Topological Solitons in the Supersymmetric Skyrme Model

Home , Chiral model, Skyrmion

JHEP01(2017)014 Springer August 16, 2016 January 4, 2017 : : December 10, 2016 December 27, 2016 : : Received 10.1007/JHEP01(2017)014 Published Revised Accepted doi: , c Published for SISSA by and Shin Sasaki b [email protected] , Muneto Nitta a . 3 1608.03526 The Authors. c Solitons Monopoles and Instantons, Sigma Models, Supersymmetric Eﬀective

A supersymmetric extension of the Skyrme model was obtained recently, which , [email protected] [email protected] Department of Physics, andKeio Research and University, Education Hiyoshi Center 4-1-1, for Yokohama,Department Natural Kanagawa of Sciences, 223-8521, Physics, Japan KitasatoSagamihara University, 252-0373, Japan E-mail: Institute of Modern Physics,Lanzhou Chinese 730000, Academy China of Sciences, b c a Open Access Article funded by SCOAP ArXiv ePrint: vortex it has finitestates action breaking all in supersymmetries. contrast to conventional theory.Keywords: All of themTheories are non-BPS duction yields a kineticdimensions, preserving term supersymmetry. in Euclidean threethe solitons or (instantons) supersymmetric are lower Skyrme constructed dimensions model. in found and by In a Speight. four potential Scherk-Schwarz3-dimensional dimensions, dimensional term theory reduction the in in is which soliton two a then isa 3d performed Skyrmion-instanton 2d an once is theory instanton found to in and first get then which a once a more 2d to vortex-instanton get is obtained. Although the last one is a global Abstract: consists of only theby the Skyrme same term number in of the quasi-Nambu-Goldstone Nambu-Goldstone bosons. (pion) Scherk-Schwarz sector dimensional complemented re- Sven Bjarke Gudnason, Topological solitons in theSkyrme supersymmetric model JHEP01(2017)014 , ], ]. 3 S 17 11 , ]. 10 18 ]. The theory was ]. A different model 2 , 1 14 ]. Although quite a few 4 , 3 5 15 3 – 1 – ]. Sometimes this bound is called a Bogomol’nyi 8 12 limit of low energy QCD [ 8 5 c 13 7 N 9 11 ]. Supersymmetrizing the BPS Skyrme model was attempted in [ ], a withstanding problem is that the binding energies are typically 9 16 1 – , 5 15 ], which may be misleading because the target space of the Skyrme model is 13 4.1 4d pure4.2 Skyrme instanton 3d Skyrmion-instanton 4.3 2d vortex-instanton 3.1 Three-dimensional model 3.2 Two-dimensional model which is not and Kähler hence cannotis, be however, supersymmetrized. not saturable The unless mentioned thewas energy space constructed bound is later, isometric which to is aenergy by 3-sphere bound now [ [ called thewhich BPS however Skyrme is model not as possible it due has to a the saturable fact that its target space is not [ Kähler direction is based on a self-dualto Yang-Mills theory four in five dimensions dimensions, dimensionally andThis reduced in leads turn one to giving searchstate rise for to or a theory an saturates where infinite abound the tower soliton discovered BPS-like of long — energy ago the vector bound. by Skyrmion mesonsbound Faddeev — [ [ [ is The either original a Skyrme BPS model has an energy exactly the baryon inphenomenologically the appealing large- results have beenmodel, achieved see in e.g. the frameworkabout [ of an the order Skyrme of magnitudelarge body too of large, work compared attempting to at lowering experimental the data. binding energies This in motivated Skyrme-like a models. One 1 Introduction The Skyrme model wasmade originally of made a soliton as intaken a a much toy theory more model of serious for pions baryons after with where higher-derivative Witten terms the showed [ baryon that is the soliton, called the Skyrmion, is A BPS property of instanton solutions 5 Discussion and conclusion 3 Scherk-Schwarz dimensional reductions 4 Euclidean solitons or instantons Contents 1 Introduction 2 The supersymmetric Skyrme model JHEP01(2017)014 ], ]. 19 21 , 19 ] dimen- 29 , ] and indeed ] lead to the 28 26 25 not being Kähler – 3 22 ] not much progress S 19 ] in the so-called pure Skyrme 31 , which however is the target space of 1 P C ]. 20 – 2 – -symmetry, supersymmetry preserving dimensional R ]. Finally, even circumventing the auxiliary field problem, 1 30 , in the chiral multiplet will have derivatives acting on it and in F ]; this complexification is inevitable due to nonlinear realization the- 27 ), which in turn gives rise to three new bosonic degrees of freedom called C ]. 21 We derive 3-dimensional and 2-dimensional Lagrangians by supersymmetry preserving It requires an attractive term in the Lagrangian in order for a Skyrmion to be stabilized Thirty years after the problem was laid out in the seminal paper [ Supersymmetrizing the Skyrme model was attempted early on, in the literature [ For a recent work in this direction, however, see [ 1 SS dimensional reductions and in turnfour. construct solitons The in all first dimensions soliton frommodel is two through — the instanton which found isconstruct by exactly a Speight the 3-dimensional [ NG Skyrmion-instanton part in the of once the SS supersymmetric reduced Skyrme theory. model. Finally, we Next, we sional reduction (DR) inan order instanton to in construct three a Euclideanpersymmetry Skyrmion, dimensions. due although Although to our the Skyrmion twistingreductions original is of are formulation rather the possible, breaks see su- e.g. [ not been carried out yet. — i.e. a nontrivial solutiontional to Skyrme the model, virial the equation kineticwill due term induce to a and Derrick’s kinetic the theorem. term Skyrme In in term the the is conven- theory balanced. by performing In Scherk-Schwarz this (SS) paper [ we formulation. The restrictionderivatives to and as the mentioned NG above,the yields submanifold, non-supersymmetric exactly however, the Skyrme eliminates Skyrme model, term.