Topologically Stratified Energy Minimizers in a Product Abelian Field Theory

We study a recently developed product Abelian gauge field theory by Tong and Wong hosting magnetic impurities. We first obtain a necessary and sufficient condition for the existence of a unique solution realizing such impurities in the form of multiple vortices. We next reformulate the theory into an extended model that allows the coexistence of vortices and anti-vortices. The two Abelian gauge fields in the model induce two species of magnetic vortex-lines resulting from $N_s$ vortices and $P_s$ anti-vortices ($s=1,2$) realized as the zeros and poles of two complex-valued Higgs fields, respectively. An existence theorem is established for the governing equations over a compact Riemann surface $S$ which states that a solution with prescribed $N_1, N_2$ vortices and $P_1,P_2$ anti-vortices of two designated species exists if and only if the inequalities \[ \left|N_1+N_2-(P_1+P_2)\right|<\frac{|S|}{\pi},\quad \left|N_1+2N_2-(P_1+2P_2)\right|<\frac{|S|}{\pi}, \] hold simultaneously, which give bounds for the `differences' of the vortex and anti-vortex numbers in terms of the total surface area of $S$. The minimum energy of these solutions is shown to assume the explicit value \[ E= 4\pi (N_1+N_2+P_1+P_2), \] given in terms of several topological invariants, measuring the total tension of the vortex-lines.


Introduction
show that such energy arises topologically and is proportional to the sum of vortex and anti-vortex numbers of two species. In Section 6, we make some concluding remarks.

(2.4)
It is interesting to observe that (2.2) is recovered from (2.4) when taking the limit |q| → 1, |p| → 1 in the denominators 1 + |q| 2 and 1 + |p| 2 of (2.4). The Euler-Lagrange equations of the energy density are found to be

6)
D * Dp (1 + |p| 2 ) 2 = 1 (1 + |p| 2 ) 3 (Dp ∧ * Dp) +2 * 1 − |q| 2 1 + |q| 2 + |p| 2 − 1 1 + |p| 2 |p| 2 − 1 (1 + |p| 2 ) 2 p, (2.7) which appear rather complicated and intractable. In order to obtain interesting solutions of these equations, we follow [29,36] to pursue a BPS reduction. Introduce the current densities (2.9) Then we have Note also that there holds the identity So, with |Dq| 2 = * (Dq ∧ * Dq), etc, we arrive at the decomposition (2.13) The quantities 1 2π (F −F ) and 1 2π F are the first Chern numbers induced from the connectionŝ A −Ã andÃ over L → S and 1 4π K(q) and 1 4π K(p) the Thom classes over L * → S, respectively [25]. Thus, the quantity τ = 2F + 2K(q) + 2K(p) (2.14) is a topological density which leads to the topological energy lower bound measuring the tension [7][8][9][10][11] of the vortex-lines, so that the lower bound is saturated when the quartet (q, p,Â,Ã) satisfies the equations 19) may be regarded as a reduction of the system of equations (2.5)-(2.8). These reduced firstorder equations are often referred to as the BPS equations after Bogomol'nyi [5] and Prasad-Sommerfield [21] who pioneered the idea of such reduction for the classical Yang-Mills-Higgs equations. When the upper sign is taken, the system is said to be self-dual; the lower, anti-self-dual. It may also be checked that the self-dual and anti-self-dual cases are related to each other through the transformationÂ (2.20) Hence, in the sequel, we will only consider the self-dual situation. From (2.16) and (2.17), we know [15,36,37] that the zeros and poles of the sections q, p are isolated and possess integer multiplicities. For simplicity, we may denote the sets of zeros and poles of q, p by respectively, so that the associated multiplicities of the zeros and poles are naturally counted by their repeated appearances in the above collections of points.
If we interpret * F as a magnetic or vorticity field, (2.18) indicates that it attains its maximum * F = 2 at the zeros and minimum * F = −2 at the poles of q. Thus, the zeros and poles of q may be viewed as centers of vortices and anti-vortices. In other words, we may identify the zeros and poles of q as the locations of vortices and anti-vortices generated from the connection 1-formÂ. Similarly, the zeros and poles of p may be interpreted as vortices and anti-vortices generated from the connection 1-formÂ +Ã. Therefore, in what follows, the zeros and poles of q, p are interchangeably and generically referred to as the vortices and anti-vortices of a solution configuration (Â,Ã, q, p).
Here is our main existence theorem.
regarding the total numbers of zeros and poles are fulfilled simultaneously. Moreover, such a solution carries a minimum energy of the form which is seen to be stratified topologically by the Chern and Thom classes of the line bundle L and its dual respectively. In particular, in terms of energy, zeros (vortices) and poles (anti-vortices) of q, p contribute equally.
It is interesting to note that the inequalities (2.23) and (2.24) imply that the differences of vortices and anti-vortices must stay within suitable ranges to ensure the existence of a solution: However, it may be checked that the conditions (2.26) and (2.27) do not lead to (2.23) and (2.24). The latter may be called the difference of total numbers of vortices and anti-vortices and the difference of 'weighted total numbers' of vortices and anti-vortices.

