PHYSICAL REVIEW D 101, 105020 (2020)

Exhausting all exact solutions of BPS domain wall networks in arbitrary dimensions

† ‡ Minoru Eto ,1,2,* Masaki Kawaguchi ,1, Muneto Nitta,2,3, and Ryotaro Sasaki1 1Department of Physics, Yamagata University, Kojirakawa-machi 1-4-12, Yamagata, Yamagata 990-8560, Japan 2Research and Education Center for Natural Sciences, Keio University, Hiyoshi 4-1-1, Yokohama, Kanagawa 223-8521, Japan 3Department of Physics, Keio University, Hiyoshi 4-1-1, Yokohama, Kanagawa 223-8521, Japan

(Received 15 April 2020; accepted 15 May 2020; published 28 May 2020)

We obtain full moduli parameters for generic nonplanar Bogomol’nyi-Prasad-Sommerfield networks of domain walls in an extended Abelian-Higgs model with N complex scalar fields and exhaust all exact solutions in the corresponding CPN−1 model. We develop a convenient description by grid diagrams which are polytopes determined by mass parameters of the model. To illustrate the validity of our method, we work out nonplanar domain wall networks for lower N in 3 þ 1 dimensions. In general, the networks can have compact vacuum bubbles, which are finite vacuum regions surrounded by domain walls, when the polytopes of the grid diagrams have inner vertices, and the size of bubbles can be controlled by moduli parameters. We also construct domain wall networks with bubbles in the shapes of the Platonic, Archimedean, Catalan, and Kepler-Poinsot solids.

DOI: 10.1103/PhysRevD.101.105020

I. INTRODUCTION equations called BPS equations. In such cases, one can often embed the theories to supersymmetric (SUSY) It sometimes happens that systems have multiple discrete vacua or ground states, which is inevitable when a discrete theories by appropriately adding fermion superpartners, symmetry is spontaneously broken. In such a case, there in which BPS solitons preserve some fractions of SUSY. appear domain walls (or kinks) in general [1–3] which are Their topological charges are central (or tensorial) charges inevitably created during second order phase transitions of corresponding SUSY algebras. The BPS domain walls 3 1 [4–7]. They are the simplest topological solitons appearing in þ dimensions were studied extensively in field N 1 – N 2 in various condensed matter systems such as magnets [8], theories with both ¼ SUSY [20 35] and ¼ graphenes [9], carbon nanotubes, superconductors [10], SUSY [36–56]; see Refs. [57–60] as a review. They 1 atomic Bose-Einstein condensates [11], and helium super- preserve a half of SUSY and thereby are called 2 BPS fluids [6,7,12], as well as high nuclear matter states, accompanied by SUSY central (tensorial) charges m [13,14], quark matter [15], and early Universe [4,16]. Zm ðm ¼ 1, 2; labeling spatial coordinates x perpen- In cosmology, cosmological domain wall networks are dicular to the domain wall) as domain wall topological suggested as a candidate of dark matter and/or dark charges [22,24,61]. In general, if several domain walls meet energy [17]. along a line, it forms a planar domain wall junction. In As the cases of other topological solitons, domain walls SUSY models, the planar domain wall junctions preserve a can become Bogomol’nyi-Prasad-Sommerfield (BPS) 1 quarter SUSY [62–64], therefore are called 4 BPS states, states [18,19], attaining the minimum energy for a fixed accompanied by a junction topological charge Y in addition boundary condition and satisfy first order differential 1 to Zm (m ¼ 1, 2). The 4 BPS domain wall junctions have been studied in theories with N ¼ 1 SUSY [32,65–73] and *[email protected] † N ¼ 2 SUSY [74–83]. In the N ¼ 2 SUSY gauge models, [email protected] ‡ not only planar domain wall junctions, but also planar [email protected] 1 domain wall networks as 4 BPS states were constructed Published by the American Physical Society under the terms of [75,76]. The low-energy effective action for normalizable the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to modes within the networks was obtained [78] and applied the author(s) and the published article’s title, journal citation, to study of low-energy dynamics [79]. Non-BPS planer and DOI. Funded by SCOAP3. domain wall networks were also studied in Refs. [84,85].

2470-0010=2020=101(10)=105020(22) 105020-1 Published by the American Physical Society ETO, KAWAGUCHI, NITTA, and SASAKI PHYS. REV. D 101, 105020 (2020)

Recently, the present authors proposed a model, a D þ 1- II. THE MODEL 1 dimensional Uð Þ gauge theory [86] admitting novel We study a Uð1Þ gauge theory with N charged complex analytic solutions of the BPS single nonplanar domain scalar fields HA (A ¼ 1; 2; …;N) and N0 real scalar fields wall junctions. This model cannot be made supersymmetric 0 ΣA (A0 ¼ 1; 2; …;N0)inD þ 1-dimensional spacetime. but still admits stable BPS states so that we can use the The Lagrangian is given by well-known Bogomol’nyi completion technique to derive BPS equations. The model consists of N charged complex 1 1 XN0 0 1 μν A0 μ A0 scalar fields and N neutral scalar fields coupled to the Uð Þ L ¼ − FμνF þ ∂μΣ ∂ Σ 4e2 2e2 gauge field. In Ref. [86], we restricted ourself to the special A0¼1 numbers N − 1 ¼ N0 ¼ D and imposed the invariance μ † − S 1 þ DμHðD HÞ V ð2:1Þ under the symmetric group Dþ1 of the rank D þ , which are the symmetry groups of the regular D simplex. 1 XN0 In this paper, we investigate generic nonplanar networks V ¼ Y2 þ ðΣA0 H − HMA0 ÞðΣA0 H − HMA0 Þ†; 2e2 of BPS domain walls in D þ 1 dimensions. We consider the A0¼1 ≥ 1 generic case of N D þ imposing no discrete symmetry ð2:2Þ and exhaust all exact solutions with full moduli of generic BPS nonplanar networks of domain walls in the infinite where H is an N component row vector made of HA, Uð1Þ gauge coupling limit in which the model reduces to N−1 the CP model. These are the first exact solutions of H ¼ðH1;H2; …;HNÞ; ð2:3Þ nonplanar domain wall networks in D dimensions (D ≥ 3). We first derive the BPS equations for generic Abelian Y is a scalar quantity defined by gauge theories in D þ 1 dimensions. Then, we partially solve them by the moduli matrix formalism [58] and find all Y ¼ e2ðv2 − HH†Þ; ð2:4Þ moduli parameters of the generic domain wall network solutions. We then demonstrate several concrete nonplanar and MA0 (A0 ¼ 1; …;N0) are N by N real diagonal mass −1 networks in the CPN model in D ¼ 3 for N ¼ 4,5,6.In matrices defined by the case of N ¼ 4, the solution has only one junction at A0 which four vacua meet. Network structures appear for M ¼ diagðmA0;1;mA0;2; …;mA0;NÞ: ð2:5Þ N>4. In the case of N ¼ 5, we show two different types of networks exist in general. The first type has a vacuum The spacetime index μ runs from 0 to D, and Fμν is an Uð1Þ bubble (a compact vacuum domain) surrounded by semi- gauge field strength. The coupling constants in the infinite vacuum domains. Instead, the second type does not Lagrangian in Eq. (2.1) are taken to be the so-called have any bubbles but all the vacuum domains are semi- Bogomol’nyi limit. For later use, let us define mA by an infinitely extended. In the N ¼ 6 case, there are three N0 vector whose components are the Ath diagonal elements different types according to the number of the vacuum of MA0 s, namely, bubbles, two, one, or zero. Finally, we find a connection to the well-known polyhedra known from ancient times. mA ¼ðm1;A;m2;A; …;mN0;AÞ: ð2:6Þ Indeed, we find the vacuum bubbles which are congruent with five Platonic solids. In addition, the Archimedean and In the following, we will mostly consider generic masses: the Catalan solids appear as the vacuum bubbles. We also construct the Kepler-Poinsot star solids as domain wall mA ≠ mB; if A ≠ B: ð2:7Þ networks. This paper is organized as follows. In Sec. II,we Since the scalar potential V is positive semidefinite, a introduce our model. In Sec. III, we derive the BPS classical vacuum of the theory is determined by V ¼ 0: equations and clarify the moduli space of the BPS HH† ¼ v2, and ΣA0 H − HMA0 ¼ 0. In the generic case of solutions. In Sec. IV, we first consider the infinite Uð1Þ Eq. (2.7), there are N discrete vacua. The Ath vacua which gauge coupling limit in which the model reduces to the we will denote by hAi is given by HB ¼ vδB, and −1 A CPN B0 massive nonlinear sigma model. We then exhaust Σ ¼ mB0;A. Simply, the vacua can be identified to the all exact solutions with full moduli parameters for the BPS discrete points determined by the mass vectors fmAg in the equations. We further give several examples of nonplanar Σ space, networks in D ¼ 3. In Sec. V, we study relations between 3 the domain wall networks in D ¼ and the classic solids, hAi∶Σ ¼ mA: ð2:8Þ like the Platonic, Archimedean, Catalan, and Kepler- Poinsot solids. Finally, we summarize our results and give The Lagrangian (2.1) is motivated by supersymmetry. In 0 2 N 2 a discussion in Sec. VI. fact, when NF ¼ , it is a bosonic part of an ¼