(Dirichlet) the is term, A four the that twist nontrivial time if compared solution to to oneit, the leaving tries auxiliary just field to a equation turn potential will term on simply for eliminate a the standard quasi-NG kinetic bosons. Introducing a superpotential has (NG) submanifold is exactly theSU(2) Skyrme term. to SL(2, The construction isquasi-NG based bosons on [ complexifying ory. Once the quasi-NGcomplicated bosons and are does turned possess on, four the time supersymmetric derivatives, Skyrme again term not is allowing for more a Hamiltonian on constructing the Skyrme termsymmetric was four-derivative made, term until a withoutidea systematic that the investigation of a auxiliary the non-Abeliancould field super- produce nontrivial problem solution a [ supersymmetric tothe nontrivial Skyrme the solution term. non-dynamic to the auxiliary This auxiliary field field was equation equation carried gave a out term in whose Nambu-Goldstone [ turn making it a dynamicalthe field. supersymmetrized Skyrme-like termmaking turned out a to Hamiltonian have formulation fourthe impossible. baby-Skyrme time model derivatives also The [ arrived latertives at [ the attempt same at type supersymmetrizing of Lagrangian with four time deriva- Several problems were encountered along thewas way. remedied The first by problem switching of thea target baby Skyrme space model to care rather than is of taken the ini.e. Skyrme constructing the model. auxiliary the The field, model, next problem one is will that encounter unless the auxiliary field problem; JHEP01(2017)014 (2.6) (2.3) (2.4) (2.5) (2.1) (2.2) , † ), used for ¯ , quasi-NG D † ]. The con- A Φ Φ σ ˙ , 26 α ¯ . D (Φ † i but can be larger ¯ † C K C . Φ ) M ˙ α H ˙ α . Finally, we conclude . N ¯ ) ¯ D D 4 y B ( are then combined with a M SU( Φ A † ∗ α A α ¯ F D T D 2 D † Φ θ A ˙ ' α Φ M ) ¯ , quasi-NG bosons ˙ D α α ) + ˆ † C A ) is a Kählertensor with the indices H, ¯ y ¯ D D C † π / ( ) C Φ N, Φ † , A ] is M ˙ , † α Φ G α ¯ , 26 θψ D (Φ SL( ∈ D B ¯ h D (Φ ) ' MM ¯ Φ ¯ C ) + ]. The potential, Kähler A D α t y ¯ C C ( Tr ) Tr A AB D ), is implicitly defined by – 3 – 35 † 2 A π † , Φ A AB /H f i Φ iσ C Φ Λ 27 , α = G θ ) which means that the number of quasi-NG bosons 4 as (Φ D M,M ) + C d ) K = ¯ y D † A ( ¯ Λ( = exp( H C C Φ Z F A ) θ , π = 4 AB M 1 N we construct the SS dimensionally reduced Lagrangians that 16 d (Φ . The appendix discusses the BPS property of the solitons. ˆ 3 H ¯ D 5 Z ) = ¯ SU( C ) + † = are composed of NG bosons ' y, θ AB Φ ( , Λ ˆ A A H / θ (Φ Φ 4 C d . In section G ]. θ K 2 Z symmetrized pairwise. The chiral superfields Φ 4 ) is a Kählerpotential and Λ 34 d † – ¯ and auxiliary fields D Φ 2) Kählertensor, Λ Z 32 , A , ¯ C, ψ is the complex isotropy group and not necessarily equal to = (Φ L ˆ K H and The superfields Φ The plan of the paper is as follows. We start by reviewing the supersymmetric Skyrme and the (2 is equal to theconstructing number the of supersymmetric NG Skyrme bosons model [ [ fermions Our case of chiral symmetrymetrizations breaking (and falls therefore into the class of maximally realized supersym- Here, in general [ nonlinear sigma model field taking value in the coset relevant for chiral symmetry breaking: where A, B 2 The supersymmetric SkyrmeWe model will begin bystruction reviewing of the fourth-order derivative supersymmetric termsproblem Skyrme in — model, supersymmetry is — found based without on in the the [ auxiliary Lagrangian field theory. All of these instantons are non-BPS states, breakingmodel all in section supersymmetries. we will use towith construct a the discussion lower-dimensional in solitons section in section create a vortex-instanton in the twice SS anisotropically reduced Euclidean two-dimensional JHEP01(2017)014 . 6= 0. (2.9) (2.7) (2.8) 2 (2.10) (2.11) ] F † , µ i ) is deter- † M µ 2.1 † = 0 which is M ν FF µ . F Tr[ in ( M ν M ¯ † D . 2 µ 1 2 ). The equations of ¯ M C 1 , . i † M ν − 4 µ π 2.2 ] AB f 4Λ † ) arise if we do not in- − = 0 = 0 M ν M † µ µ F µ is the pion decay constant. F 2.8 M † † Tr ν µ M π † M F M 1 2 f M ν µ − µ † M µ i − M µ M M ν † µ † ν † µ M and M U µ F M † ν M M h − M M † ν U − µ − µ µ † ∂ M Tr[ 2 U ) Tr µ ) F M † † µ ≡ † µ µ F 2 U M † µ 2 1 † µ M ) F − † ] µ -invariants and thus when restricting to the M † FM ) through the relation ( ) without a superpotential was found explicitly † M ν † M,M µ G 1 2 M – 4 – + ( − U 2.7 † M ν 4 − π − µ µ F f † † M,M ν + Λ( M U ( M M,M † ν 2 µ µ M U FF FF † Tr[ † † M µ 2Λ µ M † µ ν F U i 1 2 + 2 M h ): M F ) = Λ becomes a constant. The effect of adding the kinetic ν M − Tr † F 2 † µ + 2 ] 1 M 1 Tr F ν 2.7 † − † − µ M 2 π 2 F h + M = Λ f 1 † M ν 2 = Λ Tr π ) is a function of µ ] † 2 − − † f † M U µ Λ ) Tr µ M = † 2 = π UU µ ) avoids the auxiliary field problem and hence the auxiliary field equa- Tr M f M M † µ µ M

