for multiple vortex solutions in the more complicated G = SO(2N ) model [12] for which the Higgs-field matrix takes the form   Φ12N−2 0 0 q= (5.1) 0 φ 0 , 0 0 ψ and the gauge potential is given by Aj = aj 12N + bj diag{02N −2 , 1, −1},

j = 1, 2.

(5.2)

Then, in terms of the complexified field a and b defined in (2.21), the BPS system of equations found in [12], without assuming radial symmetry, assumes the form ∂Φ = iaΦ,

(5.3)

∂φ = i(a + b)φ,

(5.4)

∂ψ = i(a − b)ψ,  e2  2 a12 = v − 2(N − 1)|Φ|2 − |φ|2 − |ψ|2 , 4N 0  g2  2 b12 = |ψ| − |φ|2 . 4 Applying (2.24), we can recast the system of equations (5.3)–(5.7) into    ln |Φ|4 =  ln |φ|2 + ln |ψ|2 ,     ln |φ|2 + ln |ψ|2 = 2(N − 1)|Φ|2 + |φ|2 + |ψ|2 − v02 , N      ln |φ|2 − ln |ψ|2 = 2g 2 |φ|2 − |ψ|2 , 2e2 

(5.5) (5.6) (5.7)

(5.8) (5.9) (5.10)

where we have stayed away from the possible zeros of the fields Φ, φ, ψ . Following [12], we consider a solution so that the zeros of Φ and φ coincide but ψ has no zero so that they enjoy the following boundary condition at infinity:

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v0 |Φ|, |φ|, |ψ| → √ 2N

as |x| → ∞.

(5.11)

Thus (5.8) and (5.11) lead us to the simple relation |Φ|4 = |φ|2 |ψ|2 ,

(5.12)

which says that the multiplicities of the zeros of φ are twice of those of the zeros of Φ, as already stated in [12] under the radial symmetry ansatz. We use N to denote the set of natural numbers. Parallel to Theorem 2.1, we can state Theorem 5.1. For the non-Abelian BPS vortex equations (5.3)–(5.7) over R2 and on a doubly periodic cell domain Ω, respectively, the following conclusions hold. (i) For any parameters e, g, v0 > 0, N ∈ N, and prescribed zero-set data (2.15), the equations have a finite-energy unique smooth solution (Φ, φ, ψ, aj , bj ) over R2 so that (2.15) is the zero set of φ but ψ has no zero so that Φ is determined by φ and ψ through (5.12) and the boundary condition (2.16) is realized exponentially rapidly. (ii) When the equations are formulated over a doubly periodic domain Ω, for any given zero-set data (2.15), there is a smooth solution (Φ, φ, ψ, aj , bj ) with φ realizing these zeros as in (i) if and only if the parameters e, g, v0 , N , the domain volume |Ω|, and the total vortex number n satisfy the condition   N 1 2πn 2 + 2 < v02 |Ω|. (5.13) e g Moreover, if a solution exists, it is unique. It may be interesting to compare (5.13) with (2.18). We shall proceed as in the previous sections. Assume that the zero set of φ is as given in (2.15) and set 2e2 γ = v02 . , β = 2g 2 , N We see that (5.8)–(5.10) with (5.8) being replaced by (5.12) become u = ln |φ|2 ,

v = ln |ψ|2 ,

α=

n    1 (u + v) = α 2(N − 1)e 2 (u+v) + eu + ev − γ + 4π δps (x),

(5.14)

(5.15)

s=1 n    δps (x), (u − v) = β eu − ev + 4π

(5.16)

s=1

subject to the boundary condition   γ u, v → ln as |x| → ∞. 2N Now introduce the new variables and rescaled parameters following     αγ γ γ , v → v + ln , → α, u → u + ln 2N 2N 2N

(5.17)

βγ → β. 2N

(5.18)

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665

We recast the problem into n    1 δps (x), (u + v) = α 2(N − 1)e 2 (u+v) + eu + ev − 2N + 4π

(5.19)

s=1 n    δps (x), (u − v) = β eu − ev + 4π

(5.20)

s=1

with the associated boundary condition u, v → 0 as |x| → ∞.

