Existence of non-Abelian vortices with product gauge groups

In this paper we establish several sharp existence and uniqueness theorems for some non-Abelian vortex models arising in supersymmetric gauge field theories. We prove these results by studying a family of systems of elliptic equations with exponential nonlinear terms in both doubly periodic-domain and planar cases. In the doubly periodic-domain case we obtain some necessary and sufficient conditions, each explicitly expressed in terms of a single inequality interestingly relating the vortex numbers, to coupling parameters and size of the domain, for the existence of solutions to these systems. In the planar case we establish the existence results for any vortex numbers and coupling parameters. Sharp decay estimates for the planar solutions are also obtained. Furthermore, the solutions are unique, which give rise to the quantized integrals in all cases. © 2013 The Authors. Published by Elsevier B.V. All rights reserved.


Introduction
Vortices are important objects in various branches of physics [36] including condensed matter physics [1,28], particle physics [25], string theory and cosmology [23,29,48]. It is well Funded by SCOAP 3 . known there admit the Abrikosov-Nielsen-Olesen vortices [1,37] for the classical Abelian Higgs model, whose static limit is also known as the Ginzburg-Landau model for superconductivity [17]. The first rigorous existence result for vortex configurations were established by Taubes [45,46] for the Ginzburg-Landau model [17]. Since then various analytic methods for studying the existence of vortices and other topological solitons have been developed [28,44,50].
During the last ten years much attention has been concentrated to vortices in non-Abelian gauge field theories since they are related to the fundamental puzzle in theoretical physics, quark confinement or color confinement [18,42]. In fact, in their famous work [38] Seiberg and Witten use non-Abelian color charged monopoles and vortices to interpret quark confinement. Motivated by the importance of non-Abelian vortices in the understanding of monopole and quark confinement, a wide class of non-Abelian gauge theories were developed in [3,21,22,39]. See [4,[7][8][9][10][11][12]15,31,40,43] for more recent progress and [14,30,41,47] for surveys on this topic. In these theories there arise many interesting and challenging systems of elliptic partial differential equations. It is interesting to carry out a rigorous analysis for these partial differential equations from both physical and mathematical points of view. In this respect, we cite the work [6,[32][33][34][35], where a series of existence and uniqueness results were established.
The purpose of this paper is to establish sharp existence theories for the non-Abelian vortex model with product moduli proposed in [7], and for the Yang-Mills-Higgs model with gauge groups U(1) × SO(2M) and U(1) × SU(N ) in [11,12,19]. We recall that, by the approach of moduli matrix [13,26], a series of systems of non-Abelian BPS vortex equations were obtained in [7,11,12,19]. For each of these systems we establish sharp existence, uniqueness, asymptotic behavior and quantized integral results. In particular, for these models over doubly periodic domain we obtain some necessary and sufficient condition, each explicitly expressed in a single inequality interestingly relating the vortex numbers, to coupling parameters and size of the domain, for the existence of solutions. Over the full plane, we obtain existence and uniqueness results for any vortex numbers and coupling parameters. Furthermore, the explicit decay estimates for planar solutions are established. Our approach is based on the direct minimization methods recently developed in [32,33].
The rest of our paper is organized as follows. In Section 2 we review a system of vortex equations from the model proposed in [7] and state a sharp existence result Theorem 2.1 for this problem over a doubly periodic-domain and the full plane. In Sections 3 and 4 we prove Theorem 2.1 for the doubly periodic-domain and planar cases, respectively. Sections 5 and 6 are devoted to establishing existence results for the BPS vortex equations [11,19] arising in the Yang-Mills-Higgs model with gauge groups U(1) × SO(2M) and U(1) × SU(N ) separately.
Under the above ansatz, the matter equations (2.6)-(2.7) are satisfied naturally. Let Ω n = S n S † n , Ω r = S r S † r . (2.14) Then the gauge field equations (2.8)-(2.10) can be expressed as which are called master equations for vortices [13].
To reduce Eqs. (2.15)-(2.17) one takes the ansatz in [19], where , and the vacuum manifold is given by Letting where z is is rewritten as p is .
For the system (2.27)-(2.29), we consider two cases. In the first case we consider the problem over a doubly periodic-domain Ω, governing multiple vortices hosted in Ω such that the field configurations are subject to the 't Hooft boundary condition [24,49,50] under which periodicity is achieved modulo gauge transformations. In the second case we consider the problem over R 2 with the boundary condition  (2.30). Furthermore, the solution satisfies the following exponential decay estimate at infinity where ε ∈ (0, 1) is an arbitrary parameter, σ 0 is a positive constant defined by In both cases, there hold the quantized integrals r|ρ 1 | 2 e 2ru 1 +(n−1)u 2 + [n − 1]e 2ru 1 −u 2 + n|ρ 2 | 2 e 2nu 1 +(r−1)u 3 + [r − 1]e 2nu 1 −u 3 − 2nr(n + r)ξ dx = − 8nr(n + r)π(n 1 + n 2 + n 3 ) where the integrals are taken over either the domain Ω or R 2 .