(4) b M,M 2.7 L ) − M M (4) b -invariant function Λ( † h M,M L G Tr[ Tr[ Tr 2 2 π + Λ( 1 2 f r M,M ∓ − = Λ( ] for the SU(2) case = (4) b The second is the non-canonical branch associated with the non-trivial solutions Two consistent possibilities of solutions to the equations ( 26 L (4) b L The prefactor Λ( NG submanifold, Λ( We note thatbranch. the Although ordinary the second-order term withquasi-NG kinetic fields four term are time turned derivatives is on, doeswhen the not canceled they above cancel are on Lagrangian in simplifies turned general the exactly off when to non-canonical the the Skyrme term However, this term does notto reduce zero and to it the also Skyrme contains term four when time-derivatives. theThe quasi-NG non-canonical fields branch for are the set theoryin ( [ troduce a superpotential.called the The canonical first branch. correspondsstraightforwardly The from to on-shell eq. Lagrangian the ( on trivial the solution canonical branch is obtained The Lagrangian ( tion is algebraic; it is, nevertheless, a nontrivial matrix equation. second term is a highermined by derivative correction. the Themotion Kählertensor Λ for the auxiliary fields are where we have introduced the shortThe notation first term is the ordinary kinetic (Dirichlet) term with two time derivatives while the Then the bosonic part of the Lagrangian is JHEP01(2017)014 ) 3.2 (3.2) (3.3) (3.4) (3.1) ,          4 4 x 4 2 x 4 , R 2 4 , 2 R 4 2 m m sin cos − ) in 4-dimensional Eu- is carried out as follows 4 4 as x x 4 2.10 4 4 2 2 } , , R R SO(4) which will be utilized 4 4 4 2 2 00 0 0 , = 1. Using this notation, the πR m m 2 ∈ , n ) 3 n , reads ν · + 2 4 n , n 4 ,O R · n 2 , x n ) µ 4 , , n a 4 x 4 O a n ∼ 1 1 x x ( 4 , τ R ( 1 n 4 4 , 2 R 4 where we have Wick-rotated the time 1 4 a = 16 for convenience. The Lagrangian ( { 4 2 x , m N 0 / m 3 in ) − n = , 4 2 2 + sin x ) n , ( cos 4 µ − – 5 – O n n 2 · = 1 1 -invariant theory, Λ must be a constant and thus 4 4 µ x ) = x 4 4 µ G n 1 = 1 µ , , R R ( 4 4 2 2 x 1 4 00 0 0 cos sin U ( m m -invariance intact throughout this paper, so the addition and lowered all the indices since the Euclidean metric is n = 0. For simplicity, we will restrict to the NG submani- = with G π n sin cos µ f µ d,b x ∂ (4) 4          L = ≡ µ ) 4 n πR + 2 4 3 runs over the non-compactified dimensions and the matrix , x ( 2 , -invariance is preserved. In the next section, we will perform Scherk-Schwarz O are the Pauli matrices and the fields satisfy G − = 1 a τ a ) = 4 The upshot is thus that the Kählerpotential cannot induce a kinetic term or a po- x ( O coordinate, and SS dimensional reduction along where 3.1 Three-dimensional model We are now readyWe will to use perform the the coordinates first SSDR by compactifying the fourth coordinate. where we have defined just the identity matrix. Weadmits have also a set symmetry Λ under =for the 1 the transformation SS reduction. where NG restricted Skyrme model on Euclidean four-space, tion. Therefore we willthe simply work latter with constant only by the setting fold fourth-order before derivative performing term Scherk-Schwarz and (SS)convenient ignore to dimensional change reduction notation (DR) to and a it 4-vector will prove 3 Scherk-Schwarz dimensional reductions Our starting point willclidean be space. the supersymmetric We will Skyrme keepof model the the ( kinetic term will only induce a constant and hence not affect the equations of mo- above Lagrangian). Ifthe we potential consider is a just a c-number that we cantential ignore. if dimensional reductions in order to induce a kinetic term. (Dirichlet) term thus only had the effect of inducing the potential (the last term in the JHEP01(2017)014 = 1. (3.8) (3.5) (3.6) (3.7) N , the SS · 6=0 N 1, where the Z 1. It is clear , , 0 ∈ 2 , a 1 1. This holds for 4 ) = (SU(2)) is trivial, 3 a N − , m 1 a · 2 , N π . 4      = a = 4 N 2 2 · , 4 4 m N 4 N 3 a a m m m − } N N 2 1) 4 4 a 4 ) π − 4 2 4 N , N R ( 1 and ) 3 4 m N 4 2 − , N x 4 4 + ( a ( + ( m = 2 and used the relation , n N 2 2 4 ) 1 1) 3 ) , ) 2 , 3 4 b m N − 4 N a ( m N N ∂ m ( 1 2 + · , , ) + ) { 4 1 1 a 1 b a ) a , = m 1 4 , N N N N a 2 4 ( 2 m 1) 2 · ( m N − a N N − equal to each other and to – 6 – ( − 2 N − 1 − 2 ) ( , , 2 , a 4 ) = 2 2 4 a a 2 4 4 − m may be metastable or unstable and could decay to 1 by the boundary conditions. We will focus on the compensates a relative minus sign. N m N N m ) that the overall sign of the two integers is physically 2 · 1 1 4 1) ) } πR a are two nonzero Kaluza-Klein (KK) integers describing a 3 m − N N 3.8 + ( = ( ( and N N 1 1 + 2 ( 4 , , 6=0 ,N 4 ·      π 2 4 4 1 4 4 , 2 4 Z m a R 4 4 x m m 4 m N ( ∈ N 2 m ) = ( n 2 4 4 2 πR , − , 4 4 π π R R 4 4 4 = m m 2 } → { , 4 πR 1 + − , d,b 4 (4) 3 = and ,N L m 