(5.21)

We have Theorem 5.2. The system of nonlinear elliptic equations (5.19) and (5.20) subject to the boundary condition (5.21) has a unique solution for which the boundary condition (5.21) may be achieved exponentially fast. This theorem establishes the part (i) in the statement of Theorem 5.1. In order to prove Theorem 5.2, we set u = u0 + w as before, where u0 is defined by (3.6), and 1 1 g = (w − v). (5.22) f = (w + v), 2 2 Then (5.19) and (5.20) become   1 1 1 f = α (N − 1)e 2 u0 +f + eu0 +f +g + ef −g − N + h, (5.23) 2 2  h β (5.24) g = eu0 +f +g − ef −g + , 2 2 where h is as defined in (3.7). It can be checked that (5.23) and (5.24) are the Euler–Lagrange equations of the action functional

 1  1 1 1 I (f, g) = dx |∇f |2 + |∇g|2 + (N − 1) e 2 u0 +f − e 2 u0 − f 2α 2β R2

 1  1  u0 +f +g − eu0 − [f + g] + ef −g − 1 − [f − g] e 2 2   g f + . +h α 2β

+

(5.25)

Note that I is C 1 over W 1,2 (R2 ) and strictly convex. Besides, decompose I as I (f, g) = I1 (f ) + I2 (f, g), where



I1 (f ) = R2

 1 u +f  1 1 2 u 0 0 dx − e2 − f , |∇f | + (N − 1) e 2 4α

(5.26)



(5.27)

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I2 (f, g) =

dx R2

+

 1 1 1 |∇f |2 + |∇g|2 + eu0 +f +g − eu0 − [f + g] 4α 2β 2

   1  f −g f g − 1 − [f − g] + h e + . 2 α 2β

(5.28)

We can use the methods in [15] and in the earlier study in the present paper to establish the coercive bounds DI1 (f )(f )  C1 f W 1,2 (R2 ) − C2 ,   DI2 (f, g)(f, g)  C1 f W 1,2 (R2 ) + g W 1,2 (R2 ) − C2 ,

(5.29) (5.30)

for I1 and I2 , respectively. Consequently, I satisfies (3.32) as well. Therefore, it follows that the functional I has a unique critical point in W 1,2 (R2 ) which establishes the existence and uniqueness of a classical solution to the system of equations (5.19) and (5.20) subject to the boundary condition (5.21). We now turn our attention to the existence of multivortex solutions over a doubly periodic domain Ω. Let u0 be given by (4.1). Then, setting u = u0 + w, we see that Eqs. (5.15) and (5.16) over the doubly periodic domain Ω become  4πn  1 1 , (w + v) = α 2(N − 1)e 2 u0 + 2 (w+v) + eu0 +w + ev − γ + |Ω|  4πn  . (w − v) = β eu0 +w − ev + |Ω| Theorem 5.3. The system of equations (5.31) and (5.32) has a solution if and only if   1 1 + < γ |Ω| 4πn α β

(5.31) (5.32)

(5.33)

is satisfied. Furthermore, if a solution exists, it must be unique, which can be constructed through solving a multiply constrained minimization problem provided that n = 1, 2, 3 and 3α < β; or n = 1, 2 and α < β; or n = 1 and α < 3β. This theorem establishes the part (ii) stated in Theorem 5.1. To proceed in the formalism of calculus of variations, we use the new variables f = 12 (w + v) and g = 12 (w − v) again. Then (5.31) and (5.32) take the form   1 1 u0 +f +g 1 f −g γ 2πn u0 +f 2 + e + e − + , (5.34) f = α (N − 1)e 2 2 2 |Ω|  2πn β g = eu0 +f +g − ef −g + , (5.35) 2 |Ω| which are the Euler–Lagrange equations of the action functional

1 1 1 I (f, g) = dx |∇f |2 + |∇g|2 + (N − 1)e 2 u0 +f 2α 2β Ω    2πn 1 γ 2πn + eu0 +f +g + ef −g + − f+ g . 2 α|Ω| 2 β|Ω|

(5.36)

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667

As before, this functional is not bounded from below when 4πn > γ |Ω|. α So we need to employ a constrained minimization approach again. Integrating (5.34) and (5.35), we are led to the two constraints

1 1 1 2πn γ dx (N − 1)e 2 u0 +f + eu0 +f +g + ef −g = |Ω| − ≡ η1 > 0, 2 2 2 α Ω  

γ 1 1 1 dx (N − 1)e 2 u0 +f + eu0 +f +g = |Ω| − 2πn + ≡ η2 > 0. 2 α β

(5.37)