105020-2 EXHAUSTING ALL EXACT SOLUTIONS OF BPS DOMAIN … PHYS. REV. D 101, 105020 (2020) supersymmetric theory with eight supercharges in four direction to the domain wall interpolating the vacua hAi and dimensions. hBi is given by

Z v2 m − m : 3:6 III. SOLVING BPS EQUATIONS FOR DOMAIN j j¼ j A Bj ð Þ WALL NETWORKS On the other hand, the domain wall junction charge can A. Derivation of the BPS equations be defined from the topological charge density Ymn as L Z From now on, we investigate BPS states of in Eq. (2.1) ξ ξ in the case that the number N0 of flavors for real adjoint ξ ξ m nY − m n ≤ 0 Ymn ¼ m n dx dx mn ¼ 2 Smn ; ð3:7Þ scalars is equal to the number of the spatial dimensions D.1 e In what follows, the Roman index m stands not only for the spacial index as m ¼ 1; 2; …;Dbut also for the index of N0 negatively contributing to the BPS energy. Here, we have (m ≡ A0). Then, the standard Bogomol’nyi completion for defined this system goes as follows: Z ∂mΣm ∂mΣn S ≡ dxmdxn det : ð3:8Þ 1 X mn ∂ Σ ∂ Σ E ¼ fF2 þð∂ Σ − ξ ξ ∂ Σ Þ2g n m n n 2e2 mn m n m n n m m>n xm − xn X 2 The integrand is a Jacobian of a map from the whole 1 Σm − Σn þ ξ ∂ Σ − Y plane to a region in the plane defined by the function 2e2 m m m m Σ m n Σ m n k X ð mðx ;x Þ; nðx ;x ÞÞ with all other coordinates x (k ≠ m, n) fixed. S can be either positive or negative, þ fDmH þ ξmðΣmH − HMmÞg mn m and its absolute value is the area of the image. Nevertheless, † Ymn is always negative since Smn is accompanied with × fDmH þ ξmðΣmH − HMmÞg X X X ð−ξmξnÞ, and so it should be understood as a sort of binding þ ξmZm þ ξmξnYmn þ ∂mJ m; ð3:1Þ energy among domain walls [67–69]. m m>n m Since the first three terms of Eq. (3.1) are positive semidefinite, the Bogomol’nyi energy bound is given by with ξm ¼1, and we have defined the domain wall Z X X X topological charge density m, the domain wall junction E ≥ ξ Z ξ ξ Y ∂ J Y J m m þ m n mn þ m m; ð3:9Þ charge density mn, and m by m m>n m

2 Zm ¼ v ∂mΣm; ð3:2Þ and it is saturated by the BPS states satisfying the BPS equations, 1 ∂mΣm ∂mΣn Y ¼ − det ; ð3:3Þ mn 2 F 0; : e ∂nΣm ∂nΣn mn ¼ ð3 10Þ † ξ ∂ Σ − ξ ∂ Σ 0 J m ¼ −ξmðΣmH − HMmÞH ; ð3:4Þ n m n m n m ¼ ; ð3:11Þ J respectively. The contribution by m vanishes under the ξmDmH þ ΣmH − HMm ¼ 0; ð3:12Þ space integrations since it is asymptotically zero because of X Σ − 0 the vacuum condition mH HMm ¼ . ξ ∂ Σ − Y ¼ 0; ð3:13Þ m m m m The domain wall tension Zm measured along the x m direction can be defined from Zm by Z where m; n ¼ 1; 2; …;D. One can verify that all solutions ∞ mξ Z 2ξ Σ − Σ ≥ 0 of the above BPS equations solve the full equations of Zm ¼ dx m m ¼ v mð mjxm¼þ∞ mjxm¼−∞Þ ; 1 −∞ motion. This is the D-dimensional extension of the 4 BPS 2 ðno sum over mÞ: ð3:5Þ equations of the planar domain wall junction in D ¼ cases studied in Refs. [74,75,78,79]. In the previous work [86] by the present authors, we Note that Zm is always positive regardless of the choice of generalized the topological charge densities Z and Y ξm. Hence, the genuine tension measured along the normal m mn under the observation that Zm and Ymn are nothing but one- m → Σ 1 and two-dimensional Jacobians of maps x m and In Ref. [86], we considered the same model with restricted to m n → Σ Σ a special case in which the flavor number N is also related to the ðx ;x Þ ð m; nÞ, respectively. For a domain wall Z spatial dimensions, N ¼ D þ 1. Instead, this work studies the interpolating the hAi and hBi vacua, an integral of m case with generic Nð≥ D þ 1Þ. (with an appropriate rescale to make it dimensionless)

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1 measures a covering number of a map from R ð−∞ < ðS; H0Þ → VðS; H0Þ;V∈ C ; ð3:20Þ m x < ∞Þ onto the one-dimensional interval ðmm;A < Σm < mm;BÞ. The charge is topological and indeed takes values where V is constant. This is called the V transformation either þ1,0,or−1. Similar arguments hold for Ymn, and it under which any physical information is independent. is straightforward to generalize it in D dimensions as To solve the master equation, we first need to fix the moduli matrix H0. Once H0 is given, the boundary W ðm1;m2; …;m Þ condition at jxj → ∞ is automatically specified. Namely, d 0 d 1 ∂ Σ ∂ Σ ∂ Σ we should solve the master equation with the boundary m1 m1 m1 m2 m1 m B d C condition B ∂ Σ ∂ Σ ∂ Σ C B m2 m1 m2 m2 m2 md C 2ξ m † detB C; 3:14 Ω mMmx ¼ B . . . . C ð Þ lim ¼ H0e H0: ð3:21Þ @ . . . A jxj→∞ ∂ Σ ∂ Σ ∂ Σ md m1 md m2 md md As a trivial example, let us consider a homogeneous 1 1 1 ≤ ≤ ∈ 1 2 … α 1 2 … vacuum, say the first vacuum h i. The h i vacuum where d D, mα f ; ; ;Dg ( ¼ ; ; ;d), configuration is given by the moduli matrix and mα >mβ if α > β. We have W1ðmÞ ∝ Zm and W2 m; n ∝ Y D ð Þ mn. In general, the BPS solutions in H0 ¼ðh1; 0; …; 0Þ;h1 ∈ C : ð3:22Þ dimensions have d-dimensional substructures, and Wd provides us topological numbers associated with the The corresponding master equation becomes substructures.

