Brane nucleation in supersymmetric models

JHEP10(2023)061
Published o SISSA by Sp inge
Recei ed:July 10, 2023
Re ised:Sep embe 20, 2023
Accep ed:Oc obe 3, 2023
Published:Oc obe 11, 2023
B ane nuclea ion in supe symme ic models
Igo Bandos,a,b,c Jose J. Blanco-Pillado,a,b,c Kepa Sousadand Mikel A. U kiolab,e
aDepa men o Physics, Uni e si y o he Basque Coun y UPV/EHU,
Ba io Sa iena s/n, 48940 Leioa, Spain
bEHU Quan um Cen e , Uni e si y o he Basque Coun y UPV/EHU,
Ba io Sa iena s/n, 48940 Leioa, Spain
cIKERBASQUE, Basque Founda ion o Science, 48011, Bilbao, Spain
dDepa men o Physics and Ma hema ics, Uni e si y o Alcalá,
28805 Alcalá de Hena es (Mad id), Spain
eDepa men o Applied Ma hema ics, Uni e si y o he Basque Coun y UPV/EHU,
Plaza Ingenie o To es Que edo 1, 48013 Bilbao, Spain
E-mail: [email p o ec ed],[email p o ec ed],
[email p o ec ed],[email p o ec ed]
Abs ac : This pape explo es he p ocess o acuum decay in supe symme ic mod-
els ela ed o lux compac i ica ions. In pa icula , we desc ibe hese ins abili ies wi hin
supe symme ic Lag angians o a single h ee- o m mul iple . This mul iple combines
scala ields, ep esen ing he moduli ields in ou dimensions, wi h 3- o m ields ha in-
luence he po en ial o hese moduli ia he in ege lux o hei associa ed 4- o m ield
s eng h. Fu he mo e, using supe symme y as a guide we ob ain he o m o he cou-
plings o hese ields o he memb anes ha ac as sou ces o he 3- o m po en ials. Adding
small supe symme y b eaking e ms o hese Lag angians one can ob ain ins an on solu-
ions desc ibing he decay o he acua in hese models by he o ma ion o a memb ane
bubble. These ins an ons combine he usual Coleman-de Luccia and he B own-Tei elboim
o malisms in a single uni ied model. We s udy simple nume ical examples o heo ies wi h
and wi hou g a i y in his new amewo k and gene alize known Euclidean me hods o
accomoda e he simula aneous inclusion o scala ields and cha ged memb anes o hese
ins an on solu ions. Mo eo e , we show explici ly in hese examples how one eco e s he
s a ic supe symme ic solu ions in he limi ing case whe e he supe symme y b eaking
e ms anish. In his limi , he bubble becomes in ini e and la and ep esen s a hyb id
be ween he usual supe symme ic domain walls o ield heo y models and he b ane so-
lu ions in e pola ing be ween he supe symme ic acua; a so o d essed supe memb ane
BPS solu ion. Finally, we b ie ly commen on he implica ions o hese solu ions in cos-
mological models based on he S ing Theo y Landscape whe e hese ype o 4d e ec i e
heo ies could be ele an in in la iona y scena ios.
Keywo ds: Cosmological models, S ing and B ane Phenomenology, Supe g a i y Models
A Xi eP in : 2306.09412
Open Access,c
The Au ho s.
A icle unded by SCOAP3.h ps://doi.o g/10.1007/JHEP10(2023)061
JHEP10(2023)061
Con en s
1 In oduc ion 1
2 Rigid supe symme ic scala ield heo y models 4
2.1 Field heo y supe symme ic domain walls 4
2.1.1 Example. Double well po en ial 5
2.2 S abili y and acuum decay 6
3 Rigid supe symme ic models wi h 3- o m po en ials 8
3.1 Supe symme ic memb anes coupled o 3- o m po en ials 9
3.2 Example wi h quad a ic supe po en ial 11
3.3 Memb ane nuclea ion 12
3.3.1 Nume ical example 14
4 Supe g a i y models 16
4.1 Fla memb ane solu ions in supe g a i y models 18
4.2 Example: qua ic supe po en ial 20
5 Memb ane nuclea ion in supe g a i y 21
5.1 Example: qua ic supe po en ial 22
5.1.1 AdS/Minkowski o AdS ansi ions 23
5.1.2 dS o AdS ansi ions 26
6 Conclusions 27
A Th ee- o m mul iple s in supe symme y 29
A.1 Including 3- o ms in o he chi al supe ields: special chi al supe ields 30
B Rigid supe symme y, bosonic memb anes, and BPS equa ions 33
C Th ee- o m mul iple s in supe g a i y 36
C.1 Supe g a i y in e ac ing wi h 3- o m mul iple s 38
C.1.1 Supe -Weyl ans o ma ions 39
1 In oduc ion
Many high ene gy ex ensions o he S anda d Model make use in some way o ano he
o supe symme y. Fu he mo e, he p esence o mul iple acua is qui e gene ic in hese
models. Some o hese acua p ese e pa o he o iginal supe symme y while o he s
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b eak i comple ely. Unde s anding he s abili y o hese acua and hei possible decay
p ocesses is, he e o e, an essen ial aspec o he low ene gy desc ip ion o hese heo ies.
Some o he mos in e es ing examples o his ype o scena ios a e he e ec i e ou
dimensional models ob ained om s ing heo y compac i ica ions. Compac i ying 10d
s ing heo y o ou dimensions lea es us wi h a low ene gy e ec i e heo y wi h a collec-
ion o scala ields ( he moduli ields) ha pa ame ize he possible de o ma ions o he
in e nal compac mani old. One can u he impose ha he compac i ica ion mechanism
p ese es some supe symme y so ha we end up wi h a low ene gy heo y ha can be
classi ied as a supe symme ic scala ield heo y.1This led he au ho s in [1] o conside
he non-pe u ba i e s abili y o a N=1 model o supe g a i y. They demons a ed ha
supe symme ic acua a e s able wi h espec o he usual p ocess o bubble nuclea ion.
