Vacua and walls of mass-deformed K\"ahler nonlinear sigma models on $Sp(N)/U(N)$

We study vacua and walls of mass-deformed K\"ahler nonlinear sigma models on $Sp(N)/U(N)$. We identify elementary walls with the simple roots of $USp(2N)$ and discuss compressed walls, penetrable walls and multiwalls by using the moduli matrix formalism.

1 Introduction models on Sp(N )/U (N ) 2 . Sp(N ) ≡ U Sp(2N ), or equivalently Sp(N ) = Sp(N, C)∩U (2N ). Unlike SU (N ) or SO(2N ), the lengths of the simple roots of U Sp(2N ) are different. Therefore the operators for the compressed walls of the nonlinear sigma models on Sp(N )/U (N ) should be newly defined. We discuss the definitions of the operators and show that some of multiwalls can be compressed.
Since Sp(1)/U (1) CP 1 Q 1 and Sp(2)/U (2) Q 3 [14] 3 , the nonlinear sigma models on Sp(N )/U (N ) with N = 1, 2 are Abelian theories. However, the nonlinear sigma models on Sp(N )/U (N ) with N ≥ 3 are non-Abelian theories, so there exist penetrable walls. We use the pictorial representations proposed in [11] to investigate the vacuum structures and the recurrence of two-dimensional diagrams to prove the vacuum structures that are connected to the maximum number of elementary walls by induction.
We follow the convention of [15,16] for the description of the root systems and corresponding Lie algebras. We also identify the elementary walls with the simple roots of U Sp(2N ) as it is done in [17]. In Section 2, we discuss the nonlinear sigma models on Sp(N )/U (N ) and the moduli matrix formalism. In Section 3, we study walls of the Kähler nonlinear sigma models on Sp(N )/U (N ) with N ≤ 6. In Section 4, we study the vacuum structures that are connected to the maximum number of elementary walls. In Section 5, we make some observations about walls of the nonlinear sigma model on Sp(5)/U (5). In Section 6, we summarize our results. In Appendix A, we prove the vacuum structures that are connected to the maximum number of elementary walls.

Model
The Kähler nonlinear sigma models on SO(2N )/U (N ) and Sp(N )/U (N ) can be represented as quadrics in the Grassmann manifold G 2N,N . The Lagrangian in four dimensions is written in the N = 1 superfield formalism [13,14,18]: where Φ is an N ×2N chiral superfield with the flavor indices i, j = 1, · · · , 2N and the color indices a, b = 1, · · · N , V is an N × N matrix vector superfield in the adjoint representation 2 The result of this paper is different to the result of [10]. In [10] we did not use the root system of U Sp(2N ) to analyse the vacua and the walls of the nonlinear sigma models on Sp(N )/U (N ). In this paper we identify the elementary wall operators with the simple root generators of U Sp(2N ) and find that the elementary wall operators in [10] are not correct. The result of this paper seems to be consistent with the result of [15] where kink monopoles are studied in similar models with U Sp(2N ) global symmetry. 3  The superfields are written in terms of component fields: 3) The mass-deformed Lagrangian is obtained by dimensional reduction [19]. The bosonic part of the Lagrangian in three dimensions is The last term (h.c) stands for the Hermitian conjugate.
The Cartan generators of SO(2N ) and U Sp(2N ) are H I = e I,I − e N +I,N +I , (I = 1, · · · N ), (2.5) where e I,I (e N +I,N +I ) is a 2N × 2N matrix whose (I, I)((N + I, N + I)) component is one [15,16]. The mass matrix can be formulated as with vectors m := (m 1 , m 2 · · · , m N ), The mass matrix in the basis (2.5) is Since we are interested in generic mass parameters, we can set m 1 > m 2 > · · · > m N without loss of generality.
Equations of motion for D and F yield the constraints for the Lagrangian (2.4) We eliminate the auxiliary fields. The potential term of the model is The vacuum conditions are The condition (2.13) gives φ 0 = 0 or φ = 0. Since the latter solution is inconsistent with (2.9), we have φ 0 = 0. The scalar field Σ can be diagonalized by a U (N ) gauge transformation as (2.14) Since M and Σ in (2.12) are both diagonal matrices the vacuum solutions to (2.12) are labelled by (Σ 1 , Σ 2 , · · · , Σ N ) = (±m 1 , ±m 2 , · · · , ±m N ).  [20].
