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 density 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.