Fu he mo e, hey also showed ha his s abili y is due o he es ic ion imposed by su-
pe symme y on he ension o he wall ha in e pola es be ween he wo acua. In ac ,
his quenching phenomenon is no hing mo e han he Coleman-deLuccia [2] supp ession.
In his limi he bubble adius would be in ini e and he solu ion would be a plana domain
wall ha p ese es pa o he supe symme y [3].
On he o he hand, ecen de elopmen s in models o s ing compac i ica ion based on
he use o luxes along he in e nal dimensions has led us o he idea o an ex emely ich
Landscape o possible 4dE ec i e Field Theo ies (EFTs), also e e ed o as lux acua [4].
Each o hese acua is cha ac e ized by he p esence o a se o in ege luxes ha h ead
some cycles in he in e nal mani old. The s abilizing po en ial o he moduli in each
o hese sec o s o he heo y is also qui e complica ed and could easily ha e many local
minima i sel . One could he e o e use he conclusions discussed ea lie o he scala ield
po en ial in each o hese sec o s o s udy hei s abili y and ind he bounce solu ions
using he echniques de i ed by Coleman and collabo a o s [5,6].
One can hen ask whe he he e is an analogous p ocess ha would ake us om one
sec o wi h some se o luxes o ano he one whe e one o se e al o hose luxes ha e
been changed. Nai ely his would seem impossible since he lux is quan ized and he e o e
canno be con inuously changed. Howe e , simila ly o wha happens in he Schwinge
p ocess [7], one could educe he lux by he c ea ion o sou ces cha ged wi h espec o
he same ield ha p oduces i .2In ac , his ype o p ocess had been al eady discussed
in a pu ely ou dimensional con ex a numbe o yea s ago by B own and Tei elboim
(BT) [9,10]. In his model he p esence o a 4- o m ield s eng h in ou dimensions
induces an e ec i e cosmological cons an ha can only be changed by he nuclea ion o
memb anes cha ged wi h espec o i s 3- o m po en ial. The ins an on solu ions desc ibing
his ype o ins abili y o he model we e s udied in de ail in connec ion o he possible
sel - uning mechanism o he cosmological cons an .
1Mo e igo ously, one should speak abou supe mul iple s in ol ing scala ields; in he case o N=1
supe symme y, hese include no only he scala supe mul iple bu also he so-called h ee o m supe -
mul iple s (see below).
2See [8] o a simple desc ip ion o se e al ield heo y models in di e en dimensions o space ime ha
exhibi a simila beha iou o he one desc ibed in his pape .
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JHEP10(2023)061
In models o s ing heo y compac i ica ion he si ua ion is qui e simila , and one can
assume ha each o he o ms p esen in he 4d heo y would ha e an associa ed b ane
cha ged wi h espec o ha speci ic o m. Indeed one can iden i y 4dmemb ane objec s by
w apping highe dimensional b anes along some in e nal cycles. Simila ly, one can also ind
he e ec i e 4- o m ield ha couples o he 4-dimensional memb ane and unde s and i s
highe dimensional o igin. Taking his poin o iew, hese models o lux compac i ica ion
seem o lead o a model closely ela ed o he B own-Tei elboim idea [4].
The e is howe e an impo an di e ence be ween hese wo models. In he o iginal
B own-Tei elboim’s model he nuclea ion o he memb anes would lead o a jump in he
alue o he 4dcosmological cons an . In models wi h highe dimensions his change in
he lux would lead us o a di e en sec o wi h a di e en moduli po en ial. This is
in e es ing since i allows us o a oid he so called “emp y uni e se p oblem” in he pu ely
4dBT model by pos ula ing a pe iod o in la ion a e he bubble nuclea ion d i en by he
compac i ica ion po en ial (see o example [11]).
The p e ious a gumen s sugges he idea o combining bo h models in o a uni ying
pic u e whe e we can desc ibe wi hin he same heo y he p esence o he scala ield
moduli and he 4- o m ields. Mo eo e , ollowing wha was done be o e in he pu ely
scala ield model, we will look o guidance in supe symme ic models ha include bo h
ypes o deg ees o eedom, he 4- o m ield as well as he moduli ields. The inclu-
sion o o m ields in supe symme ic and supe g a i y mul iple s has been done be o e
in [12–18]. Fu he mo e, hei in e ac ion wi h supe symme ic memb anes (supe mem-
b anes) was s udied in [14,19–21] and hei ole in he low ene gy desc ip ion o lux
compac i ica ions has been ecen ly discussed in [18,21–23]. In his pape we will s udy
he non-pe u ba i e s abili y o hese models and hei modi ied e sions once we in oduce
small supe symme y-b eaking e ms. Mo e conc e ely, we will de o m he supe symme -
ic ield- heo e ic pa o he ac ion by including so supe symme y b eaking e ms [24],
while keeping he (supe )memb ane pa o he ac ion un ouched.
This pape is o ganized as ollows. In sec ion 2we will e iew he esul s o global
supe symme ic scala ield heo ies (mo e conc e ely, sel -in e ac ing scala supe mul iple
models which can be desc ibed in e ms o a gene ic chi al supe ield), and he exis ence
o supe symme ic domain wall solu ions in hese models. We will show wi h explici
examples how hese solu ions appea as limi ing cases o a bubble decay p ocess om a
non-supe symme ic acua. In sec ion 3we will explain how o in oduce 3- o m gauge
ields in ou supe symme ic heo y, as well as memb anes ha na u ally couple o hese
o ms. This will lead us o discuss he so-called 3- o m supe mul iple s desc ibed by a
special ype o chi al supe ields. Fu he mo e, we will ob ain he ins an on solu ions
desc ibing he decay p ocesses in hese heo ies, all he while explici ly showing how hey
connec in a p ope limi wi h he supe symme ic solu ions ound in he p e ious sec ion.