(5.38)

(5.39)

Ω

It is clear that η1 > 0 is ensured by η2 > 0. In particular, we see that (5.37) will never happen. It is also interesting to note that the condition η2 > 0 is the same as (4.10). We need to check whether the constraints (5.38) and (5.39) give rise to a multiplier problem. To this end, let (f, g) be a critical point of (5.36) subject to the constraints (5.38) and (5.39). Then there are real numbers λ1 and λ2 (the Lagrange multipliers) such that 

   1 2πn 1 1  u0 +f +g γ u0 +f f −g ˜ 2 + dx + e +e ∇f · ∇ f + (N − 1)e − f˜ α 2 α|Ω| 2 Ω

1 1 1 = λ1 dx (N − 1)e 2 u0 +f + eu0 +f +g + ef −g f˜ 2 2 Ω

1 (5.40) + λ2 dx (N − 1)e 2 u0 +f + eu0 +f +g f˜, ∀f˜, Ω

Ω



   2πn 1 1  u0 +f +g f −g + dx −e ∇g · ∇ g˜ + e g˜ β 2 β|Ω|

λ1 dx eu0 +f +g − ef −g g˜ + λ2 dx eu0 +f +g g, ˜ = 2 Ω

∀g. ˜

(5.41)

Ω

Inserting f˜ ≡ 1 and g˜ ≡ 1 in (5.40) and (5.41) and using (5.34) and (5.35) or (5.38) and (5.39), we arrive at 2πn λ1 (5.42) − λ2 eu0 +f +g dx = 0, λ1 η1 + λ2 η2 = 0, β Ω

where η1 and η2 are as defined in (5.38) and (5.39). From (5.42), we deduce λ1 = λ2 = 0. Applying this result in (5.40) and (5.41), we see that (f, g) solves the original system of equations, (5.34) and (5.35). In other words, the Lagrange multipliers do not present a problem. Thus, in order to solve (5.34) and (5.35), it suffices to prove the existence of a solution to the constrained minimization problem 

(5.43) min I (f, g)  f, g ∈ W 1,2 (Ω), f, g satisfy (5.38) and (5.39) . For any functions f, g in W 1,2 (Ω), we decompose them by (4.15). Thus (5.38) and (5.39) become

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1 f +g 1 f −g   u0 +f  +g  (N − 1)e e dx + e e dx + e ef −g dx = η1 , 2 2 Ω Ω Ω 1    (N − 1)e f e 2 u0 +f dx + e f +g eu0 +f +g dx = η2 , 1  2 u0 +f

f

Ω

Ω

which may be solved to give us   1    f = − ln (N − 1) e 2 u0 +f dx + e g eu0 +f +g dx + ln η2 , Ω

(5.44)

Ω

 2   2πn 1  +g    u0 +f  u +f 0  2 g = ln dx + η2 η3 e dx ef −g dx (N − 1) e β −

2πn (N − 1) β



Ω

 

1

e 2 u0 +f dx − ln



Ω

Ω

eu0 +f

 +g 



Ω

dx − ln(η3 ),

(5.45)

Ω

where η3 = η 1 +

2πn . β

(5.46)

Using the Schwartz inequality, we have  e

1  2 u0 +f

2 dx

 =

Ω

e



1 1 1     2 u0 + 2 (f +g )+ 2 (f −g )

Ω

e

u0 +f  +g 

Ω

ef

dx

 −g 

2 dx

dx.

(5.47)

Ω

Inserting (5.47) into (5.45), we obtain     1    e 2 u0 +f dx − ln eu0 +f +g dx g  ln Ω

Ω

   2  2πn 2πn + ln (N − 1) + η2 η3 − (N − 1) − ln(η3 ). β β On the other hand, applying the Jensen inequality as before, we have  

 1 1 1 1  e 2 u0 +f dx  |Ω| exp u0 + f  dx = |Ω|e 2|Ω| Ω u0 dx . |Ω| 2 Ω

(5.48)

(5.49)

Ω

In view of (5.49), we obtain from (5.44) the lower bound  1   −f  ln (N − 1)|Ω|e 2|Ω| Ω u0 dx − ln η2  −C,

where, as before, C denotes an irrelevant positive constant. Using (5.50) in (5.36), we get

