BPS Skyrmions of Generalized SkyrmeModel In Higher Dimensions

Emir Syahreza Fadhilla], and Ardian Nata Atmaja‡, Bobby Eka Gunara[,]∗

] Theoretical High Energy Physics Research Group,

Faculty of Mathematics and Natural Sciences,

Institut Teknologi Bandung

Jl. Ganesha no. 10 Bandung, Indonesia, 40132

and‡Research Center for Physics, Indonesian Institute of Sciences (LIPI),

Kompleks PUSPIPTEK Serpong, Tangerang 15310, Indonesia

email: emirsyahreza@students.itb.ac.id, ardi002@lipi.go.id, bobby@fi.itb.ac.id

August 20, 2021

Abstract

In this paper, the generalized Skyrme model in higher dimensional flat spacetimeis considered. The formulation of spherically symmetric Skyrme model is establishedby using invariants of strain tensor of the field and generalized hedgehog ansatz. Theresult is that there are two independent BPS submodels which are the extensionsof known BPS Skyrme model in four and five dimensions. Some features of bothmodel are described and numerical solutions are shown for some values of spatialdimension.

1 Introduction

The Skyrme model was initially designed to be a model for strongly interacting particles.The construction of this model is based on non-linear sigma model which is slightlymodified by addition of higher order differential term known as the Skyrme term [1, 2].This modification is necessary because in standard four dimensional spacetime the twoterm scale in opposite ways under coordinate scaling ~x → µ~x which means that thesolution of Skyrme model field equations are stable under the conditions of Derrick-typestability [3, 4]. The classical solution of Skyrme model field equations, called skyrmion,is considered as one of the earliest Lorentz-invariant topological soliton of non-linear

∗Corresponding author

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models. The Skyrme model, although eventually failed to become a general theory forstrong interactions, has many uses, one of which is to be an effectively good model forstrong interactions in low energy regions. This low energy scheme is quite interesting tobe studied extensively because there is no scheme through QCD that can predict hadronicphenomena in low energy regions [5]. The growing interest on Skyrme model emergedafter the discovery of correspondence on large Nc limit between such effective field theoryin low energy regime and general formulation of QCD [6, 7].

Although Skyrme formulated this model as a model for strong interactions, currentlySkyrmion is a phenomenon that can be encountered in many other non-linear systems,especially at low dimensions. Examples of skyrmion phenomena that can be found insolids are superfluid polariton [8], Bose-Einstein ferromagnets [9], antiferromagnets [10],magnetic thin films [11], to chiral nematic liquid crystal [12]. In addition, the phenomenonof skyrmion is also quite common in optics such as evanescent waves [13]. This raises anew challenge in the generalization of the skyrme model in any space-time dimension. Todo this, there is a problem with Derrick’s stability conditions because the higher the space-time dimension, the more higher-order terms must be added. The consistent formulationfor this high-order skyrme model has been carried out in [14] and it is found in this waythat terms consistent with the earlier Skyrme formulation can only be carried out up tosextic order for four-dimensional space-time. This generalized model in four dimensionalspacetime with sextic term have been studied in [15, 16, 17, 18], and the same sexticmodel have also been studied in five dimensional case in [19].

As the number of terms grows for a higher dimensional model, it is one of interestto find the term which has special qualitative features. Some example of this approachcan be found in [19] which shows that omitting quartic term does not give any significantqualitative feature in five dimensional model and in [20] which shows that the quartic termis necessary for blackhole solution to be exist. Here, we discuss about which term givesa BPS submodel for an abitrary dimensional Skyrme model. The BPS Skyrme submodelitself, is a model within the generalized Skyrme model which has the maximum symmetryof the map from spatial space to target space, leading to a linear lower bound of energyin topological degree of the map. It is known that such lower bound exist in four andthree dimensional Skyrme model [21, 22, 23]. A five dimensional BPS Skyrme submodelfor a specific spherically symmetric system also have been found in [24]. Here we usethe strain tensor eigenvalue combinations as the representation of higher order Skyrmeterms which has been proposed in [25] to describe the geometrical properties of fourdimensional Skyrmion as a harmonic map of flat spatial manifold to a three dimensionalsphere manifold.