1 m 2 2 2 −1 2 2ξ m 1x ∇ log Ω ¼ e v ð1 − Ω jh1j e m m; Þ; ð3:23Þ B. The moduli matrix method 2 Let us solve the BPS equations (3.10)–(3.13). To this which can be solved by end, we first introduce a complex scalar function SðxmÞ by m 2 2ξ m 1x Ω ¼jh1j e m m; : ð3:24Þ Am − iξmΣm ¼ −i∂m log S: ð3:15Þ

Plugging this into Eq. (3.18), we immediately find Σ ¼ m1. Then, Eqs. (3.10) and (3.11) are automatically satisfied. It is also straightforward to verify H ¼ðv; 0; …; 0Þ. Note Plugging this into Eq. (3.12), we have that the constant h1 in the moduli matrix does not play any role in the above example. This redundancy comes ∂ H ∂ log S H − ξ HM 0: 3:16 m þð m Þ m m ¼ ð Þ from the V transformation in Eq. (3.20). Indeed, we could, from the first point, fix the moduli matrix (3.22) as This can be solved by H0 ¼ðh1; 0; …; 0Þ → ð1; 0; …; 0Þ. −1 ξ M xm Nontrivial inhomogeneous solutions including a single H ¼ vS H0e m m ; ð3:17Þ domain wall connecting the hAi and hBi vacua are gen- erated by the moduli matrix with the nonzero constants where H0 is an arbitrary complex constant N vector. H0 is only in the Ath and Bth entries as called the moduli matrix because the elements of H0 are moduli (integration constants) of the BPS solutions. 0 … 0 1 0 … 0 0 … 0 Finally, we are left with the fourth equation (3.13).To H0 ¼ð ; ; ; ; ; ; ;hB; ; ; Þ Σ solve this, let us express the neutral real scalar fields m in ∼ ð0; …; 0;hA; 0; …; 0; 1; 0; …; 0Þ: ð3:25Þ terms of S, The nonzero moduli parameters h and h are related by 1 A B Σ ξ ∂ Ω Ω ≡ 2 the V transformation as h h ¼ 1. m ¼ 2 m m log ; jSj ; ð3:18Þ A B Similarly, if we have H0 with three nonzero constants, where Ω is the gauge-invariant quantity. Using this, the we will have a domain wall junction dividing the corre- fourth BPS equation (3.13) is cast into the following sponding three vacua. The moduli matrix is given by Poisson equation in D dimensions: H0 ¼ð0;…;0;1;0;…;0;hB;0;…;0;hC;0;…;0Þ: ð3:26Þ 1 2 2 2 −1 2ξ m † ∇ Ω 1 − Ω mMmx 2 log ¼ e v ð H0e H0Þ: ð3:19Þ Maximally complex solutions dividing N vacuum domains in D dimensions are, then, obtained by the moduli We call this the master equation for the BPS states. matrix which has no zeros in any elements. Such compli- We note that the original fields Am, Σm, and H are intact cated extended objects in D dimensions are not easy for us under the following transformation: to handle without exact solutions. Unfortunately, the master

105020-4 EXHAUSTING ALL EXACT SOLUTIONS OF BPS DOMAIN … PHYS. REV. D 101, 105020 (2020) equation (3.19) does not seem analytically solvable, except In the next section, we will see that the position of the d for the very special cases possessing the highest discrete wall estimated by the weight is related to the generalized S W symmetry group Dþ1, the symmetric group of the degree topological charge d defined in Eq. (3.14). D þ 1, in the model with the special number of the flavor N D 1 and finely tuned parameters g, c, and m 0 ,as ¼ þ A ;A IV. EXHAUSTING ALL EXACT SOLUTIONS demonstrated in Ref. [86]. Even in the fine-tuned models, OF DOMAIN WALL NETWORKS the exact analytic solution were only found for a single IN THE CPN − 1 MODEL junction of the domain walls since N ¼ D þ 1 is the minimum number. However, as we show in the next A. General solutions section, the solutions include domain wall networks when There is a great simplification allowing us to obtain all N>Dþ 1. exact solutions for generic domain wall networks in D To see shapes of domain wall networks, let us define a dimensions. It is the infinite gauge coupling limit in which weight for each vacuum by we formally send the gauge coupling e to infinity in the Lagrangian (2.1). There, the kinetic terms of Aμ and Σm hAi m˜ ·xþa w ¼ e A A ; ðA ¼ 1; …;NÞ: ð3:27Þ vanish to become Lagrange multipliers. At the same time, the first term in the potential (2.2) forces the charged fields Since the weight is the exponential function of the spatial HA to take their values in the restricted region S2N−1 − 1 coordinate, only one weight dominates the rest N defined by HH† ¼ v2. Furthermore, the overall phase of x weights at each point . Suppose the weight of the hAi H is gauged. Therefore, the physical target space in the x A vacuum is dominant in vicinity of a point 0. There, we infinite gauge coupling limit is reduced to the complex hAi 2 have Ω ∼ ðw Þ , and then Σ reads from Eq. (3.18), projective space,

1 2 −1 ˜ hAi 2 N Σjx∼x ¼ ∇ logðw Þ ¼ m : ð3:28Þ N−1 SUðNÞ S 0 2 A CP ≃ ≃ : ð4:1Þ SUðN − 1Þ × Uð1Þ S1 This implies that the region where the weight whAi is Indeed, if we eliminate the gauge field Aμ from Lj →∞,it dominant corresponds to the vacuum hAi. e reduces to the standard Lagrangian of the nonlinear CPN−1 Let us next consider a situation that the vacua hAi and model. Similarly, if we eliminate the neutral scalar fields hBi are next to each other. We can estimate where the two Σ vacua transit by comparing the weights of those vacua. The m, we get a nontrivial potential which lifts all the points of the CPN−1 target space leaving N discrete points as vacua. transition occurs at points on which the two weights are Σ equal. This condition determines a hyperplane which is a Here, we do not eliminate the auxiliary fields Aμ and m. The BPS equations in Eqs. (3.10)–(3.13) remain the same, subspace of codimension 1 in RD, and the first three equations (3.10)–(3.12) are solved by Eqs. (3.15) and (3.17). The fourth equation (3.13) rewritten hA; Bi∶whAi ¼ whBi ⇔ ðm˜ − m˜ Þ · x þ a − a ¼ 0: A B A B as Eq. (3.19) reduces to an algebraic equation, easily ð3:29Þ solved by

2ξ m † 2 Ω mMmx This is a straightforward generalization of D ¼ [75] to je→∞ ¼ H0e H0: ð4:2Þ generic D dimensions. This hyperplane is nothing but a domain wall interpolating the vacua hAi and hBi, and we Thus, we have completely solved the BPS equations for call it a 1-wall. arbitrary moduli matrix H0 in the infinite gauge coupling The three vacua, say hAi, hBi, and hCi, can happen to be limit. We would like to emphasize that this is the first adjacent at a hyperplane of codimension 2, which is solutions of the domain wall networks in D ≥ 3. conventionally called the domain wall junction. We call For later convenience, let us rewrite this in a more useful it a 2-wall. The position of the 2-wall corresponds to the form. First, let us denote H0 as region where the weights are equal as a1þib1 a2þib2 a þib H0 ¼ðe ;e ; …;e N N Þ; ð4:3Þ hA; B; Ci∶whAi ¼ whBi ¼ whCi: ð3:30Þ where faAg and fbAg are N realP parametersP which we 0 These can be naturally generalized to a d wall which is a restrict to satisfy the conditions A aA ¼ A bA ¼ by d codimensional intersection dividing d þ 1 vacua. The using the V transformation. Furthermore, let us define the position of the d wall is defined by new mass vectors

… ∶ hA1i hA2i hAdþ1i m˜ ξ ξ … ξ hA1;A2; ;Adþ1i w ¼ w ¼¼w : ð3:31Þ A ¼ð 1m1;A; 2m2;A; ; DmD;AÞ: ð4:4Þ

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Then, we have

XN 2 m˜ x Ω ð A· þaAÞ je→∞ ¼ e : ð4:5Þ A¼1 By using this, the BPS energy density can be simply expressed by 2 XN 2 v 2 2 m˜ x E ∇˜ Σ ∇ ð A· þaAÞ je→∞ ¼ v · ¼ 2 log e ; ð4:6Þ A¼1 where we have defined