(5.50)

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I (f, g) 

1   2 1   2 2πn ∇f + ∇g g − C, dx + 2α 2β β

669

(5.51)

Ω

where we have used the fact that the constant η1 in (5.38) is positive. Without loss of generality, we may also assume that u0  0. In view of this fact, the Trudinger– Moser inequality (4.24), (5.48), (5.49), (5.51), and a simple interpolation inequality, we have  

1   2 1   2 2πn f  +g  I (f, g)  e dx − C ∇f + ∇g ln dx − 2α 2β β Ω Ω   2   2 (5.52)  δ1 (ε) ∇f  dx + δ2 (ε) ∇g  dx − C, Ω

Ω

where ε > 0 is arbitrary and   1 n 1 − 1+ , δ1 (ε) = 2α 8β ε 1 n − (1 + ε). δ2 (ε) = 2β 8β In order to ensure δ1 (ε) > 0 and δ2 (ε) > 0, we require     β 1 <4 , n(1 + ε) < 4. n 1+ ε α

(5.53) (5.54)

(5.55)

As in the previous section, we see clearly that the condition (5.55) guarantees the solvability of the problem (5.43). It is obvious that the largest possible vortex number n permitted under (5.55) is n = 3. Hence, if we impose 3α < β,

(5.56)

we can always choose ε in (0, so that (5.55) is fulfilled for 1  n  3. In other words, (5.56) ensures the existence of an n-vortex solution for n up to n = 3. By the same method, we may draw the conclusion that the condition 1 3)

α<β

(5.57)

ensures the existence of an n-vortex solution for n up to n = 2. Finally, the existence of 1-vortex solutions follows from the more relaxed condition α < 3β. We now prove that the condition (5.39) is actually also sufficient for the existence of a solution. 1 To proceed, we shall use a homotopy flow to eliminate the term (N − 1)e 2 u0 +f in (5.34). In order to do so, we modify the constraints (5.38) and (5.39) into

1 1 u0 +f +g 1 f −g u0 +f 2 dx t (N − 1)e + e + e (5.58) = η1 , 2 2 Ω

1 dx t (N − 1)e 2 u0 +f + eu0 +f +g = η2 , (5.59) Ω

where 0  t  1, and replace (5.44) and (5.45) with

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    1      a t, f  , g  = − ln t (N − 1) e 2 u0 +f dx + eb(t,f ,g ) eu0 +f +g dx + ln η2 , Ω

(5.60)

Ω

 2     2πn 1      b t, f  , g  = ln  t (N − 1) e 2 u0 +f dx + η2 η3 eu0 +f +g dx ef −g dx β Ω Ω Ω    1 2πn    eu0 +f +g dx − ln(η3 ). (5.61) t (N − 1) e 2 u0 +f dx − ln − β Ω

Ω

| F = 0} and Y = X × X , define a family of maps Tt : Y → Y by For X = {F ∈ setting         f˜ , g˜ = Tt f  , g  , f , g ∈ Y, (5.62) W 1,2 (Ω)

where (f˜ , g˜  ) ∈ Y is the unique solution of the system of the equations   1 1 u0 +a+b+f  +g  1 a−b+f  −g  γ 2πn   u +a+f 0 ˜ + e + e − + , f = α t (N − 1)e 2 2 2 2 |Ω| 2πn β     g˜  = eu0 +a+b+f +g − ea−b+f −g + , 2 |Ω|

(5.63) (5.64)

where a = a(t, f  , g  ) and b = b(t, f  , g  ) are determined through the expressions (5.60) and (5.61), respectively. The existence and uniqueness of a solution of the system of equations (5.63) and (5.64) may easily be seen since the right-hand sides of (5.63) and (5.64) have vanishing integrals over Ω as a consequence of the definitions of a and b given by (5.60) and (5.61). Denote by (ft , gt ) any fixed point of the operator Tt defined through (5.62). We assert that there is a constant M > 0 independent of t such that     f  1,2   (5.65) t W (Ω) + gt W 1,2 (Ω)  M, 0  t  1. In order to establish (5.65), we first show that  1      Wt (x) ≡ 2t (N − 1)e 2 u0 +a+ft + eu0 +a+b+ft +gt + ea−b+ft −gt (x)  γ ,

(5.66)

for any t ∈ [0, 1] and x ∈ Ω. For this purpose, set Ut = u0 + a + b + ft + gt ,

Vt = a − b + ft − gt .

