Junctions of mass-deformed nonlinear sigma models on the Grassmann manifold

We study vacua and walls of the mass-deformed nonlinear sigma model on the Grassmann manifold $G_{N_F,N_C}=\frac{SU(N_F)}{SU(N_C)\times SU(N_F-N_C)\times U(1)}$ and discuss three-pronged junctions for $N_C=1,2,3$ in four dimensions.


Introduction
Topological defects play important roles in model building. Our world can be realized on extended topological defects such as walls or junctions in brane world scenarios [1,2]. Wall networks are applied to study dark matter and dark energy [3].
Walls in Abelian gauge theories are studied in [4,5]. The moduli matrix formalism is proposed to analyse walls of N = 2 non-Abelian gauge theories [6]. Walls interpolating isolated supersymmetric vacua preserve 1/2 supersymmetry. In the strong coupling limit, the model with N F > N C becomes the mass-deformed hyper-Kähler nonlinear sigma model on T * G N F ,N C .
In the moduli matrix formalism, walls are algebraically constructed from elementary walls. The elementary walls can be identified with simple roots of the global symmetry [7]. In [8], a pictorial representation is proposed to study vacua and walls of mass-deformed Kähler nonlinear sigma models on SO(2N )/U (N ) and Sp(N )/U (N ), which are quadrics of the Grassmann manifold G 2N,N .
Intersecting walls form junctions, which preserve 1/4 supersymmetry [9,10,11,12,13]. The energy density of a wall junction is bounded from below by central charge densities Z α (α = 1, 2) and Y. Z α are components of the tension vector of the wall, which pulls the junction along the wall direction outwards and Y is the charge density for the junction. An Abelian junction has a negative Y -charge while a non-Abelian junction has a positive Y -charge. In the strong coupling limit, the charge density Y vanishes.
A three-pronged junction is formed by a set of three vacua that are interpolated by non-penetrable walls. Three-pronged junction solutions are discussed in the moduli matrix formalism [14,15,16]. The moduli matrix formalism can be easily applied to Abelian junctions, but it becomes complicated with non-Abelian junctions. In [15], three-pronged junctions of mass-deformed nonlinear sigma models on the Grassmann manifold G N F ,N C are analyzed by Plücker embedding. The models are embedded into the complex projective space CP N F C N C −1 , which produce Abelian gauge theories.
The pictorial representation, which is introduced by [8], is applied to vacua and walls of mass-deformed nonlinear sigma models on the Grassmann manifold in [17]. The vacua and the walls of the mass-deformed nonlinear sigma models on G N F ,N C with (N F , N C ) = (4, 2),(5, 2),(5, 3),(6, 2),(6, 3), (6,4) are explicitly shown in the representation. In the representation, the duality of the Grassmann manifold N C ↔ N F −N C is realized as a π-rotation of the diagram. The diagram of G N F +1,N C repeats the diagram of G N F ,N C . It is also shown that we can compute junction positions without the Plüker embedding, since a set of nonpenetrable walls can be read from diagrams in the pictorial representation.
In this paper, we discuss vacua, walls and three-pronged junctions of mass-deformed nonlinear sigma models on the Grassmann manifold with flavour number N F and colour number N C =1,2,3. In Section 2, we review the model [18,19] and the moduli matrix formalism [6]. In Section 3, we review the work [17] and discuss vacua and walls of the mass-deformed nonlinear sigma models on G N F ,N C with N C = 1, 2, 3. In Section 4, we apply the method, which is introduced in [17], to three-pronged junctions of the mass-deformed nonlinear sigma model on G N F ,3 . In Section 5, we summarize our results.