To study wall solutions we assume that fields are static and all the fields depend only on the x 1 ≡ x coordinate. We also assume that there is Poincaré invariance on the twodimensional worldvolume of walls so we can set A 0 = A 2 = 0. The energy density along the x-direction is with D ≡ D µ=1 and which is the tension density of the wall. The tension is The energy density is constrained by (2.9) and (2.10). We choose the upper sign for the BPS equation and the lower sign for the anti-BPS equation in the first squared term in (2.16). Then the BPS equation is We introduce complex matrix functions S b a (x) and f i a (x), which are defined by Then the equation (2.19) is solved by Therefore the solution to the BPS equation (2.19) is The coefficient matrix H 0 is the moduli matrix. Σ, A and φ are invariant under the following transformations where V ∈ GL(N, C). The matrix V defines an equivalent class of (S, H 0 ). This is named as worldvolume symmetry in the moduli matrix formalism [8,9]. (2.9) and (2.10) correspond to In the moduli matrix formalism, walls are constructed from elementary walls. The elementary wall operators are the simple root generators of the flavor symmetry. So the elementary walls can be identified with the simple roots [17]. We summarize the simple root generators E i , (i = 1, · · · , N ) and the simple roots α i of SO(2N ) and U Sp(2N ) following the convention of [15,16]. The set of vectors {ê i } is the standard unit vectorsê i ·ê j = δ ij : In this paper, A denotes a vacuum and A ← B denotes a wall which connects vacuum A and vacuum B .
The mass matrix M (2.6), which is a linear combination of the Cartan generators, and elementary wall A ← B , which is generated by Cartan generator E i are related by where c is a constant and T A←B is the tension of wall. The moduli matrix of elementary wall H 0 A←B , which connects A and B is where E i is an elementary wall operator and r is a complex parameter with −∞ < Re(r) < +∞.
Unlike SU (N ) and SO(2N ), the lengths of the simple roots of U Sp(2N ) are different. Therefore the constant c in (2.30) can be different in some vacuum sectors of the nonlinear sigma models on Sp(N )/U (N ).
We first review the formalism for the walls of the nonlinear sigma models on G N F ,N C and SO(2N )/U (N ). In this case c is the same in all the sectors of the vacuum structure. Given the aim of the work [11], it can be fixed as c = 1 for convenience. Elementary walls can be compressed to single walls. In the nonlinear sigma models on G N F ,N C and on SO(2N )/U (N ), a compressed wall of level n which connects A and A is (2.32) A double wall moduli matrix is constructed by multiplying a single wall operator to a single wall moduli matrix. By repeating it, we get a triple wall, a quadruple wall and so on. A multiwall which interpolates A , A ,· · · , and B is where parameters r i (i = 1, 2, · · · ) are complex parameters ranging −∞ < Re(r i ) < ∞.
Elementary walls pass through each other if and these walls are named as penetrable walls [9]. Elementary walls can be identified with simple roots by (2.30) [17]. Let root vector g A 1 ←A 2 denote the wall which connects vacuum A 1 and vacuum A 2 . The corresponding tension of the wall is T A 1 ←A 2 = m · g A 1 ←A 2 . Then the elementary wall of (2.31) is The compressed wall of (2.32) is The root vectors of the two penetrable elementary walls of (2.34) are orthogonal Now we study walls of the nonlinear sigma models on Sp(N )/U (N ). In this case, c = 2 for i = 1, · · · , N − 1 and c = 1 for i = N in (2.30). An elementary wall A ← B is The moduli matrix of A ← A , which is a compressed wall of level n is The moduli matrices and the operators are the same as (2.32) for i = 1, · · · , N −1. However, the formula should change for operator E N . As an example, an elementary wall The formulas for multiwalls (2.33) and for penetrable walls (2.34) hold for walls of nonlinear sigma models on Sp(N )/U (N ). The compressed wall of (2.39) in terms of root vectors is whereas the compressed wall of (2.40) and (2.41) is In this paper we label the moduli matrices of vacua in descending order as There are two vacua in the nonlinear sigma model on Sp(1)/U (1).
