Physics Le e s B 769 (2017) 362–367
Con en s lis s a ailable a ScienceDi ec
Physics Le e s B
www.else ie .com/loca e/physle b
BPS submodels o he Sky me model
C. Adam a,∗, J. Sanchez-Guillen a, A. We eszczynski b
aDepa amen o de Física de Pa ículas, Uni e sidad de San iago de Compos ela and Ins i u o Galego de Física de Al as Ene xias (IGFAE), E-15782 San iago de
Compos ela, Spain
bIns i u e o Physics, Jagiellonian Uni e si y, Lojasiewicza 11, K aków, Poland
a i c l e i n o a b s a c
A icle his o y:
Recei ed 23 Ma ch 2017
Accep ed 3 Ap il 2017
A ailable online 5 Ap il 2017
Edi o : A. Ringwald
We show ha he s anda d Sky me model wi hou pion mass e m can be exp essed as a sum o wo BPS
submodels, i.e., o wo models whose s a ic field equa ions, independen ly, can be educed o fi s o de
equa ions. Fu he , hese fi s o de (BPS) equa ions ha e non i ial solu ions, a leas locally. These wo
submodels, howe e , canno ha e common solu ions. Ou findings also shed some ligh on he a ional
map app oxima ion. Finally, we conside ce ain gene alisa ions o he BPS submodels.
©2017 The Au ho s. Published by Else ie B.V. This is an open access a icle unde he CC BY license
(h p://c ea i ecommons.o g/licenses/by/4.0/). Funded by SCOAP3.
1. In oduc ion
The Sky me model [1] is a nonlinea field heo y suppo ing
opological soli on solu ions (“Sky mions”), which finds i s main
applica ion as a low-ene gy e ec i e heo y o QCD [2]. In he
analysis o Sky me models (and ela ed heo ies suppo ing opo-
logical soli ons), wo impo an concep s a e opological ene gy
bounds and he ela ed no ion o Bogomolnyi equa ions [3,4]. In-
deed, some imes i is possible o educe he s a ic field equa ions
o fi s -o de equa ions (“Bogomolnyi equa ions”) such ha he
co esponding solu ions sa u a e he bound. Wi hin he se o gen-
e alised Sky me models (bu all based on he same Sky me field
U∈SU(2)), wo cases o BPS Sky me models sa is ying a Bogomol-
nyi equa ion a e known. The fi s one is he “BPS Sky me model”
consis ing o a e m sex ic in fi s de i a i es and a po en ial [5]. In
his model, Bogomolnyi solu ions exis o a bi a y opological de-
g ee (“ba yon numbe ”) B. The second one consis s o he Sky me
e m (qua ic in fi s de i a i es) and a pa icula po en ial [6].
In his second case, howe e , only he |B| =1solu ions (and, o
cou se, he i ial acuum solu ion) sa u a e he bound and obey
he co esponding Bogomolnyi equa ion. Highe Bconfigu a ions
a e unbound in his model and u n in o ligh ly bound Sky mions
once u he e ms a e added wi h sufficien ly small coupling con-
s an s [7–9]. Bo h hese BPS models a e genuine Sky me models
in he sense ha hey can be ound wi hin he se o gene alised
Sky me model by an app op ia e choice o coupling cons an s.
In he p esen pape we wan o conside a sligh ly di e en
ype o BPS submodels. We will find ha he s anda d Sky me
*Co esponding au ho .
E-mail add ess: adam@ paxp1.usc.es (C. Adam).
model (consis ing o he sigma model e m and he Sky me e m),
oo, may be w i en as a sum o wo BPS submodels, whe e each
o he wo submodels sepa a ely leads o a Bogomolnyi equa ion
and non i ial solu ions (a leas , locally). These submodels, how-
e e , a e no Sky me models on hei own, i.e., i is no possible o
ge jus one o hese submodels by a choice o coupling cons an s
wi hin he se o gene alised Sky me models. In o he wo ds, he
wo submodels a e mul iplied by he same coupling cons an s, and
elimina ing one au oma ically elimina es he o he . Fu he , he
wo submodels ha e no common solu ions (excep o he i ial
acuum solu ion). This mus , o cou se, be expec ed, because i
is known ha , al hough he s anda d Sky me model has a opo-
logical ene gy bound ( he Sky me–Faddee bound [1,10]) and a
co esponding Bogomolnyi equa ion, his fi s -o de equa ion is oo
es ic i e and only allows o he i ial solu ion. Despi e hese
impedimen s, ne e heless, he exis ence o u he BPS submod-
els is o in e es and sheds new ligh on se e al issues wi hin he
Sky me model.