˜ ˜ 2 2 ∇ ¼ðξ1∂1; ξ2∂2; …; ξD∂DÞ; ∇ ¼ ∇ ð4:7Þ 1 ˜ and used Σ ¼ 2 ∇ log Ω. From this expression, we can see that the N − 1 real parameters faAg are the moduli which relate to the shape of the networks in the real space. On the other hand, the other parameters fbAg are internal moduli of Uð1ÞN−1 associated with constituent domain walls. FIG. 1. The grid diagram (Σ space): the four vacua (red points) D 1 The domain walls for ¼ and the domain wall in N ¼ 4 model are shown. networks for D ¼ 2 have been studied very well in the literature; therefore, we will concentrate on D ¼ 3 in the following subsections. Ω ¼ wh1i þ wh2i, and the exact domain wall solution is given as B. N = 4: Tetrahedron: Single domain wall junction 2ðm˜ 1·xþaÞ 2ðm˜ 2·x−aÞ m1 1e m1 2e Let us start with a model with N ¼ 4. There are N ¼ 4 Σ ; þ ; 1 ¼ 2 m˜ x− 2 m˜ x− ; ð4:12Þ vacua which are the minimal numbers for a nonplanar e ð 1· aÞ þ e ð 2· aÞ domain wall junction to exist. For simplicity, let us set the 2ðm˜ 1·xþaÞ 2ðm˜ 2·x−aÞ m2;1e þ m2;2e four mass vectors mA to be four vertices of a regular Σ2 ¼ ; ð4:13Þ 2ðm˜ 1·x−aÞ 2ðm˜ 2·x−aÞ tetrahedron as e þ e

2ðm˜ 1·xþaÞ 2ðm˜ 2·x−aÞ 1 1 m3;1e þ m3;2e Σ3 ¼ : ð4:14Þ m1 ¼ − pffiffiffi ; −1; − pffiffiffi ; ð4:8Þ 2ðm˜ 1·x−aÞ 2ðm˜ 2·x−aÞ 3 6 e þ e 2 1 Note that Σi takes its value in the finite interval Σi ∈ m2 ¼ pffiffiffi ; 0; − pffiffiffi ; ð4:9Þ 3 3 6 ½m 1;m 2 (i ¼ 1, 2) when we sweep the real space R . i; i; 1 1 Namely, Σ connects the two vertices m1 and m2, as desired. m3 ¼ − pffiffiffi ; 1; − pffiffiffi ; ð4:10Þ Thus, the domain wall solution ΣðxÞ can be seen as a 3 6 R3 Σ rffiffiffi function from the real space to the space, and its 3 image is the linear segment between m1 and m2, m 0 0 4 ¼ ; ; 2 : ð4:11Þ Σ1 − m1 1 Σ2 − m2 1 Σ3 − m3 1 ; ¼ ; ¼ ; : ð4:15Þ Then, the four vacua correspond to four vertices of the Σ1 − m1;2 Σ2 − m2;2 Σ3 − m3;2 regular tetrahedron in the Σ1 − Σ2 − Σ3 space, as shown in Fig. 1. We call drawn in the Σ space as grid The topological charges of 1-walls associated with diagrams [75]. Note that we have taken the regular Eq. (3.14) are tetrahedron (4.8)–(4.11) just for simplicity. In general, Z Z ∞ the grid diagrams do not have to be congruent with a ðxÞ W1 ðm1; m2Þ¼ dx ∂1Σ1 ¼ dΣ1 ¼jm1;1 − m1;2j; regular tetrahedron. The following arguments are valid for −∞ the generic grid diagrams. ð4:16Þ As we explained in Eq. (3.25), single domain walls Z Z (1-walls) can be described by the moduli matrix with ∞ ðyÞ m m ∂ Σ Σ − nonvanishing elements. For instance, the moduli matrix W1 ð 1; 2Þ¼ dy 2 2 ¼ d 2 ¼jm2;1 m2;2j; −∞ aþib −a−ib H0 ¼ðe ;e ; 0; 0Þ yields a domain wall h1; 2i con- necting the h1i and h2i vacua. The corresponding Ω is ð4:17Þ

105020-6 EXHAUSTING ALL EXACT SOLUTIONS OF BPS DOMAIN … PHYS. REV. D 101, 105020 (2020) Z Z ∞ 2ðm˜ 1·xþa1Þ 2ðm˜ 2·xþa2Þ 2ðm˜ 3·xþa3Þ ðzÞ m2;1e þ m2;2e þ m2;3e W1 ðm1; m2Þ¼ dz ∂3Σ3 ¼ dΣ3 ¼jm3 1 − m3 2j: Σ ; ; 2 ¼ ˜ ˜ ˜ ; −∞ e2ðm1·xþa1Þ þ e2ðm2·xþa2Þ þ e2ðm3·xþa3Þ ð4:18Þ ð4:20Þ

These are the lengths of segments of the edge connecting 2ðm˜ 1·xþa1Þ 2ðm˜ 2·xþa2Þ 2ðm˜ 3·xþa3Þ m3;1e þ m3;2e þ m3;3e h1i and h2i projected onto the axes Σ1 2 3. It is straightfor- Σ3 ¼ : ; ; 2ðm˜ 1·xþa1Þ 2ðm˜ 2·xþa2Þ 2ðm˜ 3·xþa3Þ ward to construct the other domain walls connecting e þ e þ e arbitrary pair of hAi and hBi. They correspond to the ð4:21Þ edges of the tetrahedron in Fig. 1. A domain wall junction (2-wall) connecting three vacua, These Σ satisfy the equation say h1i, h2i, and h3i, can be also constructed very easily. m − m m − m Σ − m 0 One mere needs to prepare the moduli matrix H0 ¼ fð 3 1Þ × ð 2 1Þg · ð 1Þ¼ ; ð4:22Þ ðea1þib1 ;ea2þib2 ;ea3þib3 ; 0Þ with three nonzero elements. The exact solution is given by representing the two-dimensional plane on which the three points m1, m2, and m3 are located. The image of the map ˜ ˜ ˜ 2ðm1·xþa1Þ 2ðm2·xþa2Þ 2ðm3·xþa3Þ ΣðxÞ in this case is the triangle whose vertices are h1i, h2i, Σ m1;1e þ m1;2e þ m1;3e 1 ¼ 2 m˜ x 2 m˜ x 2 m˜ x ; and h3i. The corresponding topological charges are iden- e ð 1· þa1Þ þ e ð 2· þa2Þ þ e ð 3· þa3Þ tical to the areas of the projections of the triangle onto the : ð4 19Þ three planes (Σ1–Σ2, Σ2–Σ3, and Σ3–Σ1)as

Z ∞ ðxyÞ W2 ðm1; m2; m3Þ¼ dxdy ð∂1Σ2∂2Σ1 − ∂1Σ1∂2Σ2Þ Z−∞ 1 Σ Σ m − m m − m ¼ d 1d 2 ¼ 2 j½ð 1 2Þ × ð 1 3Þ3j; ð4:23Þ Z ∞ ðyzÞ W2 ðm1; m2; m3Þ¼ dydz ð∂2Σ3∂3Σ2 − ∂2Σ2∂3Σ3Þ Z−∞ 1 Σ Σ m − m m − m ¼ d 2d 3 ¼ 2 j½ð 1 2Þ × ð 1 3Þ1j; ð4:24Þ Z ∞ ðzxÞ W2 ðm1; m2; m3Þ¼ dzdx ð∂3Σ1∂1Σ3 − ∂3Σ3∂1Σ1Þ Z−∞ 1 Σ Σ m − m m − m ¼ d 3d 1 ¼ 2 j½ð 1 2Þ × ð 1 3Þ2j: ð4:25Þ

Again, it is straightforward to construct the other 2-walls dividing arbitrary set of three vacua. They correspond to the faces of the tetrahedron in Fig. 1. Finally, we come to the 3-wall h1; 2; 3; 4i consisting of the four vacua, six 1-walls, and four 2-walls. It is described by the a1þib1 a2þib2 a3þib3 a4þib4 full moduli matrix H0 ¼ðe ;e ;e ;e Þ, and the exact solution is given by