(5.67)

Then Ut and Vt satisfy Ut =

n   α β δps (x), (Wt − γ ) + eUt − eVt + 4π 2 2

(5.68)

s=1

 α β (Wt − γ ) − eUt − eVt . 2 2 Thus, rewrite Wt defined in (5.66) as Vt =

1

Wt = 2t (N − 1)e 2 (Ut +Vt ) + eUt + eVt . We have, in view of (5.68) and (5.69) and in the sense of distributions, the result

(5.69)

(5.70)

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1

Wt = t (N − 1)e 2 (Ut +Vt ) (Ut + Vt ) + eUt Ut + eVt Vt 2  1 1 + t (N − 1)e 2 (Ut +Vt ) ∇(Ut + Vt ) + eUt |∇Ut |2 + eVt |∇Vt |2 2 2 α β  (Wt − γ )Wt + eUt − eVt . 2 2 Then the maximum principle implies the pointwise bound Wt  γ ,

t ∈ [0, 1], x ∈ Ω.

(5.71)

(5.72)

Multiplying (5.63) and (5.64) with f˜ = ft and g˜  = gt by ft and gt , respectively, integrating by parts, and applying (5.72), we arrive at   2     2   ∇f  dx  1 αγ f   dx, ∇g  dx  1 βγ g   dx. (5.73) t t t t 2 2 Ω

Ω

Ω

Ω

Hence, in view of the Schwartz inequality and the Poincaré inequality, we see that there is a constant C0 > 0 depending on α, β, γ , and Ω, such that   2   2  ∇f  + ∇g  dx  C0 . (5.74) t t Ω

Using the Poincaré inequality again, we see that (5.65) follows for a suitable constant M > 0. By virtue of the standard elliptic estimates, we know that Tt : Y → Y is completely continuous. Thus, let B be the ball in Y centered at the origin and of radius M + 1 (say), where M is as defined in (5.65). Then 0 ∈ / (I − Tt )(∂B) and the Leray–Schauder degree theory gives us the homotopy invariance result deg(I − T1 , B, 0) = deg(I − T0 , B, 0) = 1,

(5.75)

where the right-hand side of (5.75) follows from a calculation in [17]. In particular, the existence of a fixed point of T1 , say (f  , g  ), follows. Set f = a(1, f  , g  ) + f  and g = b(1, f  , g  ) + g  . We see that (f, g) is a solution of the system of equations (5.34) and (5.35). The uniqueness of the solution follows from the convexity of the action functional (5.36). This completes the proof of Theorem 5.3. 6. Further extensions and remarks In the recent paper [17], we carried out a development of the existence and uniqueness theory for a system of non-Abelian multiple vortex equations [8] governing coupled SU(N ) and U (1) gauge and Higgs fields characterized by two Higgs boson masses, say me and mg , which may be embedded in a supersymmetric field theory framework. When the underlying domain is doubly periodic, using a fixed-point theorem argument, we proved the existence and uniqueness of an n-vortex solution under a necessary and sufficient condition explicitly relating the domain size to the vortex number n and the Higgs boson masses; while when the underlying domain is the full plane, we adopted an iterative approach to establish the existence and uniqueness of an nvortex solution. Here we remark that our variational methods of the present paper may readily be extended to those cases. For example, for the doubly periodic case, the multiple vortex equations [8,17] over the cell domain Ω governing n vortices are

672

C.-S. Lin, Y. Yang / Nuclear Physics B 846 [PM] (2011) 650–676 v1 (N−1) v1 v2   4πn + m2e eu0 + N + N v2 + [N − 1]e N − N , |Ω| v1 (N−1) v1 v2   4πn v2 = + m2g eu0 + N + N v2 − e N − N , |Ω|

v1 = −Nm2e +

(6.1) (6.2)

which are the Euler–Lagrange equations of the action functional

v1 (N−1) (N − 1) 1 I (v1 , v2 ) = |∇v1 |2 + |∇v2 |2 + N eu0 + N + N v2 2 2 2me 2mg Ω   v1 v2 4πn 4πn(N − 1) − v2 dx. + N (N − 1)e N N − N − 2 v1 + me |Ω| m2g |Ω| Integrating (6.1) and (6.2), we obtain the constraints v1 (N−1) 4πn(N − 1) 4πn ≡ η1 , eu0 + N + N v2 dx = |Ω| − 2 − me N m2g N Ω   v1 v2 4πn 4πn − e N N dx = |Ω| − 2 + 2 ≡ η2 . me N mg N