Finally, in sec ions 4and 5we will desc ibe a simila si ua ion in he case o supe g a i y
and ind explici solu ions o he equa ions o mo ion desc ibing he bubble nuclea ion. We
end wi h some conclusions in sec ion 6.
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2 Rigid supe symme ic scala ield heo y models
As a wa m up exe cise, in his sec ion, we will conside he s abili y o acua in a scala
ield Lag angian o an N=1 ou dimensional globally supe symme ic ield heo y o a
single chi al supe ield, which desc ibes he so-called scala supe mul iple .3Many o he
esul s p esen ed in his sec ion can be ound in he li e a u e, in pa icula in [1,3]. In he
ollowing sec ions we will discuss mo e complica ed models ollowing a simila easoning o
he one p esen ed he e.
2.1 Field heo y supe symme ic domain walls
The model we will be conside ing he e is gi en by he mos gene al Lag angian o a complex
scala ield (ϕ)which is gi en by he leading componen o a gene ic chi al supe ield (which
desc ibes he so-called scala mul iple ). This Lag angian eads
L=−Kϕ¯
ϕ∂µϕ∂µ¯
ϕ−Kϕ¯
ϕ|Wϕ|2(2.1)
whe e we ha e in oduced he wo unc ions ha de ine he model: he eal Kahle po-
en ial K(ϕ, ¯
ϕ)and he complex holomo phic supe po en ial W(ϕ). We will deno e hei
de i a i es as ∂ϕ∂¯
ϕK=Kϕ¯
ϕ= 1/Kϕ¯
ϕand Wϕ=∂ϕW(ϕ), hus ollowing he con en-
ions o [25].4
The equa ions o mo ion o his heo y a e
Kϕ¯
ϕ∂µ∂µϕ−Kϕϕ¯
ϕ∂µϕ∂µ¯
ϕ+Kϕ¯
ϕ¯
ϕ(Kϕ¯
ϕ)2|Wϕ|2−Kϕ¯
ϕWϕ¯
W¯
ϕ¯
ϕ= 0 (2.2)
and i s complex conjuga e. These educe o he usual Klein-Go don equa ion o a complex
scala ield wi h a scala po en ial gi en by V(ϕ, ¯
ϕ) = |Wϕ|2i he kine ic e m is canonical
(which pa icula ly equi es K=ϕ¯
ϕ).
We a e looking o a domain wall solu ion in his model ha in e pola es be ween wo
supe symme ic minima, in o he wo ds, be ween wo poin s whose supe po en ial sa is ies
Wϕ(ϕ±) = 0. Recall ha all supe symme ic minima ha e a anishing po en ial, and
he e o e degene a e in ene gy. Fo ha ma e , le us conside a la domain wall whose
ans e se di ec ion is gi en by he coo dina e z. One can hen show [3] ha he s a ic
solu ion p ese ing hal o he N=1 supe symme y sol es he i s -o de equa ion
∂zϕ(z) = eiθKϕ¯
ϕ¯
W¯
ϕ(¯
ϕ(z)),(2.3)
known as he BPS equa ion, whe e he phase θis gi en by
eiθ =∆W
|∆W|,wi h ∆W≡W(ϕ(z=∞)) −W(ϕ(z=−∞)).(2.4)
O cou se, gi en app op ia e bounda y condi ions, bo h he i s -o de and second-
o de equa ions should yield he same s a ic solu ion o he supe symme ic domain wall.
3In he main pa o his pape we will only deal wi h he bosonic componen s o he supe mul iple s.
See, e.g., [25] o a mo e comple e ea men o he simples cases.
4In his pape we will use he (−+ ++) signa u e con en ion.
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JHEP10(2023)061
The ension o he domain wall in his model can be compu ed w i ing he ene gy pe uni
a ea as ollows:
σ=Z∞
−∞
dz Kϕ¯
ϕ∂zϕ(z)−eiθKϕ¯
ϕ¯
W¯
ϕ(ϕ(z))
2+ 2 Re[e−iθ∆W](2.5)
One can eadily see ha in he case o supe symme ic bosonic solu ions, whe e (2.3) holds,
he ension becomes
σBPS = 2|∆W|.(2.6)
2.1.1 Example. Double well po en ial
Le us illus a e all o he abo e by conside ing he model de ined by:
K(ϕ, ¯
ϕ) = ϕ¯
ϕ, W(ϕ) = 1
3ϕ3−a2ϕ(2.7)
whe e we a e assuming ha a > 0. Fo his pa icula model, he Lag angian (2.1) simpli-
ies o
L=−∂µϕ∂µ¯
ϕ−|ϕ2−a2|2.(2.8)
The po en ial o he heo y, es ic ed o he eal pa o ϕ, has been plo ed in igu e 1(a)
(dashed line) whe e we see he double well po en ial o m wi h he wo supe symme ic
minima loca ed a ϕ±=±a. The second-o de equa ions o mo ion (2.2) ead, in his case,
∂2
zϕ(z)−2¯
ϕ(ϕ2−a2)=0.(2.9)
On he o he hand, he i s -o de BPS equa ion (2.3) eads
∂zϕ(z) = −(¯
ϕ2−a2),(2.10)
whe e we ha e chosen W(ϕ(z=∞)) < W(ϕ(z=−∞), which implies5eiθ =−1. The
solu ion o his equa ion is gi en by he eal ield con igu a ion,
ϕ(z) = a anh(az),(2.11)
while he ension o his domain wall is gi en by
σBPS = 2|∆W|=8
3a3.(2.12)
As we men ioned ea lie , any solu ion o he BPS equa ion p ese es some supe sym-
me y by cons uc ion, so i is clea ha i canno ep esen he decay o he acuum. We
can also see his no ing ha bo h acua a e supe symme ic, degene a e in ene gy, and he
wall is la and in ini e, so he e is no way hese acua can decay.