3 ( d,b N 6=0 3 (4) { Z L ∈ 1 , 4 m Let us note that the lowest energy state comes from the lowest KK mode and thus the A further remark about the KK momenta is in store. Because After the dust settles, we obtain = 0 is distinguished from = 0 while for odd integers, they may decay to the states with 4 4 higher even momentum numbers m both the KK integers.m The states with minimum energylatter are in thus this paper. compactified momenta correspond tofrom the the SS integers reduced Lagrangianunobservable. ( One may naivelythe two think fields that the relative sign could matter, but renaming Then the last termthe in 3-dimensional the Euclidean first theory. linederivative We is term stress a of that kinetic theterm) the term BPS kinetic SS with Skyrme term. reduction a of model prefactor Thisterm the of produces appears is fourth-order the by in the SS KK contradistinction usual reduction mass to (second-order from in the the derivatives ordinary usual case kinetic term. where the potential where we again haveIf defined we the set notation thedimensionally reduced two Lagrangian integers simplifies to which is just an SO(4) transformation of the fields. towers of higher-momentum statesapplied twisted along boundary the conditions (TBC) compactified such circle. that Notice that we have with where JHEP01(2017)014 + + 3 3 i x x (3.9) ( (3.12) (3.11) (3.10) M ∼ N , · 3 i 3 i          x i M M 4 3 . M 3 x 3 2 2 x · M 3 , R 2 3 2 i , 2 R 3 ) 3 i 2 − m 4 m M 4 i M 4 M sin M 3 4 cos 3 ), we get M − R R + ( 2 3 M 2 − 3.7 ) m 3 4 3 i 3 x x 3 3 1 2 i 2 + , , R M R M 2 3 3 2 2 3 , 4 3 ( 00 0 0 M 4 m m 2 R R ). M m 2 4 is an integer; we allow it to be 1 , 2 2 M , 4 3 + m 4 3.6 Z 1 3 2 1 i i R R − , i m ) ) ) 2 2 2 i 4 i 3 ∈ , 1 M M 2 M , 3 x x π 3 3 2 4 2 2 3 1 x ( 3 M 2 , M R , a great simplification occurs , 1 m 3 1 , 2 R m + 4 3 M M 1 M 3 2 , M m 3 ) m m m + ( M − − − 2 − 3 ) 2 m 2 2 4 x , 2 i i = j ) sin i 2 ( 2 2 3 3 cos ) 2 e 4 3 , M M − – 7 – M O M j m − 4 1 3 1 M · R R 1 ( , m i M 2 , 2 3 M M 3 + ( M 3 · ) = 3 x x 3 3 = M i m a 3 4 1 2 1 , , R m ( R 2 ) x 3 1 , 3 again given by eq. ( 2 2 R R 1 , 2 ( , 2 4 00 0 0 cos sin M 4 3 4 3 4 3 m 4 ) m − 3 ( 4 3 1 m R R R R N , R R M 2 m m R R 4 2 1 2 ) − 2 3 , , i 1 sin + cos , 2 3 2 3 m 2 + m 2 3 + ( 2 ) , m m M i 3 4 and          2 2 3 − m + · ) R R ), with 3 2 + + i m 1 M , 3 4 ≡ 3 + 2 1 4 2 · , , m 4 4 3 3 x ) R R 4 2 4 M i 3 4 M ( m 3 ( R R R R ( 2 4 ( = m m R R 1 2 N M , , 1 m 1 2 , 4 ) ( , 2 4 2 4 , πR 4 is a nonzero integer while × 2 2 4 4 3 , R m m + . In the case the two momenta on each compactified circle are equal, 4 4 m 3 2 R m m 2 +2 2 3 R π 6=0 m R 3 π M 3 , 2 2 2 2 = i R 2 Z x 2 2 2 1 π π 2 π ∂ π − , ( R π 1 , 3 2 runs over the non-compactified dimensions and 2 e ∈ 3 O , = − − − + π + m = m ( − 1 , i = 3 d,b = 1 (4) 2 m M i ) = as L d,b ) = 3 (4) 2 3 3 Starting now from the 3-dimensional Euclidean Lagrangian ( x L ( e πR O πR viz. when where making the last two fields independent2 of the circle coordinate). The TBC are then where vanishing in order to get anto anisotropic a SS dimensional compactified reduction momentum (this does not not being correspond quantized, but merely formally to the option of Now we will perform2 another consecutive SS dimensional reduction, but along where 3.2 Two-dimensional model JHEP01(2017)014 (4.2) (4.1) (3.13) ) while the 4 m . 2 = 2 i 3 i ) 2 , 4 4 ) may yield. Again M M 4 4 · 6= 0 and one vanishing M m i 3 M πR = + ( m M − 2 1 , ) = 4 i + 2 4 3 2 1 , 4 ) m M 3 ]. Note that the pure Skyrme 2 M x 3 ( m 2 31 M M ) ∼ j 2 2 + 4 ) + ( 2 M x 1 1 , i i 2 · a ) M i ˆ 1 x M M ) to , 2 2 a 1 M M + ( ( iτ − ( M M 2 3.11 ) − + − − 4 3 1 qηq 2 2 2 2 – 8 – i i R R ) 1 i M 2 3 = 4 ( M M SU(2) as x m 1 1 M U 2 3 · = ˆ + → M M i m 2 4 q 3 4 3 M 3 4 4 3 S R R m ( R R R R 2 4 4 2 4 2 3 4 R m 2 3 m m R π 3 2 2 2 R 2 2 2 π π 2 π R 2 + π + − − . 1. The two signs are obviously not observable in the above La- } 3 = ), which again can easily be compensated by renaming the two fields = d,b ,M (4) 2 4 3.13 3 L m M ). One may however ask whether what consequences a relative sign between ) and use the Ansatz for the Skyrme field ), since the action is classically conformal in said number of dimensions. It is 3.13 ] and the bosonic sector of the NG restricted submanifold of the supersymmetric } → { 2.11 is the identity map from 1 and 3.2 4 31 q = 0. This will simplify the Lagrangian ( ,M In the next section we will construct Euclidean solitons in the above Lagrangians. In order to get a relatively simple Lagrangian with a potential, let us consider the = 3 2 , 4 3 M density ( where theory ( therefore an instanton-like soliton,model first [ constructed inSkyrme [ model are identical.just make a Hence swift the review solution of is the pure directly Skyrme-instanton. applicable Let and us here start we with will the Lagrangian In this section weand will its consider derivatives coming Euclidean from solitons SS in dimensional reduction. the4.1 supersymmetric Skyrme 4d model pureThe Skyrme first instanton and simplest case is to consider a Euclidean soliton directly in the 4-dimensional { 4 Euclidean solitons or instantons We note