In this paper we use early alphabets, a, b, c, . . ., as scalar multiplets indices and midalphabets, i, j, k, . . . as spacetime indices. Both indices sets take value from 0 to d withd + 1 is the spacetime dimensionality. i represents the imaginary number basis, i.e.i =√−1 and the combinatorial notation Cp

q reads ”p choose q”. We use a mostly positivemetric for flat spacetime with the following form

ds2 = −dt2 + dr2 + r2dΩ2d−1 (1.1)

with dΩ2d−1 is the metric of d− 1 dimensional sphere manifold, Sd−1.

2

2 The Skyrme Model

Let us first consider the standard Skyrme model action functional in 3+1 dimension givenby

S4 =

∫dx4√−g(γ1g

ij Bai Baj + γ2gijgkl CaikCajl

), (2.1)

with Bai = 12i

Tr(U †σa∂iU

)and Caij = εabcBb

iBcj where U = φ0I2 + i~φ.~σ is an SU(2) valued

chiral field originally proposed by T. Skyrme [1, 2] and εabc is Levi-Civita symbol. The

vector ~σ = (σ1, σ2, σ3) are the Pauli matrices, while (φ0, ~φ) are real scalar field satisfyingO(4) model condition, φaφa = 1, see also for example, [19, 3]. Now, we introduce the straintensor which is defined as D ≡ JJT where J is the Jacobian of generalized harmonic mapφ = (φ0, ~φ). In the recipe given in [25] we can rewrite the two term in the Lagrangian asthe invariants of harmonic map, namely λ2

1 +λ22 +λ2

3 and λ21λ

22 +λ2

1λ23 +λ2

2λ23 with λ2

1, λ22, λ

23

are the eigenvalues of D. We can observe that the use of the strain tensor eigenvaluesreduces the problem of finding the BPS bound into an algebraic problem. We discuss thedetails of this strain tensor for higher dimensional cases in the following subsections.

2.1 Generalization to Higher Dimensions

Suppose we have the eigenvalue ofD from a d+1 dimensional Skyrme Model as λ21, λ

22, . . . , λ

2d.

The corresponding Lagrangian can be written in the most general way as

L = −γ0V −d∑

n=1

γnL2n , (2.2)

where γi ∈ R are coupling constants, V ≡ V (φ) is an arbitrary potential and L2n satisfy

L2 ∝ λ21 + · · ·+ λ2

d , (2.3)

L4 ∝ λ21λ

22 + λ2

1λ23 + · · ·+ λ2

d−1λ2d , (2.4)

...

L2d ∝ λ21λ

22 . . . λ

2d . (2.5)

In four dimensional case (d = 3), this gives the generalized Skyrme model which is just thetraditional model from Skyrme added with BPS-Skyrme term (L6 ∝ λ2

1λ22λ

23) proposed in

[16].By following the prescription given in four dimensional model we can construct L using

L2n ≡1

(n!)2φa1[i1

. . . φanin]φa1[j1. . . φanjn]g

i1j1 . . . ginjn , (2.6)

so that the Lagrangian of the model mentioned above can be written as

L = −γ0V −d∑

n=1

γn

(n!)2φa1[i1. . . φanin]φ

a1[j1. . . φanjn]g

i1j1 . . . ginjn . (2.7)

This is just an O(d+1) model as the generalization proposed in [19] if we take the potentialto be V = 1 − φ0. As the summary, the action functional for d + 1 dimensional Skyrme

3

model takes the form

Sd+1 = −∫dxd+1

√−g

[γ0V +

d∑n=1

γn

(n!)2φa1[i1. . . φanin]φ

a1[j1. . . φanjn]g

i1j1 . . . ginjn

]. (2.8)

The topological charge density of this higher dimensional model is given by (up to anormalization constant)

ρ ∝ εi1...idεa1...ad+1φa1i1 . . . φadidφd+1 . (2.9)

In the next subsection, we cast the above model into the combination of the strain tensorD’s eigenvalue scheme using so called the generalized hedgehog ansatz.