On he o he hand, he solu ion we ound he e is pu ely eal. This is consis en wi h
he po en ial we ha e cons uc ed since i s o m is such ha pe u ba ions a ound he
5On he o he hand, eiθ = +1 would yield he mi o ed p o ile, o en deno ed as he an i-domain wall.
This simply co esponds o lipping he bounda y condi ions imposed a ±∞.
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JHEP10(2023)061
solu ion in he imagina y ield di ec ions a e s abilized. We can check his by expanding
he po en ial in he eal and imagina y pa s o he ield, namely
ϕ(z) = ψ(z) + i s(z),(2.13)
so he po en ial eads
V(ψ, s)=(ψ2−a2)2+ 2(ψ2+a2)s2+s4.(2.14)
This is why we can concen a e on he solu ion along he s= 0 line. In all o he exam-
ples we show he e, we ha e checked ha his is indeed he case; he e o e, in all o ou
illus a ions we will simply d aw he esul s conce ning he eal pa o he ields.
2.2 S abili y and acuum decay
In he p e ious sec ion we showed how one can ind supe symme ic domain wall con igu-
a ions ha in e pola e be ween supe symme ic acua in ou model. The ac ha hese
acua a e s able is no su p ising since supe symme y imposes hem o be degene a e
global minima o he po en ial. Le us now conside he case whe e he e is a small supe -
symme y b eaking e m in ou po en ial and s udy he s abili y o he esul an acua.
Fo simplici y le us assume ha we in oduce in he heo y a couple o so supe sym-
me y b eaking e ms [24] o he o m
Sso =−Zd4x√−ghµ2ϕ¯
ϕ+bϕ3+¯
ϕ3i,(2.15)
which b eak supe symme y explici ly. In he ollowing, we will conside he coe icien s o
be small enough so ha many o he p ope ies o he solu ion ound ea lie will s ill hold.
This means we will conside he case whe e µ2≪a2as well as b≪a, see he colou ed
cu es o 1(a), whe e o simplici y we ha e plo ed some cu es o he cases o b=µ.
In his egime we can see ha he heo y s ill has wo minima gi en by
ϕ±=±a+δ±(µ2, b),(2.16)
whe e he solu ions ha e only shi ed sligh ly, i.e. |δ±/a| ≪ 1. The in e es ing poin now
is ha bo h o hese minima b eak supe symme y. I one akes b > 0, he po en ial a
ϕ+becomes sligh ly highe han he o he minimum; his means ha his acuum will be
uns able wi h espec o he nuclea ion o bubbles o he ue acuum a ϕ−. Fu he mo e,
he o m o he supe symme y-b eaking e ms allows o he unneling o happen along
he eal di ec ion o he ield, see igu e 1(a).
In o de o compu e he p obabili y o he decay and i s bubble p o ile in e ms o
he scala ield, we will eso o he usual me hods de eloped by Coleman and collab-
o a o s [5,6] in he con ex o False Vacuum Decay. Thus, we only ha e o ex emize
he Euclidean ac ion associa ed o he Lag angian (2.1) comple ed by he supe symme y-
b eaking e ms. Assuming a s anda d kine ic e m (K=ϕ¯
ϕ) and an O(4) symme y in
Euclidean space, he equa ions o mo ion o he ield a e gi en by
ϕ′′ +3
ρϕ′=∂V
∂¯
ϕ(2.17)
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JHEP10(2023)061
-1.5 -1.0 -0.5 0.5 1.0 1.5
-1
1
2
(a)
-3-2-1 1 2 3
-1.0
-0.5
0.5
1.0
(b)
Figu e 1. (a) Po en ial (2.19) wi h a= 1, o se e al supe symme y-b eaking pa ame e s. (b) So-
lu ions o (2.17), cen e ed a ound he in lec ion poin co esponding o each p o ile, labeled as ρ∗.
The BPS limi is shown wi h a dashed line, ep esen ing he domain wall solu ion a ising om he
dashed po en ial in (a).
whe e ρ=√τ2+x2,τbeing he Euclidean ime, and p imes deno e de i a i es wi h
espec o ρ. This equa ion is o be sol ed conside ing he bounda y condi ions
lim
ρ→∞ ϕ(ρ) = ϕ+, ϕ′(0) = 0.(2.18)
No e ha he Euclidean O(4) symme y will u n o O(1,3) when anspo ing he solu-
ion back o Lo en zian space ime. Among o he hings, his will mean ha he p o ile
ϕ(ρ)ob ained ia (2.17) will co espond o he eme gen scala p o ile o he bubble a
o ma ion. Fu he mo e, his symme y implies ha he adius o he bubble so o med
will expand ou wa d ollowing a cons an accele a ion ajec o y.
Following he example p esen ed in he p e ious subsec ion, we will wo k wi h he
ollowing po en ial
V(ϕ, ¯
ϕ)=(ϕ2−a2)(¯
ϕ2−a2) + µ2ϕ¯
ϕ+b(ϕ3+¯
ϕ3)(2.19)
which al eady includes a con ibu ion om supe symme y-b eaking e ms wi hin i s de i-
ni ion.
In his one-dimensional se up, he p o ile o he scala ield can be easily ound in
Euclidean adial coo dina es using an unde shoo /o e shoo algo i hm [5]. Essen ially,
since he bounda y condi ions (2.18) do no speci y he ini ial s a ing poin ϕ(0), we can
i s ob ain a couple o poin s whe e he ield ei he unde shoo s o o e shoo s he alse
acuum a ϕ+. I e a i ely educing his ield ange, we will e en ually ind a s a ing
poin ϕ(0) which s ays su icien ly close o he ue acuum a ϕ− o la ge alues o ρ.