(6.3)

(6.4)

(6.5)

Ω

 In view of (6.4), (6.5), and the decomposition vi = v i + vi with v i ∈ R and Ω vi dx = 0 (i = 1, 2), we can rewrite (6.3) as

1   2 (N − 1)   2 ∇v1 + ∇v2 I (v1 , v2 ) = dx 2m2e 2m2g Ω    v v  v1 (N−1)  1 2 eu0 + N + N v2 dx + N (N − 1)η2 ln e N − N dx + N η1 ln Ω

Ω

   4πn(N − 1) η1 4πn  + N |Ω| − 2 ln . N − ln η1 η2N−1 + η2 me m2g

(6.6)

Using η1 > 0, we can then proceed as in Section 4 to establish the existence of a solution of (6.1) and (6.2) by a constrained minimization approach. More precisely, we have Theorem 6.1. The system of equations (6.1) and (6.2) has a solution, which is necessarily unique, if and only if   4πn 1 (N − 1) + < |Ω|, (6.7) N m2e m2g and the solution can be constructed by solving the constrained minimization problem 

min I (v1 , v2 )  v1 and v2 satisfy (6.4) and (6.5) ,

(6.8)

where the functional I is as defined in (6.3). On the other hand, however, the more complicated problem over a doubly periodic cell domain considered in Section 5 does not permit, so far, a complete resolution by a similar constrained minimization method, and, in order to show that the necessary condition for the existence of a solution obtained there is also sufficient, we have to rely on a non-constructive degree-theory

C.-S. Lin, Y. Yang / Nuclear Physics B 846 [PM] (2011) 650–676

673

method which allows us to eliminate a difficult nonlinear term by a homotopy flow. As a consequence, using the Leray–Schauder degree result obtained in [17] and the a priori estimates obtained here, we have shown that the necessary condition is indeed sufficient. Therefore, we have seen that the degree calculation in [17] provides some additional information about the problem, not contained in a variational approach, which may be useful in the study of related but more complicated problems. We next note that in the studies [4,18,22] of the non-Abelian multiple vortex equations (2.8)– (2.11) both the complex-valued scalar fields φ and ψ are allowed to independently generate vortices with their respectively prescribed zero sets Zφ = {p1 , . . . , pm },

Zψ = {q1 , . . . , qn }.

(6.9)

In such a context, we can similarly develop an existence and uniqueness theory for the solutions of the equations by the same variational methods. To see this, we observe that, with the prescribed zero sets given in (6.9) for the fields φ and ψ and in terms of the variables u = ln |φ|2 and v = ln |ψ|2 , the governing system of nonlinear elliptic equations (2.27) and (2.28) is modified into m n     (u + v) = α eu + ev − γ + 4π δps (x) + 4π δqs (x), s=1

m n     (u − v) = β eu − ev + 4π δps (x) − 4π δqs (x). s=1

(6.10)

s=1

(6.11)

s=1

For this system of equations, we have Theorem 6.2. For the vortex equations (6.10) and (6.11) defined over R2 or a doubly periodic cell domain Ω, the following conclusions are valid. (i) For any α, β, γ > 0, these equations over R2 have a unique solution satisfying the boundary condition u → ln γ2 and v → ln γ2 as |x| → ∞, which may always be realized exponentially rapidly. (ii) For the equations over a doubly periodic domain Ω, there is a solution if and only if the inequalities     1 1 1 1 4πm + − 4πn − < γ |Ω|, (6.12) α β α β     1 1 1 1 4πm − − 4πn + < γ |Ω|, (6.13) α β α β hold simultaneously. Moreover, if a solution exists, it is unique. Interestingly, the two inequalities (6.12) and (6.13) may be combined into the following single inequality   (m + n) |m − n| 4π + < γ |Ω|. (6.14) α β In the special case when the complex scalar field ψ has no zero, that is, n = 0 in (6.12) and (6.13), or (6.14), we recover Theorem 2.1 or Theorems 3.1 and 4.1.