The gene alised Sky me model we wan o conside consis s o
ou e ms. The sigma model (Di ichle ) e m
L2=1
2T (LμLμ), (1.1)
he Sky me (qua ic) e m
L4=1
16T ([Lμ,Lν]2), (1.2)
and he sex ic e m, which is jus he ba yonic cu en squa ed
e m
L6=λ2π4BμBμ,Bμ=1
24π2μνρσ T LνLρLσ,(1.3)
h p://dx.doi.o g/10.1016/j.physle b.2017.04.003
0370-2693/©2017 The Au ho s. Published by Else ie B.V. This is an open access a icle unde he CC BY license (h p://c ea i ecommons.o g/licenses/by/4.0/). Funded by
SCOAP3.
C. Adam e al. / Physics Le e s B 769 (2017) 362–367 363
whe e Lμ≡U†∂μUis he le in a ian cu en . The las e m,
wi hou de i a i es, is he po en ial L0(U) =−μ2U( U)which
p o ides a mass o pionic exci a ions. He e, we assume ha uni s
o ene gy and leng h ha e been chosen such ha he coupling con-
s an s o he Di ichle and Sky me e m a e scaled away (Sky me
uni s). λand μa e, hus, dimensionless coupling cons an s con-
olling he ela i e s eng hs o he co esponding e ms.
The o iginal ma ix SU(2)Sky me field Ucan be decomposed
in o a eal scala ξand complex scala uby
U=exp(iξ
τ·
n)=cos ξ1+isin ξ
τ·
n(1.4)

n=1
1+|u|2(u+u∗), −i(u−u∗), 1−|u|2.(1.5)
Fo opologically non i ial configu a ions he ull a ge space S3
has o be co e ed, which means ha ξshould ake alues in he
ull in e al [0, π]and ushould ake alues in he ull complex
plane C. Each e m o he gene alized Sky me model can hen be
ew i en in his new a ge space coo dina es as
L2=L(1)
2+L(2)
2,L(1)
2≡4sin
2ξuμ¯
uμ
(1+|u|2)2,L(2)
2≡ξμξμ
(1.6)
L4=L(1)
4+L(2)
4,
L(1)
4≡4sin
2ξξμξμuμ¯
uμ
(1+|u|2)2−ξμ¯
uμξμuν
(1+|u|2)2
L(2)
4≡4sin
4ξ(uμ¯
uμ)2−u2
μ¯
u2
ν
(1+|u|2)4(1.7)
L6=λ2sin4ξ
(1+|u|2)4iμνρσ ξνuρ¯
uσ2,L0=−μ2U(ξ). (1.8)
Fo la e con enience we ha e di ided he quad a ic and qua ic
e ms in o wo pa s. Ob iously, wi hin he Sky me model con ex ,
i is no possible o elimina e, e.g., L(1)
2wi hou elimina ing, a he
same ime, also L(2)
2.
A well-known example o a Sky me heo y wi h he BPS p op-
e y is he BPS Sky me model [5], and i will be use ul o e iew i
b iefly. I is defined as
LBPS =L6+L0(1.9)
which a e he field decomposi ion eads
LBPS =λ2sin4ξ
(1+|u|2)4iμνρσ ξνuρ¯
uσ2−μ2U(ξ). (1.10)
The Bogomolnyi equa ion o s a ic field configu a ions is
λsin2ξ
(1+|u|2)2iijkξiuj¯
uk=±μ√U.(1.11)
I is s aigh o wa d o no ice ha he e a e o he possibili ies o
dis ibu e he de i a i es o he fields and hei con ac ions wi h
he epsilon symbol o o m new fi s o de equa ions wi h he
same field con en . The e o e, hey can be ea ed as new Bogo-
molnyi equa ions o he pe inen Sky me-like models. This will
be analyzed in he es o he pape .