2ðm˜ 1·xþa1Þ 2ðm˜ 2·xþa2Þ 2ðm˜ 3·xþa3Þ 2ðm˜ 4·xþa4Þ Σ m1;1e þ m1;2e þ m1;3e þ m1;4e 1 ¼ ˜ ˜ ˜ ˜ ; ð4:26Þ e2ðm1·xþa1Þ þ e2ðm2·xþa2Þ þ e2ðm3·xþa3Þ þ e2ðm4·xþa4Þ

2ðm˜ 1·xþa1Þ 2ðm˜ 2·xþa2Þ 2ðm˜ 3·xþa3Þ 2ðm˜ 4·xþa4Þ Σ m2;1e þ m2;2e þ m2;3e þ m2;4e 2 ¼ ˜ ˜ ˜ ˜ ; ð4:27Þ e2ðm1·xþa1Þ þ e2ðm2·xþa2Þ þ e2ðm3·xþa3Þ þ e2ðm4·xþa4Þ

2ðm˜ 1·xþa1Þ 2ðm˜ 2·xþa2Þ 2ðm˜ 3·xþa3Þ 2ðm˜ 4·xþa4Þ Σ m3;1e þ m3;2e þ m3;3e þ m3;4e 3 ¼ ˜ ˜ ˜ ˜ : ð4:28Þ e2ðm1·xþa1Þ þ e2ðm2·xþa2Þ þ e2ðm3·xþa3Þ þ e2ðm4·xþa4Þ

Note that we can set a1 ¼ a2 ¼ a3 ¼ a4 by using three translational symmetries and V transformation, without loss of generality. Now, the solution ΣðxÞ maps the real three-dimensional space R3 to the tetrahedron itself in the Σ space. The corresponding topological charge reads

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0 1 Z ∂1Σ1 ∂1Σ2 ∂1Σ3 Z ∞ ðxyzÞ B C W3 ðm1; m2; m3; m4Þ¼ dxdydz det@ ∂2Σ1 ∂2Σ2 ∂2Σ3 A ¼ dΣ1dΣ2dΣ3 −∞ ∂3Σ1 ∂3Σ2 ∂3Σ3 1 m − m m − m m − m ¼ 6 jðð 2 1Þ × ð 3 1ÞÞ · ð 4 1ÞÞj: ð4:29Þ

This is nothing but the volume of the tetrahedron. C. N = 5: Dipyramid: Minimal domain wall networks W In Fig. 2, we show the topological charge densities d Let us next consider a model admitting a nonplanar for the regular tetrahedron given in Fig. 1. The top-left network structure of domain walls. The minimal number 1 panel shows the vacuum (0-wall) domains (h i¼green, for this is N ¼ 5 for which there is one additional vacuum 2 3 1 h i¼yellow, h i¼cyan, h i¼hidden). At boundaries h5i compared to the model with N ¼ 4. According to between two vacuum (0-wall) domains, there are domain where we put the fifth vacuum, the resulting network walls (1-walls), as shown in the top-right panel. The plot structures are classified into two types as shown in shows an isosurface of the sum of three 1-wall charge Figs. 4(a) and 4(b). The feature of (a) is the presence of ðxÞ ðyÞ ðzÞ densities W1 þ W1 þ W1 . Similarly, the bottom-left an inner vacuum h5i in the tetrahedron h1; 2; 3; 4i. On the panel shows an isosurface of the sum of the three 2-wall other hand, Fig. 4(b) is a dipyramid for which the fifth ðxyÞ ðyzÞ ðzxÞ 5 1 2 3 4 charge densities W2 þ W2 þ W2 . Finally, the vacuum h i is placed outside the tetrahedron h ; ; ; i It is bottom-right panel shows an isosurface of the 3-wall charge a and has no inner vacua. xyz Let us start with the former case of the tetrahedron with density Wð Þ. The figures clearly show that the general- 3 the inner vacuum. We take the same four vacua defined in ized topological charge defined in Eq. (3.14) is appropriate Eqs. (4.8)–(4.11) for simplicity, and the fifth one is set as to describe the codimension d structure in the solution. To close this subsection, we again emphasize that we m5 ¼ð0; 0; 0Þ: ð4:30Þ have chosen the symmetric arrangement of the masses corresponding to a regular tetrahedron in Fig. 1 just for The most generic moduli matrix up to the three translations simplicity, but the exact solution has been obtained for 3 in R and V transformation (3.20) is given by arbitrary mass arrangement. In order to make this point clearer, we show in Fig. 3 four solutions for randomly a5 H0 ¼ð1; 1; 1; 1;e Þ: ð4:31Þ chosen tetrahedrons.

Then Ω depends on the one moduli parameter a5 as

X4 2m˜ x 2 2m˜ x Ω ¼ e A· þ e a5 e 5· : ð4:32Þ A¼1

From this expression, we understand that the factor ea5 controls the strength (the weight) of the fifth vacuum h5i relative to the rests. When a5 → −∞, the h5i domain disappears. The h5i domain extends as ea5 increases. This moduli dependence of the configuration can be seen in Fig. 5 in which we have shown three examples with a5 ¼ −∞, 0, and 3. The large distance behaviors are the same for all these cases. However, a significant difference emerges around the origin. Reflecting the inner vacuum h5i inside the tetrahedron, a compact vacuum domain of h5i emerges by increasing a5. The whole structure of the solution is as follows. There is one inner vacuum bubble h5i surrounded by the semi-infinite four vacuum domains h1i, h2i, h3i, and h4i. There are six semi-infinite 1-walls corresponding to six outer edges (h1; 2i, h1; 3i, h1; 4i, h2; 3i, FIG. 2. The real space R3: isosurfaces of the topological charge h2; 4i, and h3; 4i) and four compact 1-walls corresponding 1 5 2 5 3 5 4 5 densities Wd for d walls (d ¼ 0, 1, 2, 3) are shown. We only to the four inner edges (h ; i, h ; i, h ; i, h ; i); see the depict the densities within the sphere of the radius 10. third row of Fig. 5. There are four semi-infinite 2-walls

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FIG. 3. Four examples of the BPS solutions: the leftmost panel shows the grid diagrams, and the second, third, and fourth from left show the corresponding Wd d ¼ 0, 1, 2. We only depict the densities within the sphere of the radius 10. and the six compact 2-walls corresponding to the four faces The moduli matrix is the same as the one in Eq. (4.31). (h1; 2; 3i, h1; 2; 4i, h1; 3; 4i, h2; 3; 4i) and the six inner When ea5 is sufficiently large, the configuration becomes a triangles (h1; 2; 5i, h2; 3; 5i, h1; 3; 5i, h1; 4; 5i, h2; 4; 5i, mixture of planar and nonplanar structures, as shown in h3; 4; 5i), respectively. Finally, there are four 3-walls which Fig. 6 for a5 ¼ 3. The planar structure comes from the correspond to the four subtetrahedrons (h1; 2; 4; 5i, bottom triangle h1; 2; 3i which are divided into three h2; 3; 4; 5i, h1; 3; 4; 5i, h1; 2; 3; 5i). The network structure subtriangles h1; 2; 5i, h2; 3; 5i, and h1; 3; 5i. As a conse- in the real R3 space can be best seen in the plot of the 2-walls quence, there are six parallel semi-infinite 1-walls corre- as shown in Fig. 5. sponding to h1; 2i, h1; 3i, h1; 5i, h2; 3i, h2; 5i, and h3; 5i. Next, let us place the fifth vacuum h5i on the plane On the other hand, the nonplanar structure is originated including the bottom triangle h1; 2; 3i, say from the vacuum h4i which is placed off the bottom triangle. In addition to the semi-infinite 1-walls correspond- ing to the edges h1; 4i, h2; 4i, h3; 4i, there is the compact 1 4 5 m5 ¼ 0; 0; − pffiffiffi : ð4:33Þ 1-wall of h ; i. Comparing the 2- and 3-wall structures in 6 Figs. 5 and 6, we see that the lower 3-walls is pushed down

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FIG. 4. The grid diagrams (Σ space): (a),(b) show the two different patterns of five vacua in N ¼ 5. toward infinity, which corresponds to the fact that the intact. Then the fifth vacuum is placed at opposite to h4i subtetrahedron h1; 2; 3; 5i gets squashed flat. with respect to the triangle h1; 2; 3i as Let us next study the second type with the dypyramid structure as Fig. 4(b), which can be obtained by further 5 5 pushing down the vacuum h i from the last case in Fig. 6. m5 ¼ 0; 0; − pffiffiffi : ð4:34Þ As before, the first four masses m1, m2, m3, and m4 are 6

FIG. 5. Exact solutions in three typical branches for the grid diagram in Fig. 4(a). Isosurfaces of the topological charge densities Wd for d walls (d ¼ 0, 1, 2, 3) are shown.