Existence of doubly periodic solutions
In this section we prove Theorem 2.1 for the doubly periodic case. We will use the direct minimization procedure developed in [32].
In the sequel we show that the condition (2.31) is also sufficient for the existence of solutions to (2.27)- (2.29). In other words, we prove that under the condition (2.31), the functional I admits a unique critical point, which solves (3.1)-(3.3).
There is a decomposition for the space W 1,2 (Ω) By the Trudinger-Moser inequality [2,16] Ω e w dx C exp 1 16π Ω |∇w| 2 dx , ∀w ∈Ẇ 1,2 (Ω), (3.12) we see that the functional I is a C 1 functional and weakly lower semi-continuous. Using the decomposition formula (3.11) for Then it follows from Jensen's inequality that where It is worth noting that the rearrangement of the right-hand sides of (3.14) is crucial for the subsequent treatment of the functional I .
We observe that under the condition (2.31) K i (i = 1, 2, 3) defined by (3.15)-(3.17) are all positive. Then, from (3.14) we obtain Therefore, it follows from (3.18) that the functional I is bounded from below and the minimization problem 3 )} of (3.19). We use the decomposition formula (3.11) for In view of the fact that the function f (t) = δe t − ηt, with δ, η > 0, satisfies the property f (t) → +∞ as t → ±∞, we conclude from (3.14) that {c Noting that the functional I is weakly lower semi-continuous, we conclude that We easily check that the functional I is strictly convex. Then it admits at most one critical point, which implies the uniqueness of the doubly periodic solutions to Eqs.

Existence and asymptotic behavior of planar solution
In this section we prove the existence result for (2.27)-(2.29) over the full plane with the boundary condition (2.30).
As in [28] we take the background functions where λ > 0 is a parameter. Then we see that Then we recast Eqs.
where we used the notation (3.4).
Our function space here is W 1,2 (R 2 ). It is easy to see that Eqs. (4.6)-(4.8) are the Euler-Lagrange equations of the following functional We easily check that the functional I is C 1 and strictly convex over W 1,2 (R 2 ). Then we can solve Eqs. (4.6)-(4.8) by finding the critical points of the functional (4.9). To this end, we use a direct approach developed in [32].
Our first step is to show I is coercive. A direct computation gives where (4.11) (4.12) (4.13) (4.14) Then we need to estimate the right-hand side of (4.12). In order to do this, we introduce the notations With the above notations, from (4.10) we obtain Now we estimate the general term M i (w i ) on the right-hand side of (4.19). Let w + = max{w, 0}, w − = max{−w, 0}. Then, we have the following decomposition In view of the elementary inequality e t − 1 t for t ∈ R and the fact w 0 for some constant C > 0. Noting the definition of u 0 j (j = 1, 2, 3) and taking λ sufficiently large, we see that X i < 1 2 for i = 1, . . . , 4. Then, as λ is suitably large, using the inequality 1 − e −t t 1+t for t 0, we have where we have used the fact e w 0 i − 1, X i ∈ L 2 (R 2 ) (i = 1, . . . , 4). Here and what follows we use C to denote a generic positive constant, which may take different values at different places.
Hence, combining (4.21) and (4.22), we find that Therefore, we conclude from (4.19) and (4.23) that To proceed further, we need the standard Sobolev inequality Using (4.25), we have which implies (4.28) Then, from (4.27) and (4.28), we see that where we used the fact Now we conclude from (4.24) and (4.29) that there exist some positive constant C 0 and C 1 such that (4.30) By the coercive lower bound (4.30), we can show that the functional I admits a critical point in W 1,2 (R 2 ). In view of (4.30), we can take R 0 > 0 sufficiently large such that (say). Since I is weakly lower semi-continuous, the minimization problem Consequently, for t > 0 sufficiently small, with which lead to a contradiction. Hence, we see that we see that the right-hand sides of Eqs. (4.6)-(4.8) belong to L 2 (R 2 ). Then it follows from the elliptic L 2 -estimate that v i ∈ W 2,2 (R 2 ), i = 1, 2, 3, which implies the desired boundary condition v i → 0, as |x| → ∞, i = 1, 2, 3. From the fact v i ∈ W 2,2 (R 2 ) we see that the right-hand sides of Eqs. (4.6)-(4.8) also belong to L p (R 2 ) for any p 2. Therefore, by the elliptic L p estimate, we In what follows we establish the exponential decay rate for this solution. Let (u 1 , u 2 , u 3 ) be the solution of (4.1)-(4.3) obtained above. We have shown that u i → 0 as |x| → ∞. Let R > max |p is |, s = 1, . . . , n i , i = 1, 2, 3 .

