Vacua, walls and junctions in $G_{N_F,N_C}$

We discuss vacua, walls and three-pronged wall junctions in the Grassmann manifold $G_{N_F,N_C}=\frac{SU(N_F)}{SU(N_C)\times SU(N_F-N_C)\times U(1)}$.


Introduction
The moduli matrix formalism is proposed to construct 1/2 BPS walls in non-Abelian gauge theories [1]. In the infinite coupling limit, the model becomes a massive hyper-Kähler nonlinear sigma model on the cotangent bundle over the Grassmann manifold T * G N F ,N C where G N F ,N C = The 1/4 BPS system reduces to the 1/2 BPS system when the x 2 dependence and the mass M 2 are turned off.
In Abelian gauge theories, the scalar fields [4] are The weight of the vacuum A is defined as The position of the wall which interpolates two vacua is determined by the condition of equal weights of the vacua. The position of the wall which connects A and B is Abelian three-pronged junctions divide sets of three vacua with different labels in one color component whereas non-Abelian three-pronged junctions divide sets of three vacua with different labels in two color components. Abelian junctions exist both in Abelian gauge theories and non-Abelian gauge theories while non-Abelian junctions exist only in non-Abelian gauge theories. In [5], Abelian junctions and non-Abelian junctions of the Grassmann manifold are studied by embedding G N F ,N C into the complex projective space CP N F C N C −1 by the Plücker embedding. Therefore the wall separating · · · A and · · · B is on which is similar to the wall positions (9) in the Abelian gauge theories. As SO(2N )/U (N ) and Sp(N )/U (N ) are realized as quadrics of G 2N,N in the moduli matrix formalism [6,7], it is useful to examine wall junctions of the Grassmann manifold in terms of N C × N F moduli matrices.
The purpose of this paper is to discuss walls and single three-pronged wall junctions in G N F ,N C by using N C × N F moduli matrices and diagrams which are similar to the tetrahedron for CP 3 in [4]. We apply the pictorial representation proposed in [7] to the Grassmann manifolds and present the diagrams of G 4,2 , G 5,2 , G 5,3 , G 6,2 , G 6,3 , and G 6,4 . We discuss single three-pronged wall junctions in G 5,2 by reformulating the diagram for G 5,2 to make polyhedra.

Vacua and elementary walls
Let A denote a vacuum and A ← B denote a wall which connects A and B . The where E i is a simple root generator of SU (N F ). Therefore elementary walls can be identified with simple roots [8]. The simple root generators and the simple roots of SU (N ) are The matrix e i,j is an N × N matrix whose (i, j) component is one. The set of vectors {ê i } is the unit vectorsê i ·ê j = δ ij . We apply the pictorial representation which is used in [7] to the Grassmann manifold. The diagram of the vacua and the elementary walls in G 4,2 is depicted in Figure 1. The diagrams of the vacua and the elementary walls in G 5,2 and G 5,3 are depicted in Figure 2. Two diagrams are related by a π rotation. This reflects the duality between G N F ,N C and G N F ,N F −N C . The configuration in Figure 1 appears in Figure 2 (a).
The diagrams of the vacua and the elementary walls in G 6,2 , G 6,4 and G 6,3 are depicted in Figure 3. G 6,2 and G 6,4 are dual to each other so the diagram in Figure 3 (a) and the diagram in Figure 3 (b) are related by a π rotation. G 6,3 is self-dual so the diagram in Figure 3 (c) is symmetric under a π rotation. The configuration in Figure 2 (a) appears in Figure 3 (a).

Three-pronged wall junctions
We study wall junctions in the moduli space G N F ,N C . The moduli matrices in G N F ,N C can be parameterized by real parameters a ij and b ij as We study three-pronged junctions in the moduli space G 5,2 . The moduli matrices in G 5,2 can be parameterized as with real parameters a ij and b ij . A single three-pronged junction is determined by three vacua which correspond to three vertices of a triangle that the 1/4 wall junction gets mapped onto. We choose two sets of triangles from Figure 2 (a) as shown in Figure 4 as an example. The diagram in Figure 4 (a) is an octahedron which is composed of eight triangles and the diagram in Figure 4 (b) is a pyramid which is composed of four triangles. The vertices, the edges and the triangular faces of the polyhedra correspond to the vacua, the 1/2 BPS walls and the three-pronged junctions in G 5,2 . The moduli matrix for the configuration in Figure 4 (a) is the limit of (13) as a i5 → −∞, (i = 1, 2). There are eight triangles in Figure 4 This is the limit of (13) as a 1i → −∞, (i = 2, · · · 5), and a 2j → −∞, (j = 1, 5). The solutions is with The wall dividing 1A and 1B , (A, B = 2, 3, 4) is on Therefore the junction position is In the same manner, the moduli matrix of the junction which divides { 12 , 23 , 24 } and the moduli matrix of the junction which divides { 13 , 34 , 23 } can be determined. The wall dividing 2A and 2B , (A, B = 1, 3, 4) is on and the wall dividing 3A and 3B , (A, B = 1, 2, 4) is on Three vacua { 12 , 13 , 23 } are divided by a non-Abelian junction. SS † in (3) are diagonal for Abelian junctions so we can calculate junction positions by comparing weights. However, SS † are not diagonal for non-Abelian junctions in general though they can be diagonalized. On the other hand, as three-pronged wall junctions are solitons which divide three vacua, junction positions can be determined by finding sets of three vacua connected by single 1/2 BPS walls, which correspond to the vertices of the triangles that junctions are mapped onto.
In [5], the wall webs in G 4,2 are studied by embedding G 4,2 to CP 5 by the Plücker embedding and the junction positions are obtained from the wall webs. Since the sector described by (14) is in G 4,2 we can compare the junction positions (18) and the junction position (23) is The moduli parameters are related to the moduli parameters of [5] by a 22 − a 23 = a 12 − a 13 , a 23 − a 24 = a 13 − a 14 , a 11 − a 13 = a 12 − a 23 , a 11 − a 12 = a 13 − a 23 .
The junction positions are the same as the junctions positions obtained in [5].
In Figure 4 (b), two triangles are divided by Abelian junctions and the other two triangles are divided by non-Abelian junctions. Parallelogram { 24 , 25 , 35 , 34 } is a two sets of penetrable walls. The same analysis can be done on the pyramid.
We have shown that the full configurations of vacua, 1/2 walls and singe three-pronged junctions in G N F ,N C can be determined by building polyhedra. We can always find the N C × N F moduli matrices which correspond to the vertices, the edges and the faces of the polyhedra.

Summary
We have presented diagrams for the Grassmann manifolds G N F ,N C in the pictorial representation which is proposed in [7]. We have observed that the duality between N C and N F −N C is realized as a π rotational symmetry in the representation. We have reformulated the diagrams to make polyhedra whose vertices, edges and triangular faces correspond to vacua, 1/2 BPS walls and single three-pronged junctions.