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FIG. 6. Exact solution for the fine-tuned grid diagram. Isosurfaces of the topological charge densities Wd for d walls (d ¼ 0,1,2,3) are shown.

A useful choice of the moduli matrix with one moduli The middle row of Fig. 7 corresponds to a45 ¼ 0 where parameter a45 is given by the strengths of all the vacua are comparable. Namely, all the vacuum domains meet at the origin. Accordingly, a45 a45 H0 ¼ð1; 1; 1;e ;e Þ: ð4:35Þ the inner 2-wall corresponding to the triangle h1; 2; 3i disappears. The parameter a45 controls the weight of the vacua h4i and As increasing a45 further, the vacuum domains h4i and h5i relative to that of h1i, h2i, and h3i. We show three h5i expand and directly meet to create new 1-wall h4; 5i;see typical solutions with a45 ¼ −4, 0, 3 in Fig. 7. the third row of Fig. 7. In this case, the whole dypyramid can The first row corresponds to a45 ¼ −4. The h4i and h5i be thought of as the sum of three subtetrahedrons h1; 2; 4; 5i, domains are relatively weak, whereas the domains of h1i, h1; 3; 4; 5i,andh2; 3; 4; 5i. This can be seen in the 2- and h2i, and h3i stick out and directly meet to form a 2-wall 3-walls depicted in the third row of Fig. 7. corresponding to the inner triangle h1; 2; 3i: The dypyramid Finally, we consider a rare case that the four vacua are is divided into the upper and the lower (upside down) placed on a plane similarly to Fig. 6. We now put the four tetrahedrons. Correspondingly, the configurations (1-, 2-, points on a plane so that they form a ; see Fig. 8. 3-walls) are constructed by joining two tetrahedral Namely, the five vacua form a square pyramid. As an solutions at the z ¼ 0 plane. example, our choice is

FIG. 7. Exact solutions in three typical branches for the grid diagram in Fig. 4(b). Isosurfaces of the topological charge densities Wd for d walls (d ¼ 0, 1, 2, 3) are shown.

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FIG. 8. Exact solutions in three typical branches for the fine-tuned grid diagram. Isosurfaces of the topological charge densities Wd for d walls (d ¼ 0, 1, 2, 3) are shown.

m1 ¼ð2; 2; 0Þ; ð4:36Þ When ða14;a23Þ¼ð0; 10Þ, the influence of h2i and h3i is the maximum, so that their domains contact 2 3 m2 ¼ð2; −2; 0Þ; ð4:37Þ and form the 1-wall h ; i, as shown in the third row of Fig. 8. m3 ¼ð−2; 2; 0Þ; ð4:38Þ

m4 ¼ð−2; −2; 0Þ; ð4:39Þ

m5 ¼ð0; 0; 4Þ: ð4:40Þ

The moduli matrix useful for this case is

a14 a23 a23 a14 H0 ¼ðe ;e ;e ;e ; 1Þ; ð4:41Þ where a14 corresponds to the weights of h1i and h4i 2 whereas a23 corresponds to the weights of h2i and h3i. We show three typical solutions with ða14;a23Þ¼ð10; 0Þ; ð0; 0Þ; ð0; 10Þ in Fig. 8. In the first row of Fig. 8, the first solution with ða14;a23Þ¼ ð10; 0Þ is shown. Since the weights of h1i and h4i are greater than those of h2i and h3i, the former domains stick out and directly meet to form the 1-wall h1; 4i. FIG. 9. Exact solutions of domain wall networks for the model with N 6. The first line shows the grid diagrams, and the When a14;a23 0; 0 , all the vacua have the same ¼ ð Þ¼ð Þ W influences, so that they meet at a point; see the middle row second line shows isosurfaces of 2 for the corresponding solutions. (a) The grid diagram is a tetrahedron and it has two of Fig. 8. inner vertices. (b1) and (b2) are two different type of the networks for the dipyramid-form grid diagram with one inner vertex. 2 The parameters a14 and a23 are not physically independent (a) has two vacuum bubbles, and (b1) and (b2) have one vacuum moduli due to the V transformation (3.20). bubble.

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FIG. 10. Exact solutions of domain wall network in the octahedral grid diagram (N ¼ 6) without inner vertices. We superpose the two kinds of graphs, the one is the W2 in the x space and the other is the grid diagram in the Σ space. There are three orthogonal directions to deform the network. The figure at the center shows the network where all the six vacua have the same influences. (a1) [(a2)] is obtained when the weight of h1i and h2i is larger (smaller) than the other vacua. Similarly, (b1) and (b2) [(c1) and (c2)] transit by controlling the relative influence of h5i and h6i [h3i and h4i].

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A difference of the two cases ða14;a23Þ¼ð10; 0Þ and controls the size of the bubble h6i. We choose ða123;a6Þ¼ (0,10) is just the difference of dividing the square pyramid ð10; 10Þ in Fig. 9(b1) and ða123;a6Þ¼ð−2; 4Þ in Fig. 9(b2). into two tetrahedrons. If we only look at the square face, it Finally, we show the octahedral case in Fig. 10.For is a transition between the s and t channels found for a simplicity, we set the eight masses mA at vertices of a planar 1-wall network, discussed in Ref. [75]. Therefore, regular octahedron. This configuration can be understood this sort of cutting the square pyramid into two tetrahedrons as follows. To understand this configuration, we arrange the is a three-dimensional analog of the transition between the six vacua to the following three pairs: h1i-h2i, h3i-h4i, and s and t channels in two dimensions. One can be convinced h5i-h6i: Then, the appropriate moduli matrix is by looking at the 2-wall (but not 1-wall) configurations in a12 a12 a34 a34 a56 a56 Fig. 8 from above (or bottom). H0 ¼ðe ;e ;e ;e ;e ;e Þ; ð4:44Þ

where only two among a12, a34, a56 are independent. D. N = 6: Octahedron When the weight of the vacua h1i and h2i are larger than Let us further increase the number of vacua, namely, the other vacua, the vacua h1i and h2i directly meet to N ¼ 6. There are six vacua. There are three different cases form the 1-wall h1; 2i. It corresponds to (a1) of Fig. 10 in according to the convex polyhedra made by connecting the which the vertices h1i and h2i are connected by the dashed vacua. The first case is a tetrahedron with two inner segment, and at the same time there is an inner loop of the vertices, the second is a dipyramid with a inner vertex, 2-wall penetrated by the segment. As the weight relatively and the third is a octahedron without inner vertices. decreases, the loop shrinks. When a12 ¼ a34 ¼ a56, all the The tetrahedral case is shown in Fig. 9(a). The whole vacua are equivalent and there are no 2-wall loops, as tetrahedron is divided into eight subtetrahedrons h1; 2;3;6i, shown in the center panel of Fig. 10. If we further reduce 3 4 5 6 h1; 2; 4; 5i, h1; 3; 4; 5i, h2; 3; 4; 6i, h1; 2; 5; 6i, h2; 3; 5; 6i, a12 as a12 a12 ¼ a34 (a56 a12 ¼ a56 (a34