2. The fi s BPS submodel
L(1)
2and L(1)
4can be combined in o he ollowing exp ession
L(1)=4sin
2ξuμ¯
uμ
(1+|u|2)2+(ξμuμ)(ξν¯
uν)−ξ2
μ(uν¯
uν)
(1+|u|2)2.
(2.1)
The ene gy o he s a ic case eads
E(1)=4d3xsin2ξ
(1+|u|2)2ui¯
ui+(iijkξjuk)(−iimnξm¯
un)(2.2)
=4d3xsin2ξ
(1+|u|2)2(ui±iijkξjuk)(¯
ui∓iimnξm¯
un)
∓8d3xisin2ξ
(1+|u|2)2ijkξiuj¯
uk
≥8d3xisin2ξ
(1+|u|2)2ijkξiuj¯
uk=8π2d3xB0=8π2|B|
whe e he bound is sa u a ed o solu ions o he ollowing Bogo-
molnyi equa ion
ui±iijkξjuk=0 (2.3)
and i s complex conjuga ion. No e ha his equa ion implies some
cons ain s o he fields, namely
uiξi=¯
uiξi=0,u2
i=¯
u2
j=0.(2.4)
I is in e es ing o conside he pa icula solu ions p o ided by
he ansa z in sphe ical pola coo dina es ξ=ξ( ), u =u(θ, ϕ). This
ansa z au oma ically sa isfies he fi s cons ain uiξi=0. To sim-
pli y he second cons ain , i is use ul o use he s e eog aphic
p ojec ion om he uni sphe e spanned by (θ, ϕ) o he plane
spanned by (x, y), say, whe e z≡x +iy = an(θ/2)eiϕ, because he
me ic on he uni sphe e is con o mally fla in he coo dina es
(x, y), such ha he cons ain u2
i=0 simplifies o uzu¯
z=0, i.e.,
umus be ei he holomo phic o an i-holomo phic in he complex
coo dina e z. The sign choice in he Bogomolnyi equa ion ( oge he
wi h he bounda y condi ions imposed on ξ) de e mine whe he u
is holomo phic o an i-holomo phic.
I is ins uc i e o inse he ansa z in sphe ical pola coo di-
na es di ec ly in o he ene gy unc ional. In a fi s s ep we ge
ijkξjukimnξm¯
un=ξ2
jui¯
ui, such ha bo h e ms a e p opo ional
o ui¯
ui. Nex , using he me ic in hese coo dina es
ds2=d 2+ 2ds2
S2,ds2
S2=4
1+x2+y2(dx2+dy2)(2.5)
we ge o he olume elemen
dR3=d 2dS2,dS2=2i
(1+z¯
z)2dzd¯
z(2.6)
and, u he ,
ui¯
ui≡gijui¯
uj=(1+z¯
z)2
2 2(uz¯
u¯
z+u¯
z¯
uz). (2.7)
As a esul , he ene gy unc ional ac o ises,
E(1)=2E(1)
ξE(1)
u(2.8)
whe e
E(1)
u=dS2
(1+z¯
z)2
(1+u¯
u)2(uz¯
u¯
z+u¯
z¯
uz)(2.9)
is jus he CP(1) (non-linea sigma) model on S2. I s fini e en-
e gy solu ions a e p o ided by all holomo phic (posi i e wind-
ing numbe ) and an i-holomo phic (nega i e winding numbe ) a-
ional unc ions. These solu ions sa u a e he Bogomolnyi bound
E(1)
u≥4π|N|, whe e he winding numbe Nis gi en by he deg ee
o he a ional map.