674

C.-S. Lin, Y. Yang / Nuclear Physics B 846 [PM] (2011) 650–676

We can show that the above results can be established in by the same methods of calculus of variations presented in the previous sections. As an example, we examine the doubly periodic case in some details. To this end, let Ω denote a cell domain and take u0 and w0 over Ω to satisfy u0 = 4π

m  s=1

δps (x) −

4πm , |Ω|

w0 = 4π

n  s=1

δqs (x) −

4πn . |Ω|

(6.15)

Using the new variables w1 and w2 so that u = u0 + w1 and v = w0 + w2 , we can modify (6.10) and (6.11) into   4π (m + n), (w1 + w2 ) = α eu0 +w1 + ew0 +w2 − γ + |Ω|  4π  (w1 − w2 ) = β eu0 +w1 − ew0 +w2 + (m − n). |Ω| As before, we then set f = 12 (w1 +w2 ), g = 12 (w1 −w2 ). Thus the governing system of equations becomes  2π α (m + n), (6.16) f = eu0 +f +g + ew0 +f −g − γ + 2 |Ω|  2π β g = eu0 +f +g − ew0 +f −g + (m − n). (6.17) 2 |Ω| Integrating these two equations and simplifying the results, we arrive at the constraints      1 1 1 1 1 eu0 +f +g dx = (6.18) γ |Ω| − 4πm + − 4πn − ≡ η1 , 2 α β α β Ω      1 1 1 1 1 ew0 +f −g dx = (6.19) γ |Ω| − 4πm − − 4πn + ≡ η2 , 2 α β α β Ω

which appear to be elegantly symmetric with respect to m, n but not quite so with α, β. In order to show that the necessary condition η1 > 0, η2 > 0, which is exactly what stated in (6.12) and (6.13), is also sufficient for the existence of a solution, we recognize that Eqs. (6.16) and (6.17) are the Euler–Lagrange equations of the action functional  

1 4π(m + n) 1 I (f, g) = |∇f |2 + |∇g|2 + eu0 +f +g + ew0 +f −g + − γ f dx α β α|Ω| Ω 4π(m − n) + g dx. (6.20) β|Ω| Ω

Now decompose f, g into f = f + f  , g = g + g  with f , g ∈ R and Thus, applying (6.18) and (6.19), we may rewrite (6.20) in the form

1   2 1   2 I (f, g) − ∇f + ∇g dx α β Ω         eu0 +f +g dx + η2 ln ew0 +f −g dx = η1 ln Ω

Ω

 Ω

f  dx =

 Ω

g  dx = 0.

C.-S. Lin, Y. Yang / Nuclear Physics B 846 [PM] (2011) 650–676

+ η1 (1 − ln η1 ) + η2 (1 − ln η2 ).

675

(6.21)