model
We study junctions of the N = 2 mass-deformed nonlinear sigma model on the Grassmann (1) in four dimensions. The Lagrangian of the mass-deformed nonlinear sigma models [18,19] is where c is the electric Fayet-Iliopoulos (FI) parameter and b, b * are the magnetic FI parameters. The chiral fields Φ i a (x, θ,θ), Ψ a i (x, θ,θ), Λ b a and the vector field V b a (x, θ,θ) are matrix valued and defined as follows: a (y) = −S b a (y) + θη b a (y) + θθK b a (y), (a = 1, · · · , N C ; i = 1, · · · , N F ; µ = 0, · · · , 3). (2.2) We diagonalize Λ for later use. The SU (N ) Cartan generators H = (H 1 , · · · , H N ) [20] are defined by H n = e n,n − 1 N I N ×N , (n = 1, · · · , N ), (2.3) where e p,q is an N × N matrix of which the (p, q) component is one. The complex mass matrix M can be formulated as a linear combination of (2.3) with complex parameters. Then the mass matrix M is a traceless diagonal matrix.
The Lagrangian (2.1) with the component fields (2.2) can be computed. The equations of the auxiliary fields are solved by Then the bosonic part of the Lagrangian is The covariant derivatives are defined by There are two cases for the constraint (2.7), b = 0 and b = 0, which are related by the SU (2) R symmetry. Therefore we can consider b = 0 case without loss of generality. The field φ parameterizes the base space of the Grassmann manifold whereas the field ϕ parameterizes the cotangent space. ϕ does not contribute to the vacuum configuration and the Bogomol'nyi-Prasad-Sommerfield (BPS) solutions of the mass-deformed nonlinear sigma models on the Grassmann manifold [6]. Then the relevant bosonic part of the Lagrangian (2.1) is The Lagrangian has a constraint By substituting M and S in the Lagrangian (2.8) with real valued matrices as we get the vacuum condition The mass matrices and the real scalar fields can be parameterized as Then the vacuum solutions are labelled by where i, j, k = 1, · · · , N F . Therefore there are N F C N C vacuum solutions as it is observed in [19]. The solutions should be constrained by (2.9). We are interested in static configurations, which are independent of the x 3 -coordinate. So we fix ∂ 0 = ∂ 3 = 0. We also assume that there is the Poincaré invariance on the worldvolume so we fix A 0 = A 3 = 0. Then the energy density is where the tension density is We use the index α = 1, 2 for codimensions and adjoint scalars as it is done in [14]. The energy density (2.14) and the tension density (2.15) are constrained by (2.9). The (anti-)BPS equation is We choose the upper sign for the BPS equation and the lower sign for the anti-BPS equation.
The BPS solution [6,14,15] is with a relation The constraint (2.9) becomes The BPS solution (2.17), Σ α and A α in (2.18) are invariant under the following transformation: This equivalent class of (S, H 0 ) is named as worldvolume symmetry in the moduli matrix formalism [6]. The moduli space, which is parameterized by H 0 , is the Grassmann manifold. 1/1 BPS supersymmetric vacua and 1/2 BPS walls can be labelled by a model with real masses and real fields [6]. The 1/4 BPS junctions are labelled by a model with complex masses and complex fields [14,15].

vacua and walls
We study vacua and walls of the mass-deformed nonlinear sigma models on the Grassmann manifold by using the moduli matrix formalism [6] and the pictorial representation [8]. We can simplify the Lagrangian (2.8) by introducing real masses and real scalar fields as follows: where M and Σ are real valued [14]. The vacuum condition is The matrices can be parameterized as We can set m 1 > m 2 > · · · > m N without loss of generality since we are interested in generic mass parameters. Then the vacuum solutions are labelled by where i, j, k = 1, · · · , N F . The vacuum solutions are the same as the ones labelled by (2.13).
We study walls. We can assume that fields are static and all the fields depend on x 1 ≡ x coordinate. We can also assume that there is the Poincaré invariance on the worldvolume of walls so we can set A 0 = A 2 = A 3 = 0. We can learn from (2.17) that the BPS solution [6] is e M x i j .
where E i is an elementary wall operator and r is a complex parameter with −∞ < Re(r)<+∞. The elementary wall operator E i is a simple root generator of SU (N F ), which satisfies where c is the electric FI parameter, which appears in (2.9), m = (m 1 , · · · , m N F ), and T A←B is the tension of the elementary wall. Elementary walls can be identified with the simple roots of SU (N F ) [7]. The simple root generators and the simple roots of SU (N ) [20] are The set of vectors {ê i } is the orthogonal unit vectorsê i ·ê j = δ ij . Elementary walls can be compressed to single walls. A compressed wall of level n which connects A and A is ]](r) , (i m = 1, · · · , N ; m = 1, · · · , n + 1). (3.9) A multiwall is constructed by multiplying a single wall operator to anther wall moduli matrix. A multiwall interpolating A , A ,· · · , and B is where parameters r i , (i = 1, 2, · · · ) are complex parameters with −∞<Re(r i )<+∞. Penetrable walls pass through each other since the wall operators commute: Let vector g A←A denote the wall that interpolates vacuum A and vacuum A . Then the elementary wall (3.6) with the relation (3.7) can be identified with The tension of the wall can be read from (3.7) as The compressed wall in (3.9) is identified with (3.14) The root vectors of the two penetrable walls of (3.11) are orthogonal The pictorial representation of [8] is applied to vacua and walls of mass-deformed nonlinear sigma models on the Grassmann manifold in [17]. In the pictorial representation, vacua and elementary walls are described by vertices and segments. It is observed that the duality N C ↔ N F − N C corresponds to a π-rotation in the pictorial representation and the diagram of G N F +1,N C repeats the diagram of G N F ,N C . The diagrams of the vacua and the elementary walls of the mass-deformed nonlinear sigma models on G N F ,N C , with (N F , N C )=(4, 2),(5, 2),(5, 3),(6, 2),(6, 3), (6,4) are shown in [17]. The diagrams of the vacua and the elementary walls of the mass-deformed nonlinear sigma models on G 7,N C with N C =1,2,3 are presented in Figure 1.
The mass-deformed nonlinear sigma model on G N F ,N C has N F C N C vacua. All the walls of the model are constructed from the elementary walls which can be described by the simple roots of SU (N F ), α i , (i = 1, · · · , N F − 1) of (3.8). As we can see in [17] and Figure  1, the diagrams for G N F ,N C repeats the diagrams for G n,N C , (N C < n ≤ N F − 1). It shows that the vacua and the elementary walls of the mass-deformed nonlinear sigma models on G N F +1,N C can be derived from the configuration of the vacua and the elementary walls of G N F ,N C , by adding N F C N C −1 number of vacua and α N F elementary walls. We can focus on the case where N C ≤ N F 2 since the case with N C > N F 2 can be determined by the duality condition G N F ,N C G N F ,N F −N C , which are related by a π-rotation in the pictorial representation.
We present diagrams for N C =1,2,3 in Figure 2. The structure of vacua and elementary walls of the mass-deformed nonlinear sigma models on G N F ,1 is constructed by adding one vacuum N F and one elementary wall α N F −1 to the structure of G N F −1,1 . The diagram of the additional structure is depicted in Figure 2(a). The structure of vacua and elementary walls of the mass-deformed nonlinear sigma models on G N F ,2 is constructed by adding the diagram in Figure 2(b), which consists of (N F − 1) vacua, (N F − 2) number of α N F −1 and one for each α i , i = 1, · · · , N F − 2, to the diagram for G N F −1,2 . In the similar manner, the diagram of vacua and elementary walls of mass-deformed nonlinear sigma models on G N F ,3 is obtained by adding the diagram in Figure 2(c). The diagram adds (N F − 3) number of each α i (i = 1, · · · , N F − 2) and N F −1 C N C −1 number of α N F −1 . The configuration of vacua and walls of the mass-deformed nonlinear sigma models on G N F ,N C can be systematically derived from the diagrams of the pictorial representation.