The solu ions ound his way ha e been con as ed wi h ones ob ained using he so wa e
AnyBubble [26], which applies an al e na i e mul iple shoo ing me hod, and ha e been
shown o be essen ially iden ical.
Using his nume ical algo i hm, we can check ha indeed he o m o he domain wall
o ming he bubble does no change quali a i ely in compa ison o he supe symme ic
case gi en abo e. In o de o isually make his compa ison we ha e plo ed in igu e 1(b)
he esul an domain wall p o iles o he bubble nuclea ion cen e ed on he same poin . I
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JHEP10(2023)061
is clea ha dialling back o ze o he supe symme y b eaking coe icien s b, µ one eco e s
he p o iles ob ained wi h he i s -o de BPS equa ion (2.10), i.e., he supe symme ic
domain wall p o ile. O cou se, his is consis en since in his case he bubble adius is
in ini e and he unnelling a e would be ze o, as his limi akes us back o he s able
supe symme ic acua p esen ed in he p e ious sec ion.
3 Rigid supe symme ic models wi h 3- o m po en ials
As we desc ibed in he in oduc ion, we a e in e es ed in s udying supe symme ic models
wi h some new deg ees o eedom beyond he complex scala ield p esen ed in he p e ious
sec ion. In pa icula , we would like o in oduce a 3- o m po en ial in o ou model.
Ac ually, using supe symme y as a guiding p inciple, we can desc ibe bo h he scala
ield and he 3- o m gauge ield in o a single mul iple desc ibed in e ms o a special
ype o chi al supe ield. We show in appendix Ahow one can cons uc such a special
chi al supe ield desc ibing he so-called single 3- o m supe mul iple om uncons ained
eal scala supe ields. In his case, he bosonic con en o he mul iple is simply gi en by
complex scala ields and auxilia y ields, he eal pa o which will be ela ed o a ield
s eng h o he 3- o m po en ial. Following his p esc ip ion we ind ha he simples bulk
ac ion o such ields is gi en by
Sbulk =Zd4x−Kϕ¯
ϕ∂µϕ∂µ¯
ϕ−1
4·4!Kϕ¯
ϕFµνσρFµνσρ +1
2·4! Wϕ+¯
W¯
ϕϵµνσρFµνσρ
+1
4Kϕ¯
ϕWϕ−¯
W¯
ϕ2,(3.1)
whe e Fµνρσ = 4∂[µAνρσ]and as be o e Kand Wdeno e he Kähle po en ial and supe -
po en ial which de ine he model o he complex scala ield ϕ(x).6
As is usually he case in models in ol ing gauge ields, his ac ion mus be supple-
men ed wi h some bounda y e ms ha a e equi ed o make he a ia ion o he ac ion
well posed [10,27]. In ou p esen case, he bounda y e m is gi en by
Sbd =1
2·3! Zd4x ∂µhAνρσ Kϕ¯
ϕFµνρσ −ϵµνρσ Wϕ+¯
W¯
ϕi,(3.2)
see appendix A o a de i a ion o his esul .
The equa ion o mo ion o he 4- o m ield s eng h can be w i en as
∂µhKϕ¯
ϕFµνρσ −ϵµνρσ Wϕ+¯
W¯
ϕi= 0 ,(3.3)
which can be in eg a ed o gi e,
Fµνρσ =Kϕ¯
ϕϵµνρσ Wϕ+¯
W¯
ϕ−2n,(3.4)
whe e n∈Ris a eal in eg a ion cons an . This exp ession allows us o in eg a e ou he
3- o m po en ial om he o iginal ac ion o ob ain a new e ec i e heo y w i en in e ms
6Fo he sake o simplici y, will s udy models consis ing o a single scala ield he e. See [18] o a s udy
o simila sys ems wi h an a bi a y numbe o scala ields.
– 8 –
JHEP10(2023)061
01234
1
2
3
4
5
6
7
Figu e 3. Scala po en ial (3.29) o n= 1,q= 2 and some alues o he supe symme y-b eaking
pa ame e s. The da ke cu e shows he po en ial inside he memb ane (ρ < R), while he ligh e
one ep esen s he po en ial ou side i (ρ>R).
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Figu e 4. (a) Euclidean ac ion o solu ions o he equa ion o mo ion (3.27), wi h n= 1,q= 2
and se e al alues o R,band µ. (b) Ene gy and Euclidean ac ion di e ence wi h espec o he
alse acuum s a e o se e al adii, in he case whe e µ=b= 0.03. The maximum o Bcoincides
wi h he oo o ∆E, as expec ed.
o he adius R. We ake his o be he co ec alue o he adius o he ins an on ha
media es he acuum decay o in e es o us.
We also checked he a ia ion in he o al ene gy o he p o ile wi h espec o he
backg ound a he ime o nuclea ion. Indeed, he e should be no ene gy loss o gain o
he ins an on solu ion a he ime o i s eme gence. Thus, he co ec bubble p o ile should
co espond o he oo s o he ene gy densi y di e ence wi h espec o he alse acuum
backg ound. This quan i y is de ined by
∆E= 4πZ∞
0
d 2"Kϕ¯
ϕ
dϕ
d 
2
+V(ϕ, ¯
ϕ)+2|qϕ|δ( −R)−V # =0
.(3.30)
This las exp ession can be easily e alua ed om he Euclidean solu ion, since he p o ile
ob ained in Euclidean space co esponds o he p o ile o he eme ging bubble in Lo en zian
space a = 0. In all o he p o iles ha we ha e compu ed, he alue o R co esponding
o no ene gy loss o gain wi h espec o he backg ound has been ound o co espond
wi h he maximum o he Euclidean ac ion, see igu e 4(b) o an explici example.