364 C. Adam e al. / Physics Le e s B 769 (2017) 362–367
The second ene gy unc ional is
E(1)
ξ=d sin2ξ(1+ξ2)(2.10)
and allows o an almos i ial Bogomolnyi bound,
E(1)
ξ=d sin2ξ(1∓ξ)2∓2ξ≥2d sin2ξξ
=2
π

0
sin2ξdξ=π(2.11)
whe e we used he bounda y condi ions ξ(0) =π, ξ(∞) =0. The
co esponding Bogomolnyi equa ion is
sin2ξ(ξ2±1)2=0,(2.12)
and he solu ion wi h he igh bounda y condi ions is he com-
pac on
ξ( )=π− o 0 ≤ ≤π
0 o >π.(2.13)
The fi s de i a i e o he compac on is no con inuous a he com-
pac on bounda y =π, bu he ene gy densi y is con inuous, and
bo h he Bogomolnyi equa ion and he ull second-o de Eule –
Lag ange (EL) equa ion hold e e ywhe e in space o his solu ion,
owing o he p esence o he ac o sin2ξ. I may be checked eas-
ily ha he ba yon numbe Bis equal o he winding numbe N
o his ansa z. Finally, he ene gy is E(1)=2 ·π·4π|N| =8π2|B|,
as i mus be.
In pa icula , i ollows ha his submodel has BPS solu ions
o a bi a y a ional maps u =R(z) =p(z)/q(z)(whe e pand q
a e polynomials wi hou common di iso ). This is in e es ing, be-
cause a ional maps ha e been employed o cons uc app oxima e
solu ions o he ull s anda d Sky me model L2+L4[11]. Ra io-
nal maps canno be genuine solu ions o his model, because he
a ional map ansa z is incompa ible wi h he EL equa ion esul -
ing om he e m L(2)
4(excep o |B| ≤1). Inse ing he a ional
map ansa z di ec ly in o he co esponding ene gy unc ional, ne -
e heless, defines a es ic ed a ia ional p oblem. The ene gy now
depends on he pa icula a ional map, and minimisa ion leads o
a ional maps wi h in e es ing disc e e symme ies (e.g. he sym-
me ies o pla onic solids) o |B| ≥3. Fu he , hese symme ies
ag ee wi h he symme ies o he ull nume ical solu ions [12], and
also he ene gies o he co esponding a ional map app oxima-
ions a e a he close o he ene gies o he nume ically calcula ed
Sky mions. He e we may conclude ha he a he good quali y o
he a ional map app oxima ion may be unde s ood om he ac
he model has a BPS submodel which is exac ly sol ed by a ional
maps, and he e is only one e m (L(2)
4) which p e en s a ional
maps om being exac solu ions.
3. The second BPS submodel
L(2)
2and L(2)
4can be combined in o he exp ession
L(2)=ξμξμ+4sin
4ξ(uμ¯
uμ)2−u2
μ¯
u2
ν
(1+|u|2)4.(3.1)
The ene gy unc ional o s a ic configu a ions eads
E(2)=d3xξ2
i+4sin
4ξ1
(1+|u|2)4(iijkuj¯
uk)2(3.2)
=d3xξi∓2isin2ξ
(1+|u|)2ijkuj¯
uk2
±4d3xisin2ξ
(1+|u|)2ijkξiuj¯
uk
≥4d3xisin2ξ
(1+|u|)2ijkξiuj¯
uk=4π2d3xB0=4π2|B|.
The bound is sa u a ed o solu ions o he Bogomolnyi equa ions
ξi∓2isin2ξ
(1+|u|)2ijkuj¯
uk=0.(3.3)
Obse e ha he Bogomolnyi equa ions lead o some cons ains o
he fields,
uiξi=¯
uiξi=0.(3.4)
Assuming he ansa z ξ=ξ( ), u =g(θ)eimϕ, we ge o u he solu-
ion
u= an θ
2eimϕ,(3.5)
which, o |m| >1, has a conical singula i y along he zaxis. The
ene gy densi y and winding numbe densi y, on he o he hand,
a e smoo h. Fu he , mis equal o he ba yon numbe , B =m, o
genuine Sky mion configu a ions (i.e., whe e ξobeys he co e-
sponding bounda y condi ions). I is in e es ing o no e ha he
BPS Sky me model leads o he same solu ion o u his ansa z.