Governing elliptic equations and basic properties
To proceed, we set , which leads us via [15,36,37] to the following equivalent governing elliptic equations where ∆ is the Laplace-Beltrami operator on (S, g) defined by and δ z denotes the Dirac measure concentrated at the point z ∈ S with respect to the Riemannian metric g over S.
In what follows, we use dΩ g to denote the canonical surface element and |S| the associated total area of the Riemann surface (S, g).
Regarding the equivalently reduced equations ( For convenience, we first need to take care of the Dirac distributions by subtracting suitable background functions. To do so, we let u 1 0 , u 2 0 , v 1 0 , v 2 0 be the normalized solutions of the equations that determine the source functions arising from the sets Z(q), P(q), Z(p), P(p), respectively. For instance, u 1 0 is the unique solution [4] to where and in what follows we use the notation We first show that the condition consisting of (2.23) and (2.24) is necessary for the existence of solutions for (3.8)-(3.9). In fact, integrating (3.8)-(3.9), we find where a, b are constants defined by On the other hand, noting

Proof of existence via a fixed-point argument
In this section, we prove that the condition comprised of (2.23) and (2.24) is also sufficient for the existence of a solution of the coupled equations (3.2) and (3.3). We will extend a fixed-point theorem argument used in [38] when treating a single equation.
To save notation, in the following we also use W 1,2 (S),Ẇ 1,2 (S) and L p (S) to denote the spaces of vector-valued functions.
We begin with the following lemma.
Hence by the compact embedding theorem we see that (4.9) holds. Denote Therefore we have whereÛ ′ k lies between U ′ k and U ′ 0 ,V ′ k between V ′ k and V ′ 0 ,ĉ 1 between c 1 (U ′ k ) and c 1 (U ′ 0 ), andĉ 2 between c 2 (V ′ k ) and c 2 (V ′ 0 ). Multiplying both sides of (4.21) and (4.22) byŨ ′ k −Ũ ′ 0 andṼ ′ k −Ṽ ′ 0 , respectively, and integrating by parts, we obtain where the property (3.11) is used. Combining (4.23) with (4.24), and using the Poincaré inequality, we arrive at for some C > 0. Then, from (4.9), Lemma 4.2, and (4.25), we see that which, with (4.9), yields Then the proof of Lemma 4.3 is complete. Before applying the Leray-Schauder fixed-point theory, we need to estimate the solution of the fixed-point equation,