a5 a6 spond to the decomposition of the octahedron into four H0 ¼ð1; 1; 1; 1;e ;e Þ; ð4:42Þ subtetrahedrons. On the other hand, (a2), (b2), and (c2) correspond to the decomposition of the octahedron into two a a 10 a a with 5 ¼ 6 ¼ . The moduli parameter 5 ( 6) controls square pyramids. the size of the bubble of h5i (h6i). For the dipyramid type case, there are two branches V. PLATONIC, ARCHIMEDEAN, according to how to divide it into subtetrahedrons. The first CATALAN, AND KEPLER-POINSOT branch is shown in Fig. 9(b1) in which the whole dipyramid VACUUM BUBBLES is divided into five tetrahedrons, h1; 2; 3; 6i, h1; 2; 4; 6i, h1; 3; 4; 6i, h2; 3; 4; 6i, and h1; 2; 3; 5i. It can be also In the previous sections, we have seen several examples regarded as the octahedron made of two tetrahedrons in which the configurations have vacuum bubbles; see bonded at the surface h1; 2; 3i: The upper tetrahedron Figs. 5 and 9. They can appear when the grid diagrams has the inner vertex, whereas the bottom one has no inner which are generally convex polytopes have inner vertices. vertices. The resulting 2-wall wire frame shown in the The number of the inner vertices is equal to the number of lower figure of Fig. 9(b1) can be indeed obtained by the vacuum bubbles. The size of a vacuum bubble is connecting the 2-walls of Fig. 2 and the third row of Fig. 5. controlled by a modulus. The bubble is sometimes invisibly On the other hand, the whole dipyramid is divided into small, and we need to take the corresponding moduli six subtetrahedrons h1; 2; 4; 6i, h2; 3; 4; 6i, h1; 3; 4; 6i, parameter sufficiently large to broaden the bubble. h1; 2; 5; 6i, h2; 3; 5; 6i, and h1; 3; 5; 6i in the second branch; see Fig. 9(b2). A. The Platonic vacuum bubbles The useful moduli matrix for describing this transition Here, we are interested in the shape of the vacuum turns out to be bubble. In Fig. 5, we met the bubble which is a regular tetrahedron. There are two conditions for a regular tetra- a123 a123 a123 a6 H0 ¼ðe ;e ;e ; 1; 1;e Þ: ð4:43Þ hedral bubble to exist. One is that the grid diagram is a regular tetrahedron, and the other is that the inner vertex The moduli parameter a123 controls strength of the vacua is located at the center of the regular tetrahedron. If we h1i, h2i,andh3i: In other words, it is related to distance relax either or both conditions, the bubble deforms between the 3-walls h1; 2; 3; 5i and h1; 2; 3; 6i.Theothera6 accordingly.

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Platonic solids

Platonic bubbles

Platonic wire frames

FIG. 11. The top row shows five grid diagrams which are congruent with the Platonic solids. The inner vertices at the center are hidden. The second row shows the vacuum bubbles for each grid diagrams, and the third row shows the 2-wall charge densities W2. The bubble shape and the grid diagram are dual each other.

Now, one notices that the two regular tetrahedrons in the For example, we consider N ¼ 6 þ 1 for the octahedron grid diagrams and in the real space in Fig. 5 are upside with the inner vertex. Let the first six flavors (A ¼ 1; 2; down. This should be so, since each face of the bubble is …; 6) correspond to the vertices of the octahedron and perpendicular to the inner edge of the grid diagram. This the seventh flavor (A ¼ 7) to the inner vertex. To get a can be rephrased as follows. The shape of the vacuum vacuum bubble of a desired shape, we consider the moduli a7 bubble is equivalent to a shape obtained by exchanging the matrix H0 ¼ð1; 1; 1; 1; 1; 1;e Þ. Then we take a suffi- outer vertices and faces of the grid diagram. Such a ciently large value for ea7 to broaden the vacuum bubble is called dual of the original polyhedron. h7i: The result is shown in the second column from the left Since the regular tetrahedron is known to be self-dual, of Fig. 11. We indeed observe that the resultant vacuum the grid diagram and the vacuum bubble are both regular bubble takes a shape of a regular , dual to a regular tetrahedrons. We show the regular tetrahedron as the grid octahedron. diagram, the dual tetrahedron as the vacuum bubble, and an Similarly, we take N ¼ 8 þ 1 for a regular cube plus an isosurface of the 2-wall density in the leftmost column inner vertex, N ¼ 20 þ 1 for a regular plus in Fig. 11. an inner vertex, and N ¼ 12 þ 1 for a regular To understand the duality better, let us next consider plus an inner vertex. We find that the shapes of the domain wall network in a shape of the other convex regular vacuum bubbles are a regular octahedron, icosahedron, polyhedra: octahedron, cube, icosahedron, and dodecahe- and dodecahedron for the regular cube, dodecahedron, and dron. Namely, we consider the grid diagrams which are icosahedron, respectively; see Fig. 11. Thus, putting an congruent with the Platonic solids and put an inner vertex at inner vertex at the centers of the Platonic solids and their centers. broadening the corresponding vacuum give us the regular

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Archimedean solids

Catalan bubbles

Catalan wire frames

FIG. 12. The top row shows 13 grid diagrams which are congruent with the Archimedean solids. The inner vertices at the center are hidden. The second row shows the vacuum bubbles for each grid diagrams, and the third row shows the 2-wall charge densities W2. The bubbles are dual to the grid diagram, so that their shapes are congruent with 13 Catalan solids. bubble polyhedra dual to the Platonic solids as the grid Figure 12 shows the vacuum bubbles and the 2-wall diagrams. charge densities for the Archimedean grid diagrams with an Here, we equally set all the weights of the outer vertices additional inner vertex at the center. As desired, the bubble to be 1 to have the regular polyhedrons. Of course, if shapes are dual to the grid diagrams, namely, we indeed we choose a generic moduli matrix, the vacuum bubbles have the Catalan bubbles. Let us explain how to construct deform accordingly, as mentioned at the beginning of this these bubbles by taking the truncated tetrahedron (the subsection. solids at the left-top corners in Fig. 12) as an example. We need N ¼ 12 þ 1 flavors and put 12 masses on 12 B. The Archimedean and Catalan vertices of the truncated tetrahedron. In addition, we put the vacuum bubbles thirteenth mass at the center of the truncated tetrahedron. a13 Let us also test the idea to the other classic polyhedra, 13 Then we take the moduli matrix H0 ¼ð1; 1; …; 1;e Þ Archimedean polyhedra and their duals called the Catalan with ea13 being sufficiently large. This way, we get the polyhedra. The Archimedean polyhedra are uniform poly- triakis tetrahedron as the vacuum bubble, dual to the hedra which are vertex transitive. Therefore, the Catalan truncated tetrahedron. All the rest solids can be obtained polyhedra are face transitive. by similar procedures.

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Catalan solids

Archimedean bubbles

Archimedean wire frames

FIG. 13. The top row shows 13 grid diagrams which are congruent with the Catalan solids. The inner vertices at the center are hidden. The second row shows the vacuum bubbles for each grid diagrams, and the third row shows the 2-wall charge densities W2. The bubbles are dual to the grid diagram, so that their shapes are congruent with 13 Archimedean solids.

Figure 13 shows the vacuum bubbles and the 2-wall polyhedra. Here, we take the small and great stellated charge densities for the Catalan grid diagrams with an dodecahedron. Namely, we consider the grid diagrams additional inner vertex at the center. As expected, the bubble which are congruent with the star polyhedra. Stellating a shapes are dual to the grid diagrams, namely, we have the dodecahedron to a stellated dodecahedron is a three- Archimedean bubbles. Let us explain how to construct these dimensional analog of stellating a to a . bubbles by taking the triakis tetrahedron (the solids at the In order to obtain an intuitive picture, let us first consider left-top corners in Fig. 13) as an example. We need N ¼ two-dimensional domain wall network for a grid diagram 8 þ 1 flavors and put eight masses on eight vertices of the which is congruent with a pentagram. The ten vacua are trikias tetrahedron. In addition, we put the ninth mass at the located on vertices of the pentagram as shown in Fig. 14. center of the trikias tetrahedron. Then we take the moduli The pentagram consists of large (the red five vertices) and 1 1 … 1 a9 a9 matrix H0 ¼ð ; ; ; ;e Þ with e being sufficiently small (the green five vertices) . The five vacua large. This way, we get the truncated tetrahedron as the corresponding to the small pentagon lead to five vacuum vacuum bubble, dual to the trikias tetrahedron. All the rest bubbles. When we equally broaden them by the moduli solids can be obtained by similar procedures. a a a a a matrix H0 ¼ð1; 1; 1; 1; 1;e ;e ;e ;e ;e Þ with suffi- ciently large ea, the bubbles (h6i, h7i, h8i, h9i, h10i) form C. The Kepler-Poinsot vacuum bubbles a stellated pentagon as shown in Fig. 14. So far, we have only studied the convex polyhedra. We are ready to study three-dimensional domain wall Before closing this section, let us briefly mention star networks with the grid diagrams congruent with the small

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equally broaden them, we have 12 vacuum bubbles which form the small stellated dodecahedron as shown in the right-bottom panel of Fig. 15. In short, when the grid diagram is the small stellated dodecahedron, the vacuum bubbles form the great stellated dodecahedron, and vice versa. Although the small and great stellated are not dual to each other (dual of the small stellated dodecahedron is the while that of the great stellated dodecahedron is the great icosahedron), here, we found they exchange via the vacuum bubbles.