2.2 Generalized Hedgehog Ansatz

In the rest of the paper we only discuss the higher dimensional Skyrme model in term ofthe generalized hedgehog ansatz case, given in [26]. This ansatz, with a modification onthe basis, takes the form

φ =(

cos ξ,x

rsin ξ

), (2.10)

with x is the coordinate vector on Rd satisfying xTx = r and ξ ≡ ξ(r) is a scalar field.Using this ansatz, the coresponding strain tensor is given by

D =sin2 ξ

r2

[Id +

((ξ′)2

sin2 ξ− 1

r2

)xxT

]. (2.11)

Once we get the strain tensor, we can straightforwardly construct the Lagrangian. First,we need to find the characteristic equation from the eigenvalue equation det(D−λ2Id) = 0,namely,

det

(sin2 ξ

r2− λ2

)Id +

((ξ′)2

r2− sin2 ξ

r4

)(

sin2 ξr2− λ2

) xxT

= 0 , (2.12)

⇒((ξ′)2 − λ2

)(sin2 ξ

r2− λ2

)d−1

= 0 . (2.13)

From the resulting characteristic equation, it is easy to observe that eigenvalues of suchcase is

λ21 = (ξ′)2 , (2.14)

λ22 = λ2

3 = · · · = λ2d =

sin2 ξ

r2. (2.15)

Since Ln’s are just combinations of eigenvalues, then in the generalized hedgehog ansatzwe have a simple expression

L2n =sin2(n−1) ξ

r2(n−1)

[Cd−1n−1(ξ′)2 +

(Cdn − Cd−1

n−1

) sin2 ξ

r2

]. (2.16)

4

3 BPS type Solutions via Bogomolny Equations

Using eq. (2.16) we can construct the effective Skyrme Lagrangian of the form

Leff =

[(ξ′)2

d∑n=1

γnsin2(n−1) ξ

r2(n−1)Cd−1n−1 + γ0V +

d∑n=1

γn sin2n ξ

r2n

(Cdn − Cd−1

n−1

)]. (3.1)

To find the BPS type solution of this model we can use BPS Lagrangian method forSkyrme model which can be found on [27, 28]. Since our metric satisfies

√−g ∝ rd−1,

the method suggests a BPS Lagrangian LBPS = − Qrd−1 ξ

′ with Q ≡ Q(ξ) does not dependexplicitly on the coordinates. At the BPS limit Leff − LBPS = 0 we have[

d∑n=1

γnsin2(n−1) ξ

r2(n−1)Cd−1n−1

](ξ′)2 −

[Q

rd−1

]ξ′ +

[γ0V +

d∑n=1

γn sin2n ξ

r2n

(Cdn − Cd−1

n−1

)]= 0 ,

(3.2)which is the quadratic equation of ξ′. Solving (3.2) for ξ′ and then imposing condition forwhich we have only single solution, it gives

Q2

4r2(d−1)−

[d∑

n=1

γnsin2(n−1) ξ

r2(n−1)Cd−1n−1

][γ0V +

d∑n=1

γn sin2n ξ

r2n

(Cdn − Cd−1

n−1

)]= 0 . (3.3)

Equation (3.3) is nothing but a polynomials in r, thus, by setting every correspondingcoefficient to be zero, we have a set of equations to determine each γn which satisfies theBPS limit. These equations are

γ0γ1V = 0 , (3.4)

(d− 1)γ0γ2V + (d− 1)γ21 = 0 , (3.5)

(d− 1)(d− 2)

2γ0γ2V + γ1γ2

[(d(d− 1)

2− (d− 1)

)+ (d− 1) (d− 1)

]= 0 , (3.6)

...

Cd−1k−1γ0γkV +

∑n+m=k

γnγmCd−1n−1

(Cdm − Cd−1

m−1

)= 0 , (3.7)

...

Q2

4− sin2(d−1)ξ

[γ0γdV +

∑n+m=d

γnγmCd−1n−1

(Cdm − Cd−1

m−1

)]= 0 . (3.8)

From this sets of equations we can deduce that for all d ∈ N there exists only one BPSsubmodel with γ1 = · · · = γd−1 = 0. This submodel is a higher dimensional extensionof BPS-Skyrme model in four dimensions [16]. On the other side, there is another BPSsubmodel if we restrict our case for only even d with coupling constants γ0 = · · · =γ d−1

2= γ d+1

2= · · · = γd = 0, i.e. only γ d

2is non-zero. This submodel is also a higher

dimensional extension of the model in five dimensions introduced in [19]. Solutions ofboth BPS submodels is discussed in the next section subsection.