three-pronged junctions
A three-pronged junction is formed by three vacua interpolated by three non-penetrable walls. A wall that interpolates vacuum A and vacuum B has a tension T AB pulling the junction along the wall direction outwards [14] T which is computed by the line integral of the tension density defined by (2.15).
We study three-pronged junctions of the mass-deformed nonlinear sigma models on the Grassmann manifold G N F ,3 . We reformulate the diagram in Figure 2(c) to produce a pyramid of which the vertices, the edges, the triangular faces and the parallelogram shaped base correspond to vacua, walls, three-pronged junctions and a pair of penetrable walls [17]. The pyramid is depicted in Figure 3. Let us use semicolons in vacuum labels i; j; · · · ; k to separate the flavour numbers and substitute coordinates x and y for x 1 and x 2 .
There are two types of junctions. Abelian junctions divide a set of three vacua with labels that differ by one components: · · · , A , · · · , B , · · · , C . Non-Abelian junctions divide a set of three vacua with labels that differ by two components: · · · , AB , · · · , BC , · · · , AC . Abelian junctions exist in Abelian gauge theories and non-Abelian gauge theories whereas non-Abelian junctions exist in non-Abelian gauge theories.
There are two Abelian junctions and two non-Abelian junctions in Figure 3: •abelian junctions •non-abelian junctions We study Abelian junction Figure 3: The vertices, the edges and the triangular faces correspond to vacua, walls and three-pronged junctions. The parallelogram shaped base correspond to a pair of penetrable walls.

The moduli matrix for the junction is
The limit of (4.4) as (4.5) The BPS solution (2.17) with (4.4) and (4.5) is The position of the junction is the solution of the equations: which produce The position of the junction is (4.10) The matrix SS † defined by (2.19) is not diagonal for non-Abelian three-pronged junctions as the vacua differ by two label components. A detour is needed to apply the moduli matrix formalism to junctions. In [15], the Grassmann manifold is embedded into the projective space by the Plücker embedding. In [17], the moduli matrix formalism is applied to each wall that form a three-pronged junction as the set of walls can be identified from the diagrams of the pictorial representation.
We study non-Abelian three-pronged junction [17]. The moduli matrix of the wall, which interpolates The wall solution is (4.12) The wall is located in the region where Re(f 2 N F −2 ) = Re(f 2 N F −1 ): The moduli matrix of the wall that interpolates N F −4;

summary
We have discussed vacua and BPS objects of the N = 2 mass-deformed nonlinear sigma model on the Grassmann manifold G N F ,N C with N C = 1, 2, 3 as an extension of [17]. We have applied the moduli matrix formalism [6] and the pictorial representation [8,17] to vacua, walls and three-pronged junctions. Since we can analyse three-pronged junctions with N C × N F moduli matrices in the pictorial representation, we can apply the moduli matrix formalism to three-pronged junctions of the nonlinear sigma models on SO(2N )/U (N ) and Sp(N )/U (N ), which are quadrics of the Grassmann manifold. We hope to report on the results elsewhere.