The moduli matrices of the vacua are There is only one wall, which is an elementary wall. The elementary wall operator is and the moduli matrix of the elementary wall is The tension of the wall is The diagram of the elementary wall is depicted in Figure 1(a). We study walls of the nonlinear sigma model on Sp(2)/U (2). The Cartan generators H I , (I = 1, 2), the simple root generators E i , (i = 1, 2), and the simple roots of U Sp(4) are (3.6) For N = 2 the vacuum condition (2.12) gives rise to 4 vacua, which have the following form The moduli matrices of (3.7) are The moduli matrices of elementary walls that connect the vacua (3.8) are The wall solution (2.22) with H 1←2 is (3.10) All the phases, which appear due to the U (1) gauge symmetry, are fixed to zero. The wall (3.10) has the limits The wall (3.12) has the limits and Φ 3 are the same vacuum. We can also see this by making use of worldvolume symmetry. The moduli matrix of φ 23 (x → −∞) is which is related to H 0 3 by worldvolume symmetry The wall solution (3.16) has the limits Tension T A←B of the wall that connects vacuum Φ A and vacuum Φ B is obtained from (3.7). The tensions of the elementary walls are (3.20) Therefore the elementary walls are identified with The diagram of the elementary walls are depicted in Figure 1(b). We omit the coefficients of the simple roots in elementary wall diagrams in this paper. From the diagram in Figure  1(b), one can see how a compressed walls is constructed. From (2.39), the compressed wall that interpolates 1 and 3 is and the compressed wall that interpolates 2 and 4 is These are the compressed walls of level one. It can be shown that these compressed walls can be obtained from double walls. Moduli matrices of double walls 1 ← 2 ← 3 and 2 ← 3 ← 4 are H 0 1←2←3 can be transformed as 1 e r 2 0 0 0 1 −e r 1 +r 2 e r 1 = 1 0 e r+ln 2 −e r−r 2 +ln 2 e −2r 2 1 + e −r 2 −e r−r 2 +ln 2 e r−2r 2 +ln 2 , (3.25) where r := r 1 + 2r 2 − ln 2. The limit of H 0 1←2←3 in (3.25) as r 2 → +∞ with finite r equals to H 0 1←3 in (3.22). Or equivalently, where r := r 1 +2r 2 −ln 2 and means the following worldvolume symmetry transformation where r := 2r 1 + r 2 − ln 2. The limit of H 0 2←3←4 in (3.28) as r 1 → −∞ with finite r equals to H 0 2←4 in (3.23). Or equivalently, where r := 2r 1 +r 2 −ln 2 and means the following worldvolume symmetry transformation and The eight vacua of the nonlinear sigma model on Sp(3)/U (3) are labelled in the descending order of (2.44): (3.34) The tensions of elementary walls that connect vacua (3.34) are Therefore the elementary walls are We make some observations of walls. One can guess existence of compressed walls from the wall diagram in Figure 1(c). Since g 1←2 · g 2←3 = 0, elementary wall 1 ← 2 and One can also see that g 2←3 · g 3←5 = 0. Therefore elementary wall 2 ← 3 and elementary wall 3 ← 5 are compressed to compressed wall 2 ← 5 , which is a compressed wall of level one. The moduli matrix of compressed wall 2 ← 5 is Let us consider the moduli matrix of double wall 1 ← 2 ← 3 and the moduli matrix of double wall 2 ← 3 ← 5 where r := r 1 + 2r 2 − ln 2 and means As r 2 → +∞ with finite r, the limit of H 0 1←2←3 equals to H 0 1←3 . Double wall 1 ← 2 ← 3 is plotted in Figure 2.
Next we discuss penetrable walls. Since g 3←5 · g 5←6 = 0, we can observe elementary wall 3 ← 5 and elementary wall 5 ← 6 pass through each other. Double wall 3 ← 5 ← 6 is plotted in Figure 3. The moduli matrix of 1 ← 5 , which is a compressed wall of level two is The moduli matrix of triple wall 1 ← 2 ← 3 ← 5 is We shall consider higher N . Elementary walls can be identified with the simple roots of U Sp(2N ) with proper coefficients. All the compressed single walls and multiwalls can be constructed from the elementary wall configuration. The elementary wall configuration for (3.47) The diagram of the elementary walls are depicted in Figure 1(d). We leave vacuum labels out of diagrams from Figure 1(d) onwards.