again thatm the lowest energy statesgrangian ( correspond to thethe lowest two KK momenta modes, onit being the simply first amounts compactifiedthird to line circle a in ( sign eq. ( in front of the last two terms (in the parenthesis) on the terms remain. case where the firstsecond compactification compactification has one has nonvanishing momentum equalm momenta ( However, we can see that this simplification also eliminates the potential, i.e. only derivative JHEP01(2017)014 1 − (4.5) (4.6) (4.7) (4.3) (4.4) ]. which 0 ) = 36 2. s , e , 1 , 2 −∞ = 2 π ( ]; to see this / 0 0 2 3 r η 31 √ [ = 1 ) is 2 64 0 s Z e 4.5 = = 2 0 0 ), see also e.g. [ r η r 3 , S 10 − ( and finally 4 1 2 π µ ) 0 x s = 1. Thus the Euclidean action µ =1/2 =1 =2 ( dη 0 0 0 2 x 2 0 3 8 r r r η , η . , √ ) Z ) 0 + − 2 r ) s = ( 2 0 s 1 π 3 ( η r 2 − η 2 0 3 s 6 η √ ) + iτ 2( − s ( √ 1 + 2 = 16 0 0 2 – 9 – η 2 1 ) 4 1 2 ) √ s r ( ( 2 0 0 η ds η ) = ) = tanh s . The Skyrme-instanton solution is − s ( = Z r ( 0 0 2 1 0 2 η η η π ) s ( = log 0 0 = 16 s 0 η E ds η 1.0 0.5 S - 1.0 - 0.5 Z 2 π and the radius of the 3-sphere is 2 √ we show 4d Skyrme-instanton solutions for various instanton sizes. Notice /r . Skyrme-instanton solutions for various instanton sizes µ 1 x = 16 ) = 0, i.e. the curve in SU(2) passes zero exactly at the values of ) = 1. The topological charge of the instanton is, however, ≡ 0 E ) and the Bogomol’nyi equation is a curve in SU(2) obeying the constraint r µ ∞ S ( ( x η 0 0 4.4 Figure 1 η η In the following, we look for instantons in three- and two-dimensional models. In figure 4.2 3d Skyrmion-instanton Skyrmions in the pureagainst Skyrme expanding theory (without themselves. a kinetic Either or a potential kinetic term) term are unstable or a potential term is needed to and requires a suspension of the Hopf map to getthat to a nontrivial we called the instanton size. The Euclidean action associated with the Skyrme-instanton solution ( where we have used the boundary conditions for the instanton solution: which solves both thetion ( second-order equation of motion derived from the Euclidean ac- where we have introduced where ˆ where can be written as JHEP01(2017)014 . + 3 2 , v (4.8) (4.9) 3 a (4.10) (4.11) N = 0 4 4 α 4 N . δ α 2 3 , δ − 4 3 a N 4 a m N − N = 3 are dimensionless − 3 and mapping 1 α 3 , a N 2 δ . 4 α )-plane and thus we α ) a x 4 δ 3 4 δ + = 0 being black. m 4 N a N N 1 ) which corresponds to ) 1 a N , x γ a c N v ). For simplicity, we will 1 N + x 3.8 2 N N + ( + 2 β 3.7 · b α 2 2 N , where ˜ ) δ α a N bb . We have colored the figures 3 1 δ − ˜ x α a 1 2 N a 4 2 N N 4 a ( N , R N m 2 N − − 1 X + ( − 1 2 2 = α α αβγδ ) ab N 1 ) X δ a a α 2 N 2 x ˜ δ x x ) 1 2 ( abc 2 3 N b a N is a density. The solution calculated in ( d ˜ x − N N 1 6 3 R p · a X d / 3 a = = 1 being white and aa ) N 3 R a | 4 · N N B ) breaks spherical symmetry we have not been 1 N – 10 – ( 4 a = v 4 | ,N i N 4.8 − N 3 corresponds to red, green and blue, respectively. 3 ] N 1 2 α a − X , π/ − [ ,N N 2 2 + B 4 4 2 a aa ) ) , b ˜ a 3 x N N N 2 3 x ( ) 3 3 N ( b d · h N N , π/ ≡ as N Z ab · + v + a 2 summed over and a N 1 = 0 a 1 π 1 aa N 2 N ( N 2 + ( not − 2 color − N α aa θ N E = is 2 ) N − L cubic lattice with a fourth-order stencil. We define the topological charge − ) a 4. We will call the soliton a 3d Skyrmion-instanton and it is very similar a B 2 b a ˜ , x 3 N 2 3 aa 3 N N · , d 1 · N 2 a 1 b , where , N Z N N N 4 ( color = 1 m is rotated such that there is an axial symmetry in the ( iθ 1 4 + ( + + 2 direction for a fixed π α e 2 a α aa = = As we can see from the figure, the solution is a squashed sphere. In order to calculate Since the last term in the action ( For convenience, we will rescale the lengths as x = is then mapped to the lightness with N E E 2 1 S L the where the index figure using a normalized 3-vector iv v the squashing of the solution, let us first define the size of the 3d-Skyrmion-instanton along The solution is showndensities as at isosurfaces their of respective the half-maximum values topological in charge figure and Euclidean Lagrangian able to reduce thedifferential equation equation. of motion We fordifferential therefore equations a turn with single to the 3d-Skyrmion-instanton numericalmethod finite to methods on difference an an and method 81 ordinary in solveof conjunction the the with full 3d-Skyrmion-instanton the partial as relaxation where to a Skyrmion indimensional (Euclidean) the configuration sense space. that it wraps a 3-sphere in the target space and lives in 3- The equations of motion derived from the above action read coordinates. The Euclidean action thus reads the SS dimensionally reducedtheory theory Skyrmion-instantons. and we Afterdimensional call Euclidean performing space, Skyrmions a we in haveconstruct single the 3-dimensional a Lagrangian SS Euclidean density soliton dimensional ( inthe the reduction latter isotropically to Lagrangian reduced 3- with theory, namely the ( two momenta on the circle set equal stabilize the Skyrmions with a finite size (and energy). Here, we have the kinetic term for JHEP01(2017)014 i x ] = E (4.12) (4.13) L [ 91. The . σ 55 , where ˜ i × ˜ x 4 4 4 m R m , 3 3 π } ) and the second R m θ 9499 and 4 . = m q E S ) sin = = 0). r (b) ] = 0 = 2 ( 2 2 9308; the topological charge , B , . f [ i 4 3 σ x m m sin ] = 0 = and the Euclidean Lagrangian θ, E 1 L , B [ 4 σ m , ) cos = i i r ] ] ( X f X X [ [ 2 2 and formally ) ) sin 2 1 3 9499 and . x x α, m ( ( h h = – 11 – ] = 0 s 1 ) sin , B r [ 3 ≡ ( σ ] m f 9992 and the Euclidean action is . X [ cos σ = 0 α, ] ) cos 39 r – numerical ( B f 37 (a) cos { , respectively; the numerical calculation gives = E L M = . 