The wo BPS equa ions (1.11) and (3.3) a e, in ac , e y simila o
his ansa z (iden ical o u, di e en o ξ( )). The esul ing equa-
ion o ξ eads
dξ
d =±m
2sin2ξ. (3.6)
Choosing he minus sign (a nega i e slope o ξ( )), he solu ion is
ξ=a cco ms0−1
(3.7)
whe e s0is an in eg a ion cons an . A →0we ha e co ξ→
−∞, i.e., ξ→π, as desi ed. In he limi →∞, howe e , co ξ
does no app oach ∞, i.e., ξdoes no app oach 0. Ins ead, he
p ofile unc ion ξ akes alues only in he in e al π≥ξ( ) ≥
a cco ms0>0. The local solu ion o he BPS equa ion, he e o e,
canno be ex ended o a solu ion on he ull a ge space, i.e., o a
Sky mion. Ins ead, i leads o a “ ac ional” Sky mion, whe e bo h
he ba yon numbe and he BPS ene gy may ake a bi a y ac-
ional alues, defined by he choice o he in eg a ion cons an s0.
4. Some gene alisa ions
We now wan o conside some gene alisa ions o he wo BPS
submodels, by mul iplying each e m by a ce ain coupling unc-
ion.
4.1. The dila on-YM like model
We conside he ollowing model
˜
L(2)=h2(ξ, u¯
u)ξμξμ+g2
2(ξ, u¯
u)sin4ξ(uμ¯
uμ)2−u2
μ¯
u2
ν
(1+|u|2)4.
(4.1)
The ene gy in eg al o s a ic configu a ions eads
C. Adam e al. / Physics Le e s B 769 (2017) 362–367 365
˜
E(2)=d3xh2ξ2
i+g2
2sin4ξ1
(1+|u|2)4(iijkuj¯
uk)2(4.2)
=d3xh2ξi∓ig2sin2ξ
(1+|u|)2ijkuj¯
uk2
±2d3xig2h2sin2ξ
(1+|u|)2ijkξiuj¯
uk
≥2d3xig2h2sin2ξ
(1+|u|)2ijkξiuj¯
uk
=2π2d3xg2h2B0=2π2|B|g2h2S3
whe e ·S3is he a ge space a e age o he a ge space unc ion
inse ed be ween he b acke s. The bound is sa u a ed o solu ions
o he Bogomolnyi equa ions
ξi∓ig2sin2ξ
(1+|u|)2ijkuj¯
uk=0.(4.3)
No e, ha his Bogomolnyi equa ion is iden ical o he one in he
dila on – SU(2)Yang–Mills model desc ibing a magne ic monopole
[13–16]. Fu he , i leads o he cons ain s (3.4), again.
An example can be p o ided by a pa icula choice o g2and h2,
g2=1
cos2ξ
2
,h2=1.(4.4)
Then, using he ansa z in sphe ical pola coo dina es, again, he
opologically non i ial solu ions a e again (3.5) o u, whe eas ξ=
ξ( )obeys
dξ
d =±2m
2sin2ξ
2.(4.5)
Imposing he ollowing bounda y condi ions: ξ( =0) =0 and
ξ( =∞) =π, he pe inen solu ion is
co ξ
2=m
⇒ξ=2 a cco m
.(4.6)
He e,
g2h2S3=2
π
π

0
dξsin2ξ·1
cos2ξ
2=4 (4.7)
and he ene gy and opological cha ge a e
˜
E(2)=8π2m,B=m.(4.8)
4.2. The holomo phic map like model
Now, we define he model
˜
L(1)=h1uμ¯
uμ
(1+|u|2)2+g2
1sin4ξ(ξμuμ)(ξν¯
uν)−ξ2
μ(uν¯
uν)
(1+|u|2)2.