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3.1 General d ∈ N Case

In this case we only have one BPS submodels as mentioned above. The Lagrangian ofthis submodel is given by

L = γ0V +γd

(d!)2φa1[i1. . . φadid]φ

a1[j1. . . φadjd]g

i1j1 . . . gidjd , (3.9)

which gives field equations as follow(δcb − φcφb

) [∇k

(2γd

(d!)2 gi1j1 . . . gidjdφa1[i1

. . . φadid]

∂

∂φbkφa1[j1

. . . φadjd]

)− ∂

∂φbγ0V

]= 0 . (3.10)

The BPS inequality can be found by introducing energy functional of static case whichsatisfies E =

∫ √−gdxdL. Recall that the second term in (3.9) is just the determinant of

strain tensor, thus our energy functional takes the following form

E =

∫ √−gdxd [γddetD + γ0V ] =

∫ √−gdxd

[(√γddetD ∓

√γ0V

)2

± 2√γ0γdV detD

].

(3.11)In the second term we can transform spacetime invariant measure dxd

√−g to the target

space invariant measure by using√

detD√−gdxd =

√−g |det J | dxd =

√hdϕd where ϕa’s

are coordinates and h is metric tensor determinant of the target manifold. The integrationon whole target manifold is actually B fold of integration on whole Sd, where B is thetopological degree of φ, hence we state the resulting BPS inequality as follows

E ≥ 2B√γ0γdVolSd

⟨√V⟩Sd

, (3.12)

with VolSd is the total volume of compact manifold Sd and 〈a〉Sd denotes the average offunction a on Sd. The BPS limit is satisfied when

√γddetD ∓

√γ0V = 0 which leads to

the Bogomolny equation of this submodel with ansatz (2.10), namely

sind−1 ξ

rd−1ξ′ ∓

√γ0

γd

√V = 0 . (3.13)

We can immediately see that vacuum solution is the trivial solution of this submodel. Thenon-trivial solution needs a specific form of potential, in this work we use the modifiedpion-mass type potential which takes the form V = (1−φ0)N = (1−cos ξ)N with N ∈ R+.Integrating equation (3.13) with mentioned pion-mass type potential, it gives

sind−N(ξ2

)d−N 2F1

(1− d

2,d−N

2;

2 + d−N2

; sin2

(ξ

2

))∓

√2Nγ0

d2γd

(r2

)d+cd = 0 , (3.14)

for d 6= N or

cosd(ξ2

)d

2F1

(1,

d

2;

2 + d

2; cos2

(ξ

2

))∓

√2Nγ0

d2γd

(r2

)d+ c′d = 0 , (3.15)

for special case where d = N with 2F1 is the hypergeometric function while cd and c′d areintegration constants. Equation (3.14) holds for all d ∈ N because the hypergeometric

6

series from 2F1 always terminate at finite term. From equation (3.14) we can find the cut-off value of radius r in which ξ takes vacuum value. This cut-off radius is

rcut−off =

2

[d

d−N

√γd

2Nγ0

Γ( d2)Γ( 2+d−N

2 )Γ(d−N

2 )

] 1d

for 0 < N < d

∞ for N ≥ d

, (3.16)

which implies that Skyrmion distributed in a smaller radius for higher dimensions and thesolution becomes a regular solution for N ≥ d. It is also can be observed that the cut-offradius is asymptotic to 1 as d goes to infinity for compacton solutions. This propertyensures the existence of positive finite energy of compacton solutions in BPS Skyrmemodel for any dimensions. Since equations (3.14) and (3.15) are transcendental in ξ weuse numerical method to find the solutions for some special cases. The first case is theordinary pion-mass potential where N = 1. This case is a higher dimensional extensionof the four dimensional case in [16]. We can see in Figure 1 that all solutions are ofcompacton type for d ≥ 2, in agreement with the special four dimensional case. The

Figure 1: Numerical solution for equation (3.13) with pion-mass type potential

next special case is N = d where the solution start to become a regular solution whichextends to infinity. We demonstrate this feature in Figure 2 where we plot the solutionsin d = 5 spatial manifold for N = 1, 2, . . . , 8. In Figure 3 we show the solution profilesfor some value of d. We can observe that the solutions become closer as d goes to infinitywhich is the same feature we have found in the compacton solutions above. Furthermore,there are no intersections between solutions with different value of d and all solutions aremonotonically decreasing, i.e. ξd ≥ ξd+1 ≥ 0 for r ∈ [0,∞) with ξd is the solution of ξ(r)in d spatial dimension. From these mentioned features and further support by the factthat energy for d = 1 case is finite, we can conclude that the energy in general is positiveand finite for all value of d.