3d-Skyrmion-instanton solution in the once SS reduced 3-Euclidean dimensional theory. X It will again prove convenient to rescale the lengths as 9308. . are dimensionless coordinates. Thewhich anisotropic we dimensional need reduction in inducesAnsatz order a for to the potential construct vortex a [ Euclidean vortex. We will employ the appropriate first one is isotropiconly and the the first second two fields dimensionalthe depend reduction case nontrivially is of on anisotropic the thereduction in second first with the circle only reduction coordinate. way one that with We momentum thus equal ( have momenta ( density 0 4.3 2d vortex-instanton We will now consider the case of two consecutive SS dimensional reductions where the can define the squashing parameter as where we will use the topological charge density Lagrangian density; both at their respectivewith half-maximum squashing values. parameter The measured solution to is a beevaluated squashed numerically sphere as colors are described in the text. Figure 2 (a) shows the isosurface of the topological charge density and (b) the isosurface of the Euclidean JHEP01(2017)014 × r 4 is a 5 m 3 (4.18) (4.14) (4.16) (4.17) (4.15) f, α 2 m , 3 π sin = 1 4 κ=1 κ=16 κ=1/ 16 f f 2 (16) = sin 2 E + cos 3 S E f df 2 in the last equality. L 4 16. (a) shows the radial / π/ Z 658, . 1 (b) sin , 4 π 1 2 2 2 16 × ) = 1 , 4 κr = ∞ m ( ]. . The action of the solutions are r 3 − = 1 f f E ) 42 m κ L 1 3 ) and it reads – f f π 2 40 and thus cancel, leaving the action , 3.13 1 sin(2 sin κr and the Lagrangian density for various − 2 (1) = . √ 1 κ 5 r E 4 3 dr f 15 10 − − S (0) = 0 and R R + Z 3 4 f Ae 2 r m m π f 1 − r and – 12 – 2 ≡ 2 π 1 10 = κ − f ∼ κ 3 2 θ f -theories, see e.g. [ M cos K 4 r 8 κ=1 κ=16 κ=1/ 16 328. . κ 4 M + × ] with a similar potential term, albeit the kinetic term has a , − 1 4 κ E 4 θ 39 m 6 L – 3 M are the standard polar coordinates in two dimensions and f 3 r m + 37 16. Vortex solutions in this model are somewhat similar to those 3 2 , 2 dr r r M π ˜ 1 x (a) f i , ) Z 4 f 4 + 16 ( and (b) the Euclidean Lagrangian density dr / 2 1 m 16) = are shown the profile function f 3 x / Z sin 3 m (1 = 1 π 3 2 = ˜ 1 2 1 E 2 r π κ S iθ . 2d vortex-instanton solutions for various values of = = = re E E N In figure Notice that although there are two potentially logarithmically divergent terms in the S L 1.5 1.0 0.5 241 and . action, they comeintegral with convergent. coefficients Let us howeverasymptotic examine behavior the of potential the divergence profile in function more is detail. The where we have used the boundary conditions values of in the Skyrme modelnontrivial [ field dependence as in The winding number is defined as where we have defined by plugging the above Ansatz into the Lagrangian ( indeed finite and are5 calculated numerically to be where U(1) modulus. Finally, the 2-dimensional Euclidean action for the vortex system is found Figure 3 profile function JHEP01(2017)014 . As (4.19) 1 -invariance, G , κr √ 4 − ], it is possible to write down e 26 O + 2 κ r + 2 1 Relevant terms Pure Skyrme term, no kinetic term Kinetic and Skyrme terms Kinetic, potential and Skyrme terms κr SU(2)) is kept. If we sacrifice the – 13 – × 1 + κr . Instantons in various dimensions. √ 2 − e 2 A 2 Table 1 ∼ -symmetry breaking potential in the theory. Another possibility, yet E G L ]. Instantons Pure Skyrme-instanton Skyrmion-instanton Vortex-instanton 43 is an undetermined constant. Substituting into the Euclidean Lagrangian 0 > ), we get to leading order (i.e. approaching zero the slowest) at asymptotically -invariance (here it is SU(2) R Dim. 4 3 2 G ∈ 4.15 : A r A comment on supersymmetry of the models is in order. We have performed SS As mentioned above, the existence of a stable soliton necessitates the existence of a symmetric Skyrme model to thehas NG quite subspace. a Although non-linear thethe form solution full on of untruncated the the Lagrangian non-canonical auxiliarytwo and field branch dimensions. perform [ We the pointed SS outthe that dimensional supersymmetric the reductions model. 