– 15 –

JHEP10(2023)061
0 5 10 15 20 25 30
1.0
1.5
2.0
2.5
(a)
-2-1 1 2 3 4 5
1.0
1.5
2.0
2.5
3.0
(b)
Figu e 5. (a) Scala ield p o iles co esponding o each maximum Euclidean ac ion o some
supe symme y-b eaking pa ame e s. (b) The same p o iles as be o e, cen e ed a ound he mem-
b ane adius co esponding o each p o ile, deno ed by ρ∗. The dashed line ep esen s he BPS
solu ion om eq. (3.22), wi h n= 1 and q= 2, which is clea ly he asymp o ic beha iou o he
p o iles as he supe symme y-b eaking pa ame e s band µ end o 0.
The p o iles co esponding o he solu ions which ex emize SEha e been plo ed in
igu e 5(a). As expec ed, as he supe symme y-b eaking pa ame e s a e made smalle ,
he adius o he eme ging memb ane inc eases and he scala ield p o iles p og essi ely
end owa ds he BPS solu ion de i ed abo e, as shown in igu e 5(b).
A cu ious ea u e o hese p o iles is ha , when conside ed as a pa icle in he in e ed
po en ial −V, hey i s end o ge away om he alse acuum, only o hen be p ojec ed
in he > R po en ial wi h enough eloci y o asymp o ically each he alse acuum.
4 Supe g a i y models
We now u n o s udy he same se up as in he p e ious sec ion, wi h g a i y aken in o
conside a ion. We will wo k in he con ex o N=1, D= 4 Supe g a i y coupled o chi al
ma e . We will be in e es ed in gene alizing he alse acuum decay discussed in he
p e ious sec ion including g a i y. Howe e , be o e analysing mo e gene ic si ua ions, we
will i s s udy he supe symme ic limi o la memb ane solu ions in supe g a i y. Thus,
in his sec ion, we will s a by analyzing he ac ion o he sys em composed by scala
ields, eal h ee- o ms and la memb anes in a space ime o Lo en zian signa u e.
The ac ion we will be conside ing is once again gi en by he sum o he ollowing e ms,
S=Sbulk +Smemb ane +Sbounda y e ms.(4.1)
As we e iew in appendix A, he bosonic pa o he bulk ac ion o he sys em, which
includes g a i y, scala ields and 3- o ms, is gi en by he ollowing supe g a i y ac ion
Sbulk =Zd4x√−g1
2R−Kϕ¯
ϕ∂µϕ∂µ¯
ϕ−1
3e−K(M+K¯
ϕ¯
F) ( ¯
M+KϕF)−M¯
W−¯
MW
+e−KKϕ¯
ϕF¯
F+FWϕ+¯
F¯
W¯
ϕ(4.2)
– 16 –
JHEP10(2023)061
whe e, as be o e, K(ϕ, ¯
ϕ)and W(ϕ)desc ibe he eal Kähle po en ial and he holomo phic
supe po en ial unc ions o he scala ield and
F=1
2(DµAµ+id) + 2
3¯
ϕM +1
3ϕ¯
M. (4.3)
In he abo e equa ions we use MPl = 1 and deno e he Ricci scala by R. Fu he mo e, M
is a complex scala auxilia y ield o he minimal supe g a i y mul iple , d is a eal scala
auxilia y ield and Aµis he Hodge dual o he h ee- o m. In ou se ing his la e ield
belongs o he ma e supe mul iple , which also includes he scala ield.8
Jus as in he non-g a i a ional case, bounda y e ms mus be included in he ull
ac ion o ensu e he a ia ion o he ac ion wi h espec o he o m ield is well posed. We
lea e hei de i a ion o appendix C. No e ha hese bounda y e ms will also include he
Gibbons-Hawking e m [35] in o de o he a ia ional p oblem o be well de ined also in
he g a i a ional sec o .
The ac ion o he memb ane can be ixed by equi ing he in e ac ing sys em (4.1),
wi h e mionic e ms aken in o accoun in he i s wo e ms, o emain in a ian unde
a hal o local supe symme y a e he bosonic memb ane is included [36,37]. I eads
Smemb. =−ZM
d3ξ√−h2eK/2|qϕ|+q
3! ZM
d3ξAµνρ
∂xµ
∂ξa
∂xν
∂ξb
∂xρ
∂ξcϵabc (4.4)
which as be o e desc ibes he coupling be ween he memb ane and he 3- o m po en ial as
well as he scala ield. No e howe e ha in supe g a i y he Nambu-Go o e m ecei es
a co ec ion om he exponen ial o he Kähle po en ial.9
I is easy o check ha se ing he o m ield and auxilia y ields on-shell and aking
in o accoun he con ibu ion o he bounda y e ms, he ac ion eads [21]
S=Zd4x√−gR
2−Kϕ¯
ϕ∂µϕ∂µ¯
ϕ−V(ϕ, ¯
ϕ)+SGH −ZM
d3ξ√−h2eK/2|qϕ|(4.5)
whe e SGH ep esen s he Gibbons-Hawking bounda y e m and V(ϕ, ¯
ϕ)is he N=1, D= 4
ma e -coupled supe g a i y scala po en ial o he o m
V(ϕ, ¯
ϕ) = eKhDϕˆ
WKϕ¯
ϕD¯
ϕˆ
¯
W−3|ˆ
W|2i(4.6)
wi h he usual Kähle -co a ian de i a i e deno ed by Dϕ=∂ϕ+Kϕ. As in he global
supe symme ic models he in o ma ion abou he 3- o m lux is encapsula ed in he o m
o he e ec i e supe po en ial:
ˆ
W=W−(n+qH(x))ϕ , (4.7)
whe e, jus as in he non-g a i a ional case, H(x)is gi en by eq. (3.11). This modi ied
supe po en ial will allow us o ind a p o ile o he d essed memb ane which in e pola es
be ween supe symme ic minima o di e en po en ials.