(4.9)
The ene gy o he s a ic case eads
˜
E(1)=d3xh1ui¯
ui
(1+|u|2)2
+g2
1
sin4ξ
(1+|u|2)2(iijkξjuk)(−iimnξm¯
un)
(4.10)
=d3xh1
(1+|u|2)2(ui±ig1sin2ξijkξjuk)
×(¯
ui∓ig1sin2ξimnξm¯
un)
∓2d3xig1h1sin2ξ
(1+|u|2)2ijkξiuj¯
uk
≥2d3xig1h1sin2ξ
(1+|u|2)2ijkξiuj¯
uk
=4π2d3xg1h1B0=4π2|B|g1h1S3
whe e he bound is sa u a ed o solu ions o he ollowing Bogo-
molnyi equa ion
ui±ig1sin2ξijkξjuk=0 (4.11)
and i s complex conjuga ion. No e ha his o mula implies he
cons ain s (2.4) o he complex and eal fields. An example o
BPS Sky mions o his ype can be ound o
h1=sin2ξ, g1=1
sin3ξ.(4.12)
We assume ha ξ=ξ(z)and u =u(x, y)in ca esian coo dina es
(x, y, z). Then, he scala obeys
g2sin2ξξz=−1⇒ξz=−sin ξ(4.13)
which is he sine-Go don kink equa ion and he e o e
ξ=2a c ane−z.(4.14)
This means ha he complex scala obeys
ui∓iijuj=0 (4.15)
whe e (i, j) ∈{1, 2}and he e o e i is jus a holomo phic (an i-
holomo phic) unc ion in x +iy. Fo example, u =ρmeimϕ, when
cylind ical coo dina es (ρ, ϕ, z)a e used. So, finally we ge holo-
mo phic 2-dimensional soli ons loca ed on a sine-Go don b ane
wi h co-dimension one, and
˜
E(1)=4π2|B|1
sin ξS3=16π|B|,B=m.(4.16)
5. Summa y
We ound he in e es ing esul ha he gene alized Sky me
model can, in ac , be exp essed as a sum o h ee BPS submod-
els,
LSk =L2+L4+L6+L0
=L(1)
2+L(1)
4+L(2)
2+L(2)
4+(L6+L0)
≡L(1)
BPS +L(2)
BPS +LBPS.(5.1)
In compa ison wi h he BPS Sky me model, howe e , he new BPS
submodels e eal some impo an di e ences.
1. The new BPS submodels canno be ob ained as a ce ain limi
(pa icula alues o he model pa ame e s) o he ull Sky me
model. Each o hem consis s o wo e ms – one eme ging
om he Di ichle pa and one om he Sky me pa .
2. The new BPS submodels a e comple ely independen o he
pion mass. The po en ial pa o he gene alized Sky me model
only con ibu es o he BPS Sky me model.
366 C. Adam e al. / Physics Le e s B 769 (2017) 362–367
3. While he BPS Sky me model possesses well-beha ed opolog-
ically non i ial solu ions i.e., BPS Sky mions, whose compac
o non-compac na u e depends on he po en ial, he o he
BPS submodels p o ide solu ions wi h di e en cha ac e is-
ics. The second BPS submodel leads o ac ional Sky mions,
i.e., o solu ions wi h a non-in ege ba yon numbe . The fi s
BPS submodel has compac Sky mions as solu ions. Ob iously,
he wo submodels canno ha e common solu ions, because
his would lead o a BPS solu ion o he ull s anda d Sky me
model L2+L4, which a e known no o exis .
4. One cha ac e is ic ea u e o he solu ions o he wo submod-
els is he opposing e ec which hey ha e on he Sky mion
size. Indeed, he model L(1)leads o fini e size Sky mions
(compac ons) whose size, in addi ion, is independen o he
ba yon numbe . The model L(2), on he o he hand, leads o
ac ional Sky mions, i.e., “Sky mions” o “mo e han infini e
size” which do no e en fi in he infini e in e al ∈[0, ∞].
The size o he Sky mions o he ull model L2+L4is a
comp omise be ween hese wo ex emes, i.e., Sky mions wi h
infini e size, which decay algeb aically o →∞.
5. The submodel L(1)leads o Sky mions wi h a shell-like s uc-
u e, whe e he ene gy densi y is ze o bo h a he cen e =0
and ou side he compac on bounda y and akes i s maximum
alue a some nonze o adius. This eflec s a known beha iou
o he Sky mions o he ull model L2+L4, which ha e a
shell-like s uc u e, as well [12].
As men ioned al eady, he submodel L(1)is sol ed by a bi a y
a ional maps u(z) o an ansa z ξ( )and u(z)in sphe ical pola
coo dina es, and he same “ansa z” p o ides a he good app oxi-
ma ions o solu ions o he ull s anda d Sky me model L2+L4.