FIG. 14. The left panel shows the grid diagram congruent with a VI. SUMMARY AND DISCUSSION pentagram. The right panel shows an exact solution with five vacuum bubbles which have the same influence. The bubbles In this paper, we have proceeded to the study on the form a stellated pentagon. nonplanar BPS domain wall junctions in the Abelian gauge theory with the N Higgs fields HA and the D neutral scalar fields Σm in D þ 1 dimensions, which were recently and great stellated dodecahedron shown in Fig. 15. The proposed in Ref. [86]. In the previous work [86],we former consists of a large icosahedron with 12 red vertices obtained the new exact BPS solutions of the single domain and a small dodecahedron with 20 green vertices. The ver- wall junctions associated with a particular symmetry tices of the small dodecahedron are inner vacua. There- S → S S breaking pattern Dþ1 D ( D is the symmetric group fore, by broadening them with the equal weight, we of rank D)inD þ 1 dimensions. We also needed to impose have 20 vacuum bubbles which form the great stellated S the symmetry Dþ1 in addition to the special relation dodecahedron as shown in the left-bottom panel of between the model parameters (the masses, the gauge Fig. 15. On the other hand, the latter is made of the coupling, and the Fayet-Illiopoulos term) for having the large dodecahedron with the green vertices and the small exact solutions in [86]. icosahedron with the red vertices. Now, the vertices of Generalizing the previous study, the present work has the dodecahedron become the inner vacua. So, when we dealt with the generic network of domain walls including multiple junctions. We have not imposed any particular Kepler–Poinsot solids discrete symmetry in this paper. We have succeeded in solving partially the BPS equations by the moduli matrix formalism and finding all the moduli parameters of the generic domain wall network solutions. While the master equation (3.19) cannot be analytically solved in general, we focused on the infinite Uð1Þ gauge coupling limit in which the model reduces to the CPN−1 model and the BPS equations including the master equation are fully solvable for any moduli parameters. These are the first analytic small stellated dodecahedron great stellated dodecahedron solutions for nonplanar domain wall networks in D dimensions (D ≥ 3). Kepler–Poinsot bubbles As demonstrated on showing how the nonplanar net- works look like, we have studied the CPN−1 model in D ¼ 3 for N ¼ 4, 5, 6 in detail. In the case of N ¼ 4, the solution has only one junction. The solution is similar to one found in [86], but is more generic. The solutions in this paper can be obtained for any grid diagrams congruent with tetrahe- dra in contrast to the previous case [86] in which the grid diagram was limited to the regular tetrahedron. The net- work structure appears for N>4.ForN ¼ 5,wehave great stellated dodecahedron small stellated dodecahedron found two different types of networks in general. The first FIG. 15. The top row: the two regular star solids categorized type has a vacuum bubble (a compact domain) surrounded into the Keplar-Poinsot solids are used as the grid diagrams. The by the four semi-infinite vacuum domains. The other type bottom row: the corresponding vacuum bubbles. The small does not have any bubbles but all the five vacuum domains (great) stellated dodecahedron yields the great (small) stellated are semi-infinitely extended. The corresponding grid dia- dodecahedral bubble. gram is the tetrahedron with an inner vertex for the former

105020-18 EXHAUSTING ALL EXACT SOLUTIONS OF BPS DOMAIN … PHYS. REV. D 101, 105020 (2020) and the dipyramid for the latter. We have shown the dimensional domain wall networks in more details network shape is controlled by one moduli parameter. elsewhere. We also have studied the special cases in which the four Second, we only have studied the Abelian gauge theories vertices are on a plane while the fifth vertex is off the plane. and the massive CPN−1 model as the infinite gauge coupling In the N ¼ 6 case, there are three different types in general. limit in this paper. The non-Abelian generalization was With respect to the grid diagram, they correspond to the obtained in D ¼ 2 [75,76]. We will study non-Abelian non- tetrahedron with two inner vertices, the dipyramid with planar domain wall networks in higher dimensions D ≥ 3 one inner vertex, and the octahedron without any inner elsewhere. vertices. Accordingly, there appear two, one, and no Third, we have met the polyhedra and well-known vacuum bubbles in the networks, respectively. As in the mathematical notions like the duality in study of the case of N ¼ 5, we have explicitly shown how the network domain walls. While the mathematical solids are sharp shape changes according with the moduli parameters. They objects, the vacuum bubbles found in this work are rounded are essentially controlled by the two moduli besides the off. We expect that the domain wall networks would translations, either the size of the bubble or the distance mathematically be useful for studying such melting poly- between junctions. hedra and polytopes as the case of Amoeba and tropical Finally, we have constructed the beautiful polyhedra geometry discussed in D ¼ 2 [80]. known from ancient times as domain wall networks. We Fourth, the grid diagrams in the Σ space and the have started with the grid diagrams congruent with five networks in x space seem to be very similar to the Platonic solids with an inner vertex at the centers. The Delaunay diagram and the Voronoi diagram appearing in network solutions have single vacuum bubbles correspond- vast area of Science. ing to the inner vertices, and we have found that shapes of We expect our exact solutions of the D-dimensional the bubble are dual to the grid diagrams. In addition to the domain wall networks and the grid diagrams would be regular polyhedra, we have also investigated the semiregular applicable to many areas. Josephson junctions of super- polyhedra, the Archimedean solids. The corresponding conductors [87,88] are one of interesting applications. vacuum bubbles are again dual to the Archimedean, namely, the Catalan solids. Conversely, the grid diagrams congruent with the Catalan solids lead to the bubbles congruent with ACKNOWLEDGMENTS the Archimedean solids. Our final examples have been star This work was supported by the Ministry of Education, polyhedra. We have taken two well-known star polyhedra Culture, Sports, Science (MEXT)-Supported Program for from the Kepler-Poinsot solids, the small and great stellated the Strategic Research Foundation at Private Universities dodecahedrons. The former (latter) has 20 (twelve) inner “Topological Science” (Grant No. S1511006) and JSPS vertices. Accordingly, the same number of the bubbles KAKENHI Grant No. 16H03984. The work of M. E. is also appears in the networks. Interestingly, the bubbles in the supported in part by JSPS Grant-in-Aid for Scientific former case form the great stellated dodecahedron and those Research KAKENHI Grant No. JP19K03839 and by in the latter case form the small stellated dodecahedron. MEXT KAKENHI Grant-in-Aid for Scientific Research Before closing this work, let us make several comments on Innovative Areas “Discrete Geometric Analysis for on future directions. First, although we have established the Materials Design” Grant No. JP17H06462 from the generic formulae for the exact solutions of the domain wall MEXT of Japan. The work of M. N. is also supported in networks in the limit of the CPN−1 model in generic D þ 1 part by JSPS KAKENHI Grant No. 18H01217 by a Grant- dimensions, we only have shown concrete configurations in in-Aid for Scientific Research on Innovative Areas the D ¼ 3 case. For D ≥ 4, the networks become more “Topological Materials Science” (KAKENHI Grant complicated and their deformations by changing moduli No. 15H05855) from MEXT of Japan. parameters are intricate. We will explain such higher-

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