8We no e ha a 3- o m ields may also be included as an auxilia y ield o he supe g a i y supe mul-
iple [14,28–34]. E en hough his p o ides a mo e s aigh o wa d supe symme ic gene aliza ion o he
B own-Tei elboim cons uc ion [9,10], we will no conside i he e since he ad an age o inclusion o he
3- o m in he ma e supe mul iple is ha his cons uc ion has a smoo h and non i ial la space limi .
9The exponen ial ac o eK/2a ises due o he supe -Weyl escaling and ield ede ini ions equi ed o
b ing he ac ion o Eins ein ame, see appendix C o mo e de ail.
– 17 –
JHEP10(2023)061
4.1 Fla memb ane solu ions in supe g a i y models
Simila ly o wha we did in he la space case, one can ind smoo h domain wall solu ions in
he supe g a i y con ex . This has been ex ensi ely s udied in he li e a u e s a ing in [3]
(see also a mo e ecen discussion abou his opic in [38] and e e ences he ein). He e we
show how one can gene alize hese solu ions o include he p esence o a supe memb ane
as i was desc ibed in [21].
Le us begin wi h he case o a la memb ane in e pola ing be ween wo supe symme -
ic acua. Le us assume his s a ic memb ane si s a z= 0. In o de o s udy he p o ile
ac oss such a memb ane, we will assume he ollowing ansa z o he me ic10 [21,38]:
ds2=e2D(z)(−d 2+dx2+dy2) + dz2(4.8)
so ha √−g=e3D(z). Le us u n o s udy he equa ion o mo ion o he scala ield,
which is
1
√−g∂µ√−gKϕ¯
ϕgµν∂νϕ=Kϕ¯
ϕ¯
ϕ∂µϕ∂µ¯
ϕ+∂V
∂¯
ϕ+eK/2hK¯
ϕ|qϕ|−qeiηiδ(z),(4.9)
whe e eiη is de ined as
eiη =−qϕ
|qϕ|z=0
.(4.10)
I is na u al o assume ha scala ield only depends on he ans e se coo dina e o he
memb ane, i.e., ϕ=ϕ(z). In ha pa icula case, he ield obeys
∂z(Kϕ¯
ϕ∂zϕ)+3Kϕ¯
ϕ∂zD ∂zϕ=Kϕ¯
ϕ¯
ϕ|∂zϕ|2+∂V
∂¯
ϕ+eK/2hK¯
ϕ|qϕ|−qeiηiδ(z).(4.11)
On he o he hand, he Eins ein equa ions o he me ic can be combined o gi e
∂2
zD+ 3(∂zD)2=−V−eK/2|qϕ|δ(z)(4.12)
No e ha he del as a z= 0 will yield jumps in he i s de i a i e o bo h ϕ, as in
he non-g a i a ional case, and in he scale ac o D.
A supe symme ic and s a ic domain wall may in e pola e be ween non-degene a e
minima, since essen ially he g a i a ional con ibu ion may compensa e he di e ence in
scala po en ial be ween bo h acua [3]. I supe symme y is pa ly conse ed ac oss he
p o ile o he domain wall, hen he minima a e bound o be ei he Minkowski o AdS
acua (no e, howe e , ha no supe symme ic domain wall may in e pola e be ween wo
Minkowski acua when g a i y is included [3]).
Wi h hese ema ks a hand, he BPS equa ions acqui e he o m [3,21,38]
ϕ′(z) = ∓eK/2eia g( ˆ
W)K¯
ϕϕD¯
ϕˆ
W(4.13)
D′(z) = ±eK/2|ˆ
W|(4.14)
10No e ha wi h his choice o me ic and in he so-called s a ic gauge xa(ξ) = ξa, o la memb ane,
z(ξ) = 0, we will ha e √−h=√−g, whe e he .h.s. is calcula ed a z= 0.
– 18 –
JHEP10(2023)061
whe e p imes deno e de i a i es wi h espec o z, The second o de equa ions a e obeyed
by he solu ions o he i s -o de BPS equa ions when sui able bounda y condi ions a e
imposed. In pa icula , aking he de i a i e o eq. (4.14) and using (4.13), we ind ha
he second o de equa ion (4.12) is sa is ied p o ided11
eia g( ˆ
W)z=0 =∓eiη .(4.15)
I is con enien , ollowing [38], o w i e hese equa ions in e ms o
Z ≡ eK/2ˆ
W. (4.16)
No e ha he alue o he scala po en ial a supe symme ic c i ical poin s is hen gi en
by Vsusy =−3|Z|2. In e ms o Z, he BPS equa ions simpli y o
ϕ′(z) = ∓2K¯
ϕϕ∂¯
ϕ|Z| (4.17)
D′(z) = ±|Z| .(4.18)
The sign o use in he in eg a ion o he BPS equa ions is de e mined by he alue o |Z|
a z→ ±∞. In o de o see his, le us i s ly no e ha
d|Z|
dz = (ϕ′∂ϕ|Z|+¯
ϕ′∂¯
ϕ|Z|)∓eK/2|qϕ|δ(z) = ∓h4Kϕ¯
ϕ∂ϕ|Z|∂¯
ϕ|Z|+eK/2|qϕ|δ(z)i(4.19)
whe e, in he second s ep, we ha e used (4.17). Then i is easy o obse e ha , i o
example we choose he lowe sign in eqs. (4.17) and (4.18), which hen become
ϕ′(z)=2K¯
ϕϕ∂¯
ϕ|Z| (4.20)
D′(z) = −|Z| ,(4.21)
we ind ha he de i a i e o |Z| is posi i e
d|Z|
dz = 4Kϕ¯
ϕ∂ϕ|Z|∂¯
ϕ|Z|+eK/2|qϕ|δ(z)>0.(4.22)
The e o e, |Z| mus inc ease mono onically, which implies |Z|−∞ <|Z|+∞. On he o he
hand, i |Z|−∞ >|Z|+∞, he lowe sign o he BPS equa ions eqs. (4.17)–(4.18) will apply.