Yang-Mills-Higgs model with gauge group U(1) × SO(2M)
In this and the following sections we study the Yang-Mills-Higgs model with gauge group U(1) × G introduced in [11,12,19]. The concrete case with G = SO(2M) and G = SU(N ) will be studied this section and next section, respectively. The Lagrangian density takes the form where field strength, gauge fields and covariant derivative are defined as A 0 μ is the gauge field of U(1), A a μ are the gauge fields of G , t 0 and t a are the standard generators of U(1) and G . The matter scalar fields are written as a color-flavor mixed matrix H . Here e and g are the U(1) and G coupling constants, respectively.

Doubly periodic case
In this subsection we will prove Theorem 5.1 for the doubly periodic case. We use the argument of Section 3. Let u 0 be the solution of the problem and u 0 i be the solution of the problem (5.24) which are the Euler-Lagrange equations of the following functional We first prove the necessity of the condition (5.18) for the existence of solutions to (5.15)-(5.16) . Let (v, v 1 , . . . , v M ) be a solution of (5.23)-(5.24). Then, integrating Eqs. (5.23)-(5.24) over the domain Ω, we obtain Then both the right-hand sides of (5.26) and (5.27) should be positive, which concludes the necessity of the condition (5.18).
In what follows we show that the condition (5.18) is also sufficient for the existence of solutions to (5.15)-(5.16) . For (v, v 1 , . . . , v M ) ∈ W 1,2 (Ω), using the decomposition formula (3.11) we have v = c + w, v i = c i + w i , i = 1, . . . , M. Then using Jensen's inequality we obtain where Noting the condition (5.18), we see that K 1 > 0, K 2 > 0. Then, from (5.28) we obtain Hence, from (5.29), we see that the functional I is bounded from below and the minimization problem is well-defined.
Using (5.28) and a similar argument in Section 3, we can get a critical point of the problem (5.30), which is a weak solution to (5.23)-(5.24). The uniqueness of the solution follows from the strict convexity of the functional I .
The quantized integrals follow from a direct integration. Then the proof of Theorem 5.1 for the doubly periodic case is complete.