He e we jus wan o ema k ha inse ing he a ional map ansa z
in o he s a ic ene gy unc ional o he gene alised Sky me model
(5.1) leads exac ly o he same es ic ed a ia ional p oblem o
he a ional map u =R(z). Tha is o say, he es ic ed ene gy
unc ional is
Egen =4πd  2ξ2+μ2 2U(ξ) +2Bsin2ξ(ξ2+1)
+Isin4ξ
21+λ2
4ξ2 (5.2)
whe e he a ional map R(z)mus minimize he unc ional [11]
I=1
4πdS21+z¯
z
1+R¯
R4
Rz¯
R¯
z2(5.3)
exac ly as in he a ional map app oxima ion o he model
L2+L4. This ac was poin ed ou ecen ly in [9] and used he e
o a de ailed s udy o he B =4Sky mion ( he helium nucleus)
wi hin he gene alised Sky me model.
In addi ion o in oducing he p ope BPS submodels o he
s anda d (massless) Sky me model, we also conside ed some gen-
e alisa ions, whe e each e m in he submodels is mul iplied by
a field-dependen coupling unc ion. I he coupling unc ions only
depend on he p ofile unc ion ξ, hen he gene alisa ions based on
L(1)con inue o be o he holomo phic ype, i.e., he ufield has
a CP(1)-model ene gy densi y in a sepa a ion-o - a iable ansa z.
The second model, L(2), gene alises o a field heo y which sha es
i s Bogomolnyi equa ions wi h he ones in he dila on – SU(2)
Yang–Mills model o a magne ic monopole [13–16], whe e ξplays
he ole o he dila on. Bo h gene alised models lead o genuine
opological soli ons obeying he equi ed bounda y condi ions o
app op ia e choices o he coupling unc ions.
The e exis s a lowe dimensional coun e pa o he Sky me
model, he baby Sky me model [17] (he e in he complex field o -
mula ion)
Lbaby =λ2L2+λ4L4+λ0L0
=λ2
uμ¯
uμ
(1+|u|2)2−λ4
(uμ¯
uμ)2−u2
μ¯
u2
ν
(1+|u|2)4−λ0U(5.4)
whe e one can dis inguish wo BPS submodels. The CP1(sigma)
model
LCP1≡L2=uμ¯
uμ
(1+|u|2)2(5.5)
and he baby BPS Sky me model [18]
LBPS ≡L4+λ0L0=(uμ¯
uμ)2−u2
μ¯
u2
ν
(1+|u|2)4−λ0U.(5.6)
Bo h BPS submodels a e genuine baby Sky me models. Fu he ,
hey a e ue BPS heo ies, i.e., he e exis Bogomolnyi equa ions
and solu ions which sa u a e a opological bound in ol ing a opo-
logical index. These equa ions a e
ui±iijuj=0 (5.7)
o he CP(1) model and
iijui¯
uj
(1+|u|2)2±λ0U=0 (5.8)
o he BPS baby Sky me model. Hence, we ecognize a simila
pa e n o he one ound in he (3 +1)-dimensional Sky me model.
Le us ema k ha he e exis u he possibili ies o find BPS
e sions o he Sky me model. One pa icula example, based on
he same field con en s bu a di e en Lag angian was cons uc ed
in [19]. I he opology o he base space mani old is changed
and allows o define addi ional opological indices, hen u he
BPS sec o s o he Sky me model ela ed o hese new opological
indices may be ound [20,21]. Ano he op ion which equi es, how-
e e , o change he field con en s by adding an infini e numbe o
ec o mesons was p oposed and de eloped in [22,23].
Finally, he hidden BPS s uc u e o he s anda d Sky me model
e ealed he e migh be ela ed o possible supe symme ic e -
sions o he heo y [24,25], because he e is a close ela ion be-
ween BPS sec o s and supe symme y, in gene al (see [26] o he
baby Sky me model).
Acknowledgemen s
The au ho s acknowledge financial suppo om he Minis y o
Educa ion, Cul u e and Spo s, Spain (G an No. FPA 2014-58-293-
C2-1-P), he Xun a de Galicia (G an No. INCITE09.296.035PR and
Conselle ia de Educacion), he Spanish Consolide -Ingenio 2010
P og amme CPAN (CSD2007-00042), and FEDER. AW was sup-
po ed by NCN g an 2012/06/A/ST2/00396.
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