Explicit calculation of minimum energy
In this section we establish the minimum energy formula (2.25) and show how it is stratified topologically.
By the equations (2.16)-(2.19), the fact * 1 = dΩ g , and (3.5)-(3.6), we see that are valid, which give us To calculate the lower bound of the energy, we need to compute the fluxes contributed by the current densities K(q) and K(p).
Take a coordinate chart {U j } of S. Assume z ′′ 1,j ∈ U j , j = 1, . . . , P 1 . In local coordinates, we have D i q = ∂ i q − i(Â i −Ã i )q, i = 1, 2 and the density K(q) in U j can be written as Besides, in K(q) = dJ(q), we have Then it follows from the Stokes formula that where B(z, r) denotes a disc centered at z with radius r > 0 and all the line integrals are taken counterclockwise. Note that near z ′′ 1,j ∈ P(q), the section q has the representation where h j is a non-vanishing function defined near z ′′ 1,j . From the equation (2.16) we see that which, with u = ln |q| 2 , implies Noting (5.7), near z ′′ 1,j ∈ P(q), we see that where w j is a smooth function. Thus we obtain lim r→0 ∂B(z ′′ 1,j ,r) J(q) = 4π, (5.13) which, with (5.6), gives S K(q) = 4πP 1 . (5.14) Following a similar procedure, we have S K(p) = 4πP 2 . (5.15) As described in [25], the normalized integrals 1 4π K(q) and 1 4π K(p), counting the numbers P 1 , P 2 of anti-vortices of the two species, are the Thom classes of the dual bundle L * → S, of two respective classification (Chern) classes, 1 2π (F −F ) and 1 2π F .

Conclusions and remarks
In this work we have extended the formalism of Tong and Wong [29] of a product Abelian Higgs theory describing a coupled vortex system with magnetic impurities to accommodate coexisting vortices and anti-vortices of two species realized as topological solitons governed by a BPS system of equations. In additional to the usual first Chern classes suited over a complex Hermitian line bundle, the presence of anti-vortices switches on the Thom classes over the dual bundle, as in [25]. When the underlying Riemann surface S where vortices and anti-vortices reside is compact, we have established a theorem which spells out a necessary and sufficient condition, consisting of two inequalities, (2.23) and (2.24), for prescribed N 1 , N 2 vortices and P 1 , P 2 anti-vortices, of two respective species, to exist. This necessary and sufficient condition contains a few special situations worthy of mentioning.
(i) When N 2 = P 2 = 0 (only vortices and anti-vortices of the first species are present), the condition becomes (6.1) (ii) When N 1 = P 1 = 0 (only vortices and anti-vortices of the second species are present), the condition reads (6.2) (iii) When N 1 = N 2 = N and P 1 = P 2 = P (there are equal numbers of vortices and anti-vortices, respectively, of two species), the condition is In all these situations, the numbers of vortices and anti-vortices may be arbitrarily large, provided that the differences of these numbers are kept in suitable ranges as given.
Although the vortices and anti-vortices of the two species do not appear in the model in a symmetric manner as seen in the field-theoretical Lagrangian density and the governing equations, they make equal contributions to the total topologically stratified minimum energy as stated in (2.25) of an elegant form.
Let M(N 1 , P 1 , N 2 , P 2 ) denote the moduli space of solutions of the BPS equations (2.16)-(2.19) with N 1 + N 2 and P 1 + P 2 prescribed vortices and anti-vortices, of two respective species. Since these solutions depend on at least 2(N 1 + N 2 + P 1 + P 2 ) continuous parameters which specify the locations of zeros and poles of the two sections q, p, respectively, we obtain the following upper bound for the dimensionality of M(N 1 , P 1 , N 2 , P 2 ): dim(M(N 1 , P 1 , N 2 , P 2 )) ≥ 2(N 1 + N 2 + P 1 + P 2 ). (6.4) Since we have not established the uniqueness of a solution with N 1 + N 2 and P 1 + P 2 prescribed vortices and anti-vortices of the two species yet, we do not know whether the inequality (6.4) is actually an equality. In this regard, it will be interesting to carry out an investigation along the (well-known classical) index theory work of Atiyah, Hitchin, and Singer [2,3]