4d to Skyrme-instantons This three by is Speight and obvious are because not of BPS the in fact that the topological charge is only if then we can make a to be explored, isequation to again [ include a superpotential in thedimensional theory reduction and on solve the the pure auxiliary Skyrme field model which is just a truncation of the super- that this is — to the best of ourpressure knowledge term. — the first Twopossibility global common is vortex to terms with use finite areit the tension. does the potential not kinetic that provide is term a induced potential and by for the the the potential kinetic NG bosons, term; term. but we only mentioned One for that the quasi-NG bosons: this term. First we reviewedmodel. Speight’s instanton Then in we thewhich performed 4-dimensional we constructed Euclidean a a pure Scherk-Schwarz 3-dimensional Skyrme we dimensional instanton that performed reduction looks yet to like another a 3 anisotropicwe squashed constructed SS dimensions sphere. a dimensional in Finally, Euclidean reduction vortex-instanton to with 2 finite action. dimensions, in We would which like to point out different codimensions from two throughmentioned, four. the The supersymmetric results Skyrme arefield model summarized equation lacks in eliminates a it, table. kinetic leavingIn only term this a because paper, potential the term wewhich auxiliary for concentrated the the on supersymmetric quasi-NG the bosons Skyrme theory behind. term restricted is to exactly the equal to NG the submanifold standard only, bosonic for Skyrme where density ( large which leaves the action integral convergent, as promised. 5 Discussion and conclusion In this paper we have constructed solitons in the supersymmetric Skyrme model in three JHEP01(2017)014 2 ], Z 39 – ], by 2 ]. 37 47 ]. A 3d – 52 44 , [ 50 , charge is ]. Instantons 1 51 4 , 49 ]. The vortex- S 48 π 30 × 50 , 3 ) is twisted to make 49 R Z with TBC were con- 1 ]. Difficulties, however, ' ]). The application of bions S 36 54 × , )[ 1 2 53 and that the with TBC in [ R Z 1 [SO(3)] 1 3 ' S S π model [ × 2 2 also with TBC, which for chains of 1- P R 1 C model in [SO(3)] S charge. This was discussed in the context 4 n 4 × π P π 2 C R – 14 – ]. 57 – 55 ]. Analogously to this paper, Scherk-Schwarz dimensional 23 ] if we perform it twice. . Consequently, two instantons annihilate each other and an anti- ]. A lump can be interpreted as a domain wall ring with the U(1) Z 57 , ], where the kinetic term also vanishes when a nontrivial solution of the 52 , 60 52 – , 51 , 51 58 , 30 24 , 23 and hence not A story, similar to the one in this paper, plays out in the supersymmetric baby Skyrme These relations should hold for our 4d instanton and 3d instanton. In four dimensions, Lumps (sigma model instantons) in the In this paper we have performed SS dimensional reductions, which can be obtained as For instance, this has been considered inSkyrme Yang-Mills chains have theory been in constructed in 2 This was used to construct bions, a pair of a kink-instanton and an anti-kink-instanton with zero total 2 Z work of S. B.(Grant No. G. 11675223). was supported The by work the of National M.instanton Natural charge N. (exact Science bion is Foundationto solutions supported resurgence are of has available in China been for extensively the part studied by [ a Grant-in-Aid for reduction can be carried out alsowall in solution that [ case, yielding a kinetic term and possibly a domain Acknowledgments S. B. G. thanks the Recruitment Program of High-end Foreign Experts for support. The are that SU(2) moduliso cannot twisting be twice uniquely should twisted be along equivalent to untwisting. model [ auxiliary field equation is used [ a 3d Skyrmion-instanton ismoduli a should string be with twisted somehow SU(2) toin moduli. induce Helium-3; a a When Shanker monopole we string makea (characterized a by ring producing ring an the instanton SU(2) (characterized by instanton in this paper is somewhat similar to thesestructed two in cases. [ modulus twisted once, and inan small anti-kink compactification limit with it U(1) is modulus decomposed twisted into half a (having kink half and lump charges) [ Skyrmions are well-approximated by thein holonomy of the Yang-Mills principal calorons [ chiralinstanton model is were first constructed interpretedand as on in a the vortex small ring compactificationwith limit, with the it U(1) a is modulus U(1) decomposed twisted modulus into half twisted a (having once vortex half and [ Skyrmion an charges) anti-vortex [ the small compactification limit of the(TBC). compactified This circle with reduction twisted can boundary give conditions a relation between solitonswhich in an different instanton dimensions. (caloron) is decomposed into a set of BPS monopoles. instanton is the same astwo the solitons have instanton the itself. mass Thisalso exactly is possible equal in to to contrast twice show to the thatA BPS mass all brief solitons of the discussion a for instantons on single which discussed thefound soliton. in BPS in Indeed, this appendix. properties it paper of is are the non-BPS instantons solitons. in three and two dimensions is is JHEP01(2017)014 , : ¯ ¯ ξ ξ ξ, (A.5) (A.2) (A.3) (A.4) (A.1) and ) with are the ξ 2 2.8 1 µν δ . ! = 2 are independent of } M , M ¯ ν E ¯ F ξ ¯ ˙ F ¯ 2 σ ! ˙ ¯ 1 , ξ . , ¯ i ξ µ E i M M and σ + F F . { = 0 + 1 2 = 0 , vanishes for some ξ 1 2 iξ iξ M M M = 0 ¯ ¯ F . Mξ ¯ − − F † Mξ ) α ˙ ˙ 2 1 ) 2 M ξ ¯ ¯ 2 ξ ξ ) M 2 i∂ 4 µ i∂ M M , √ i∂ ) ) + − 2 2 1 + M∂ + 1 ∂ = 0 i∂ i∂ µ 3 ∂ ∂ M ∂ ) which satisfy − + ( µ + ( M 2 1 1 + ( ) ∂ 1 2 2 ˙ ]. The BPS states satisfy the condition that ∂ ∂ α 1 , ¯ = Tr ξ = const. The other choices of nonzero compo- i∂ ¯ 25 ˙ i~τ, α Mξ ¯ M + ( + ( – 15 – † α Mξ ) M + − ˙ ˙ F ) ) 1 2 M 4 F ¯ ¯ 1 4 ξ ξ µ E i∂ ∂ = M σ = ( i∂ ( M M F i − ) ) 6= 0. Let us consider the following 1/4-BPS condition µ E + , the fermionic partner of M 2 4 4 = ( 3 σ ) 3 ∂ 2 M √ M i∂ i∂ ( ∂ ¯ ), ¯ M ψ F ( i∂ ) 2 = 0, lead to the condition = − + − , 4 0 = Tr 1 2 3 3 − α M