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Figure 2: Numerical solutions for equation (3.13) with modified pion-mass type potentialfor d = 5 spatial manifold and N = 1, 2, . . . , 8.

3.2 Even d Case

For this case there exists two type of BPS submodels. The first one is from the generald ∈ N case which has been described in Section 3.1 and the second one is obtained bytaking only γ d

2to be non-zero. The Lagrangian of the second submodel is given by

L =γd/2

((d/2)!)2φa1[i1. . . φ

ad/2id/2]φ

a1[j1. . . φ

ad/2jd/2]g

i1j1 . . . gid/2jd/2 (3.17)

which gives field equations as follow(δcb − φcφb

)∇k

(gi1j1 . . . gid/2jd/2φa1[i1

. . . φad/2id/2]

∂

∂φbkφa1[j1

. . . φad/2jd/2]

)= 0 (3.18)

Here, we take a slightly different approach to find the BPS inequality of this submodel.We know that from the construction of lagrangian terms via strain tensor eigenvalue wehave

Ld ∝ λ21 . . . λ

2d2

+ · · ·+ λ2d2

+1. . . λ2

d (3.19)

There are Cdd/2 terms in (3.19), hence by employing arithmetic-geometric means inequality,

which reads∑

iwifi ≥∏

i fwii with

∑iwi = 1, to (3.19) with identical wi = 1

Cdd/2

for every

term we can proceed as follow

λ21 . . . λ

2d2

+ · · ·+ λ2d2

+1. . . λ2

d

= Cdd/2

(1

Cdd/2

λ21 . . . λ

2d2

+ · · ·+ 1

Cdd/2

λ2d2

+1. . . λ2

d

)

≥ Cdd/2(λ2

1)

Cd−1(d/2)−1

Cdd/2 . . . (λ2

d)

Cd−1(d/2)−1

Cdd/2

= Cdd/2λ1 . . . λd = Cd

d/2

√detD (3.20)

8

Figure 3: Numerical solutions for equation (3.13) with modified pion-mass type potentialfor some values of d = N .

From here, it is straightforward to comclude that this BPS submodel satisfies a BPSinequality of the form

E ≥ B Cdd/2volSd (3.21)

with BPS limit saturation happens if and only if λ21 = · · · = λ2

d, i.e. all the eigenvaluesare identical. The Bogomolny equation of this second submodel is

ξ′ ± sin ξ

r= 0 (3.22)

Solution and features of this submodel is well-known and has been discussed in detail in[19, 24].

4 Conclusions and Remarks

We have considered the higher dimensional Skyrme model and find two independent BPSSkyrme submodel within the static spherically symmetric ansatz, known as generalizedhedgehog ansatz. The first BPS submodel is the model with only L2d term and potentialterm which can be found in any d ∈ N dimensional spatial manifold. The second BPSsubmodel is the model with only Ld term which can only be found in even dimensionalspatial manifold.

The first BPS submodel has compacton solutions for pion-mass type potential andlower bound of energy which is proportional to topological degree as expected. Theprofile of the solution of this submodel highly depends on the form of potential and thedimension of spatial manifold. This submodel is the higher dimensional extension of theknown BPS Skyrme model in four and five dimensions [16, 24].

The second submodel only has a regular solution which is identical with the one foundin five dimensional case [19]. The solution and the energy of this submodel are scale

9

invariant and the solution is identical for all value of d. BPS limit saturation happens ifand only if all strain tensor eigenvalues are identical.

In a more general case where a wider sets of symmetry is considered with highertopological degree, we expect that BPS features of the first submodel are maintainedbecause the key term, which is the L2d term, is just a product of all eigenvalues. This isfurther supported by the expression of the energy which can only be altered if we chooseanother kind of compact manifold other than Sd as the target space. In contrast with thefirst one, the second submodel needs to satisfy a strong condition where all eigenvalues arethe same. This might lead to a smaller sets of symmetries where BPS solution exist forthis submodel. It is interesting to point out some, or all, symmetries which are compatiblewith this second submodel and such topic will be addressed in future works.

Acknowledgments

The work in this paper is supported by GTA Research Group ITB and WCR RistekBRIN-ITB 2021. BEG is also supported by PDUPT RistekBRIN-ITB 2021 and Riset ITB 2021.

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