Planar case
In this subsection we consider (5.15)-(5.16) over the full plane with the boundary condition (5.17). Let where λ > 0 is a parameter. Then we see that where which are the Euler-Lagrange equations of the functional Since the functional is differentiable and strictly convex, as in Section 4, to get the solution of (5.34)-(5.35), we need to show that the functional is coercive over W 1,2 (R 2 ). Although we can follow a similar argument as in Section 4 to prove the coerciveness of the functional, here we use a new direct approach recently developed in [33].
Taking α > 0 such that α 2 > max{e 2 , g 2 }, we rewrite the functional I as where To show the coerciveness of I , in view of (5.37), we consider a generic functional of the following form where w 0 (taking the place of u 0 ± u 0 i ) and H (taking the place of H 1 i and H 2 i ) are defined as below where m (taking the place of n 0 + Mn or n) is a positive integer.
Let w + = {w, 0}, w − = max{−w, 0}. We decompose J as (5.43) which will be estimated separately in the sequel. Denote by B R a disc centered at the origin with radius R. From the definition of w 0 , we may choose R 0 > 0 with R 0 > 2 max |p s |, s = 1, . . . , m , (5.44) such that, for any λ > 1, there exists a positive constant a 0 > 1 such that It is easy to see that When |x| > R 0 , noting that 1 − e w 0 ∈ L 2 (R 2 ), we have where C λ is a generic positive constant depending only on λ.
When |x| < R 0 , we decompose w + as Then from (5.49) and Young's inequality, we have Using the Poincaré inequality By the Hölder and Young inequalities, we have In what follows we estimate J (−v − ). In view of the elementary inequality 1 − e −s s 1+s for s 0, we get Then from the inequality (5.56) and an inequality similar to (4.26), we may obtain .
(5.57) Using Hölder's inequality again, one has Hence we infer from (5.57) and (5.58) that where we have used the inequality At this point, by taking λ suitably large we infer from (5.55) and (5.59) that there exist positive constants C 1 , C 2 such that (5.60) Therefore, using the estimate (5.60) on the right-hand side of (5.37), we conclude that where C 1 , C 2 are two positive constants. Now using the coercive lower bound (5.61), we can obtain a critical point for the functional I by a routing argument. The critical point is also unique since the functional I is strictly convex.
To establish the behavior at infinity, decay estimate of the solutions and the quantized integrals, we can use a similar argument as in Section 4. Then the proof of Theorem 5.1 for the planar case is complete.
Theorem 6.1. Consider the problem (6.9)-(6.10) with arbitrary distribution of points p i1 , . . . , p in i , and n i 0 are integers, i = 1, 2. For any ρ ∈ C and ξ > 0 satisfying (6.11), and any coupling parameters e, g > 0, we have the following conclusion: Over a doubly periodic-domain Ω, there exists a solution for (6.9)-(6.10) if and only if Moreover, if a solution exists, it must be unique. Over R 2 , there exits a unique solution for (6.9)-(6.10) satisfying the boundary condition (6.12). Furthermore, the solution satisfies the following exponential decay estimate at infinity where ε ∈ (0, 1) is an arbitrary parameter, σ 0 is a positive constant defined by In both cases, there hold the quantized integrals where the integrals are taken over either the domain Ω or R 2 .

Doubly periodic solution
In this subsection we prove Theorem 6.1 for the doubly periodic-domain case. We argue as in Section 3.
Let u 0 be the solution of the problem and u 0 2 be the solution of the problem As previous section, setting u i = u 0 i + v i , i = 1, 2, we may rewrite (6.9)-(6.10) as v 1 = e 2 N |ρ| 2 e u 0 1 +v 1 +(N−1)(u 0 We observe that Eqs. (6.18)-(6.19) are the Euler-Lagrange equations of the following functional To show the necessity of condition (6.13) for existence of solutions to (6.9)-(6.10), we integrate Eqs. (6.18)-(6.19) over the domain Ω to find which conclude Hence, if there exits a solution for (6.9)-(6.10), the right-hand sides of (6.21)-(6.22) should be positive, which implies the necessity of the condition (6.13).
In what follows we prove that the condition (6.13) is also sufficient for the existence of solutions to (6.9)-(6.10).
We see that the functional I is a C 1 functional and weakly lower semi-continuous. As previous section, to find the critical point of I , we need to show the coerciveness of I .
Using Jensen's inequality we find where K 1 and K 2 are defined by (6.21) and (6.22), respectively. We see that both K 1 and K 2 are positive under the condition (6.13). Then, from (6.23) we obtain 2N e 2 ∇w 1 2 2 + 2N(N − 1) g 2 ∇w 2 2 2 + K 1 ln which implies the functional I is bounded from below and the minimization problem is well-defined. Now we may use a similar argument as Section 3 to get the existence of a critical point of I , which is also unique since I is strictly convex.

Planar solution
In this subsection we prove the existence result for (6.9)-(6.10) over the full plane with the boundary condition (6.12). We use a similar argument as in Section 4.
We introduce the background functions where λ > 0 is a parameter. Then we see that We observe that the functional I is C 1 and strictly convex over W 1,2 (R 2 ). To solve (6.26)-(6.27), as in Section 4, we just need to find the critical points of the functional (6.28).
Then we need to show the coerciveness of I . A simple computation leads to where Now estimating the right-hand side of (6.29) as that in Section 4, we obtain that there exist some positive constant C 0 and C 1 such that Then the existence of critical point follows a standard argument. Hence we see that the system (6.18)-(6.19) admits a solution, which is also unique since I is strictly convex.
The behavior at infinity, the decay estimates of the solution and the quantized integrals can be established as in Section 4. Then the proof of Theorem 6.1 for the planar case is complete.