ξ ∂ ∂ i∂ ) 1 i F ( ( i~τ, ∂ 2 M = − (

ψ ˙ 1 √ 3 ( ¯ 2 = ( ξ ∂ δ ( − √ µ E = = 0, then we have σ 1 = = 2 ξ iξ M M = ¯ − ψ 1 ˙ δψ δ 2 2 and ξ ¯ ξ , result in the same condition. The other possible 1/4-BPS combinations of = ¯ ξ ˙ = 1 2 , ¯ ξ ξ α We are interested in codimension-four instanton-like configurations. For the non- = 0 implies the following condition: 6= 0, ˙ π 1 ¯ The only solution tonent these of conditions is for example The last condition togetherf with the equation of motion of the auxliary field ( each other in Euclidean spaces. canonical branch, we have ξ We stress that the supersymmetry transformation parameters where sigma matrices in Euclidean space. More explicitly, the variation is found to be We remark that solitons inpreserve supersymmetric fractions field of theories supersymmetry. are BPS potentiallyderivative states theories BPS in states have 4-dimensional which been supersymmetric classifiedthe higher- supersymmetry in variation [ of the Strategic Research FoundationNo. at S1511006). Private Universities TheGrant work “Topological for of Science” Young S. Researchers. (Grant S. is supported inA part by Kitasato University BPS Research property of instanton solutions No. 15H05855) and “NuclearAstronomical Matter Observations” in (KAKENHI Neutron GrantEducation, No. Stars Culture, Sports, 15H00841) Investigated Science by from (MEXT) of Experiments thein Japan. part and the by The the Ministry work Japan of of Society M.Research for N. the is (KAKENHI Promotion also of supported Grant Science (JSPS) No. Grant-in-Aid for 25400268) Scientific and by the MEXT-Supported Program for Scientific Research on Innovative Areas “Topological Materials Science” (KAKENHI Grant JHEP01(2017)014 . 4 , x . 3 x ]. is a mass ]. . associated m (1983) 433 G SPIRE (1962) 556 ) in which the IN SPIRE [ IN 31 ]. A.2 B 223 ][ ) where (1961) 127 ]. The combinations 25 . If we combine, for M 25 ( and its dependence in M is independent of 4 (1983) 422 A 260 mG ix M Nucl. Phys. = + , Nucl. Phys. , 3 Light nuclei of even mass number M x B 223 3 ∂ on the right-hand side in the first arXiv:0905.0099 = [ M z F are replaced by Killing vectors Nucl. Phys. 4 , , x Proc. Roy. Soc. Lond. 3 – 16 – , x (2009) 034323 C 80 ), which permits any use, distribution and reproduction in , this leads to the condition that 2 = 0 give lump-lump type configurations. This is true even if iξ Phys. Rev. 2 , = is a holomorphic function of iξ = 0 which results in a vacuum condition of CC-BY 4.0 ˙ 1 charge. Although it is not a BPS solution in the sense that it does A unified field theory of mesonsA and nonlinear baryons field theory ¯ ξ − M 2 M , This article is distributed under the terms of the Creative Commons ˙ 1 F Z 1 ¯ ξ iξ Global aspects of current algebra Current algebra, baryons and quark confinement = ]. = 1 ˙ 2 plane is determined by the source term ¯ iξ ξ 2 − SPIRE , x ˙ . One finds that the Skyrmion-instanton and the vortex-instanton discussed in the 2 1 IN ¯ [ in the Skyrme model 2 ξ The same analysis can be applied even to the lower-dimensional models. In lower The conclusion is that the pure Skyrme-instantons constructed by Speight is not BPS For the 1/2-BPS condition, if we keep two of the four parameters, it inevitably leads x E. Witten, E. Witten, R.A. Battye, N.S. Manton, P.M. Sutcliffe and S.W. Wood, T.H.R. Skyrme, T.H.R. Skyrme, , x [3] [4] [5] [1] [2] 1 References main body of this paper do not belongOpen to this Access. class andAttribution they License are ( thus non-BPSany solitons. medium, provided the original author(s) and source are credited. parameter. These Killing vectorsUsing generate this central fact, charges in itlower dimensions. the is supersymmetry We obvious find algebra. that thatlumps only the possible in above BPS the instantons discussion are masslessx in two-dimensional 1/2 theory, four BPS which dimensions are holds characterized true by in a holomorphic dependence on dimensions, the supersymmetry variationderivatives in of the compactified fermions directions iswith given isometries in by the eq. target space.direction ( induces Namely, a the motion derivative with along respect the Killing to vectors, the e.g. compactified in the supersymmetric Skyrme model.since In they hindsight, carry it isnot obvious preserve that fractions they of cannot supersymmetry,satisfy be the the BPS instantons equation by of Speight motion in and four saturate dimensions an do energy bound on the NG subspace. to the condition example, Therefore codimension-four solitons are inconsistentthat with this the 1/2 is BPS theBPS condition. story configurations We note of for codimension codimension-four two solitons. on the We non-canonical can, branch [ however, find 1/4 and 1/2 the equation. 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