It is seen immediately that the right-hand side of (6.21) has a uniform lower bound in view of the Jensen inequality again. So the existence of a critical point of (6.20) subject to the constraints (6.18) and (6.19) follows as before. The uniqueness of a critical point of (6.20) results from the convexity of the functional. Note that (6.12) and (6.13) are also equivalent to the single inequality   1 1 8π 4π + (m + n) < γ |Ω| + min{m, n}. (6.22) α β β The same methods may be applied to (5.3)–(5.7) when both φ and ψ are allowed to have independently prescribed sets of zeros as given in (6.9) and the discussion is skipped here. References [1] A.A. Abrikosov, On the magnetic properties of superconductors of the second group, Sov. Phys. JETP 5 (1957) 1174–1182. [2] T. Aubin, Nonlinear Analysis on Manifolds: Monge–Ampére Equations, Springer, Berlin and New York, 1982. [3] R. Auzzi, S. Bolognesi, J. Evslin, K. Konishi, Nonabelian monopoles and the vortices that confine them, Nucl. Phys. B 686 (2004) 119–134. [4] R. Auzzi, S. Bolognesi, J. Evslin, K. Konishi, A. Yung, Nonabelian superconductors: vortices and confinement in N = 2 SQCD, Nucl. Phys. B 673 (2003) 187–216. [5] A. Bezryadina, E. Eugenieva, Z. Chen, Self-trapping and flipping of double-charged vortices in optically induced photonic lattices, Optics Lett. 31 (2006) 2456–2458. [6] E.B. Bogomol’nyi, The stability of classical solutions, Sov. J. Nucl. Phys. 24 (1976) 449–454. [7] M. Eto, T. Fujimori, S.B. Gudnason, K. Konishi, M. Nitta, K. Ohashi, W. Vinci, Constructing non-Abelian vortices with arbitrary gauge groups, Phys. Lett. B 669 (2008) 98–101. [8] M. Eto, T. Fujimori, T. Nagashima, M. Nitta, K. Ohashi, N. Sakai, Multiple layer structure of non-Abelian vortex, Phys. Lett. B 678 (2009) 254–258, arXiv:0903.1518. [9] M. Eto, Y. Isozumi, M. Nitta, K. Ohashi, N. Sakai, Moduli space of non-Abelian vortices, Phys. Rev. Lett. 96 (2006) 161601. [10] L. Fontana, Sharp borderline Sobolev inequalities on compact Riemannian manifolds, Comment. Math. Helv. 68 (1993) 415–454. [11] V.L. Ginzburg, L.D. Landau, On the theory of superconductivity, in: D. Ter Haar (Ed.), Collected Papers of L.D. Landau, Pergamon, New York, 1965, pp. 546–568. [12] S.B. Gudnason, Y. Jiang, K. Konishi, Non-Abelian vortex dynamics: effective world-sheet action, J. High Energy Phys. 1012 (2010) 1008. [13] A. Hanany, D. Tong, Vortex strings and four-dimensional gauge dynamics, J. High Energy Phys. 0404 (2004) 066. [14] K. Huang, R. Tipton, Vortex excitations in the Weinberg–Salam theory, Phys. Rev. D 23 (1981) 3050–3057. [15] A. Jaffe, C.H. Taubes, Vortices and Monopoles, Birkhauser, Boston, 1980. [16] K. Konishi, New results on non-Abelian vortices – further insights into monopole, vortex and confinement, Int. J. Mod. Phys. A 25 (2010) 5025–5039. [17] C.S. Lin, Y. Yang, Non-Abelian multiple vortices in supersymmetric field theory, Commun. Math. Phys., in press. [18] A. Marshakov, A. Yung, Non-Abelian confinement via Abelian flux tubes in softly broken N = 2 SUSY QCD, Nucl. Phys. B 647 (2002) 3–48. [19] M.K. Prasad, C.M. Sommerfield, Exact classical solutions for the ’t Hooft monopole and the Julia–Zee dyon, Phys. Rev. Lett. 35 (1975) 760–762. [20] N. Sakai, D. Tong, Monopoles, vortices, domain walls and D-branes: the rules of interaction, J. High Energy Phys. 0503 (2005) 019. [21] M. Shifman, M. Unsal, Confinement in Yang–Mills: elements of a big picture, Nucl. Phys. Proc. Suppl. 186 (2009) 235–242. [22] M. Shifman, A. Yung, Non-Abelian string junctions as confined monopoles, Phys. Rev. D 70 (2004) 045004. [23] M. Shifman, A. Yung, Supersymmetric Solitons, Cambridge University Press, Cambridge, UK, 2009.

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[24] J. Spruck, Y. Yang, On multivortices in the electroweak theory I: existence of periodic solutions, Commun. Math. Phys. 144 (1992) 1–16. [25] G. ’t Hooft, A property of electric and magnetic flux in nonabelian gauge theories, Nucl. Phys. B 153 (1979) 141– 160. [26] C.H. Taubes, Arbitrary N -vortex solutions to the first order Ginzburg–Landau equations, Commun. Math. Phys. 72 (1980) 277–292. [27] C.H. Taubes, On the equivalence of the first and second order equations for gauge theories, Commun. Math. Phys. 75 (1980) 207–227. [28] D. Tong, Monopoles in the Higgs phase, Phys. Rev. D 69 (2004) 065003. [29] A. Vilenkin, E.P.S. Shellard, Cosmic Strings and Other Topological Defects, Cambridge University Press, 1994. [30] S. Wang, Y. Yang, Abrikosov’s vortices in the critical coupling, SIAM J. Math. Anal. 23 (1992) 1125–1140. [31] Y. Yang, Topological solitons in the Weinberg–Salam theory, Physica D 101 (1997) 55–94. [32] Y. Yang, Solitons in Field Theory and Nonlinear Analysis, Springer, New York, 2001.

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