While the elementary wall diagrams are planar for N ≤ 4, the diagrams are non-planar for N ≥ 5. The elementary wall configurations for N = 5 and N = 6 are as follows: (3.48) The diagrams of the elementary walls of N = 5 and N = 6 cases are depicted in Figure  4 and Figure 5. . The left-hand side is the limit as x → +∞ and the right-hand side is the limit as x → −∞. . The left-hand side is the limit as x → +∞ and the right-hand side is the limit as x → −∞.

Vacua connected to the maximum number of elementary walls
We study the vacua that are connected to the maximum number of elementary walls. We denote the vacua A and B . Let A be the vacuum near 1 and B be the vacuum near 2 N . From Figure 1, Figure 4 and Figure 5, we make the following observations where m denotes simple root α m : •N = 6 6 ← · · · ← { 2, 4, 6} ← A ← { 1, 3, 5} ← · · · · · · ← { 1, 3, 5} ← B ← { 2, 4, 6} ← · · · ← 6 (4.6) From Figure  The vacuum labels are not unique since we can change them as we please. Therefore let us label the vacua that are connected to the maximum number of elementary walls as A and B . The vacuum structures that are connected to the maximum number of elementary walls are as follows: 2m ← · · · · · · ← N (4.14) where means As r 1 → −∞ with r 1 + r 2 = r(finite), H 0 7←11←19 → H 0 7←19 . Double wall 7 ← 11 ← 19 is compressed to compressed wall 7 ← 19 , which is a compressed wall of level one. This is depicted in Figure 6. In (5.1) α 2 · α 5 = 0. Therefore elementary wall 7 ← 11 and elementary wall 11 ← 12 are penetrable. The moduli matrix of double wall 7 ← 11 ← 12 , which consists of two penetrable elementary walls 7 ← 11 and 11 ← 12 is This is depicted in Figure 7.

Conclusion
We  In this appendix, we prove (4.11), (4.12), (4.13) and (4.14). The vacuum structures that are connected to the maximum number of elementary walls in the nonlinear sigma models on SO(2N )/U (N ) are studied by decomposing the diagrams into two-dimensional diagrams in [11]. We use the same method in the nonlinear sigma models on Sp(N )/U (N ). The rule for the decomposition is that the simple roots that have already appeared in the previous diagrams should not be repeated. The vacuum structure of N = 5 case is depicted in Figure 4. The vacuum structure near 1 ( 32 ) decomposes into two diagrams, as is shown in Figure 8. The circle indicates A ( B ). The letter 'X' indicates the vacuum that is connected to the both diagrams. The left-hand side(the right-hand side) of the each diagram is the limit as x → +∞(x → −∞) for the vacuum structure near 1 whereas the left-hand side(the right-hand side) of the each diagrams is the limit as x → −∞(x → +∞) for the vacuum structure near 32 . Figure 5, which describes the vacuum structure of N = 6 case decomposes into two diagrams as is shown in Figure 9. In the same manner the vacuum structures of N = 7 and N = 8 cases are presented in Figure 10 and Figure 11. The vacuum structures repeat the four diagrams in Figure 1.
All the vacuum structures can be decomposed into two dimensional diagrams in Figure  12, where only the first two diagrams are shown and then fall into four categories. The vacuum that is connected to the maximum number of elementary walls is circled in each diagram in Figure 13.
The vacuum structures that are connected to the maximum number of elementary walls can be obtained from the repeated diagrams.   denotes the vacuum near 2 N . The common parts of each vacuum structure near A and B are shown in Figure 14 and Figure 15. The rest of the vacuum structures are obtained from Figure 13. The remaining parts of each vacuum structure near A and B are shown in Figure 16 and Figure 17 for N = 4m − 3, N = 4m − 2, N = 4m − 1 and N = 4m. The vacuum structure of A is derived from Figure 14 and Figure 16 as follows: Each case with m = 1 is shown in Figure 1. Each case with m = 2 is shown in Figure 8, Figure 9, Figure 10 and Figure 11.  •N = 4m, (m ≥ 2) Each case with m = 1 is shown in Figure 1. Each case with m = 2 is shown in Figure 8, Figure 9, Figure 10 and Figure 11. Let us assume that (A.5), (A.6), (A.7) and (A.8) are true. Then these are true for m = m + 1 as it corresponds to adding one more diagram in Figure 15. Therefore (A.5), (A.6), (A.7) and (A.8) are true. For any N the vacuum structure is N ← · · · ← A ← · · · ← B ← · · · ← N (A.9) Therefore (4.11), (4.12), (4.13) and (4.14) are proved.