Finally, i should be no ed ha in case |Z| has some oo along z, he si ua ion be-
comes a bi mo e complica ed. Indeed, he signs mus be swapped a e c ossing a oo o
he supe po en ial in o de o ha e a posi i e o e all ension o he domain wall [21,38].
Howe e , as no ed in [3], such cases do no co espond o he limi ing case o a bubble
ins abili y. In ac he space ime induced by hese solu ions is asymp o ically qui e di -
e en and esembles he one in he Randall-Sund um scena io [40] wi h a single posi i e
ension b ane.12
11Ac ually his iden i ica ion can be ob ained s aigh o wa dly as i implies ha on he wo ld olume o
he memb ane he supe symme y p ese ed by he solu ion coincides wi h κ-symme y o he supe mem-
b ane ac ion [39].
12We will de e he explo a ion o hese ypes o solu ions in ou con ex o a u u e publica ion.
– 19 –
JHEP10(2023)061
0.2 0.4 0.6 0.8 1.0 1.2
-0.4
-0.3
-0.2
-0.1
0.1
0.2
Figu e 6. Scala po en ial o he model desc ibed in his sec ion, o wo di e en alues o he
lux alue n.
-4-2 0 2 4
0.95
1.00
1.05
1.10
(a)
-4-2 2 4
-1.0
-0.5
0.5
(b)
Figu e 7. Solu ion o he BPS equa ions o he model de ined in (4.23), o (a) he scala ield
(b) he scale ac o , o he me ic de ined in (4.8). The po en ial a z < 0co esponds o he case
n= 3, while z > 0co esponds o n= 2; hus, he memb ane is cha ged wi h q=−1.
4.2 Example: qua ic supe po en ial
The abo e equa ions can be used o ind he p o iles o a scala ield in e pola ing be ween
di e en minima o he model de ined by
K(ϕ, ¯
ϕ) = ϕ¯
ϕ, W(ϕ) = (10MPl)−1ϕ4.(4.23)
whe e we ha e es o ed he Planck mass momen a ily in o de o show explici ly he ene gy
scales in ol ed in his example. The scala po en ial de ined by his Kähle po en ial and
supe po en ial is shown in igu e 6, once he 3- o m has been in eg a ed ou . The minimum
ea u ed by each b anch can be shown o be supe symme ic, i.e., i sa is ies Dϕˆ
W= 0.
Going back o uni s whe e MPl = 1, he nume ical BPS p o ile a ising om his
po en ial o bo h he scala ield and he scale ac o D is shown in igu e 7, whe e we
ha e placed he lowe minimum o he le ( o easie compa ison la e on). We ha e also
checked ha he second-o de equa ions (4.11) and (4.12), which explici ly inco po a e
he i s -de i a i e jumps as Di ac del as, yield exac ly he same p o iles. No e ha o
his pa icula solu ion he supe symme ic minima on bo h sides o he wall desc ibe an
an i-deSi e acua wi h di e en alues o hei cosmological cons an .
– 20 –

JHEP10(2023)061
5 Memb ane nuclea ion in supe g a i y
As we discussed in he in oduc ion memb ane nuclea ion in a model wi h a 4- o m lux has
been s udied in he li e a u e in he con ex o models simila o B own-Tei elboim [9,10].
Ou supe g a i y model includes a 3- o m ield and a b ane cha ged wi h espec o i ,
howe e , as we ha e shown in he p e ious sec ion i can also be cas ed exclusi ely in
e ms o a scala ield by in eg a ing ou he 3- o m po en ial. We can hen ask whe he
he e will be bounce solu ions simila o he la space ones whe e he ield in e pola es
be ween wo minima o he lux-dependen po en ial o he scala ield as one c osses
he wall.
In o de o illus a e hese acuum decay p ocesses in ou model, we will ollow a
simila p ocedu e o he one we p esen ed in he la space limi in sec ion 3.3 whe e we
in oduced small co ec ions o he supe symme ic Lag angian. These e ms will allow
he possibili y o ha ing supe symme y b eaking acua ha could be suscep ible o decay.
In pa icula , we will s udy he Euclidean ac ion gi en by13
SE=Zd4x√g−R
2+Kϕ¯
ϕgab∂aϕ∂b¯
ϕ+˜
V(ϕ, ¯
ϕ)+ 2 ZM
d3ξ√heK/2|qϕ|+SGH (5.1)
whe e,
˜
V(ϕ, ¯
ϕ) = eKKϕ¯
ϕDϕˆ
W
2−3ˆ
W
2+µ2ϕ¯
ϕ, ˆ
W≡W−(n+qH(x))ϕ. (5.2)
As we will see a e wa ds, he Gibbons-Hawking bounda y e m will become impo an
once we e alua e he ac ual alue o he Euclidean ac ion.
Fu he mo e, assuming an O(4) symme ic Euclidean solu ion o he ins an on, we
in oduce he ollowing ansa z o he me ic
ds2=dχ2+ρ(χ)2dΩ2
3.(5.3)
Using hese coo dina es we sea ch o a solu ion ea u ing a sphe ical memb ane si ing a
a ixed alue o he adial coo dina e, which we call χ=R. Wi h his se ing, one a i es
o he ollowing equa ions o mo ion o he me ic unc ion and he scala ield,
∂χKϕ¯
ϕ∂χϕ+3∂χρ
ρKϕ¯
ϕ∂χϕ=Kϕ¯
ϕ¯
ϕ|∂χϕ|2+∂˜
V
∂¯
ϕ+eK/2hK¯
ϕ|qϕ|−qeiηiδ(χ−R)(5.4)
ρ′′ =−1
3ρ2Kϕ¯
ϕ|ϕ′|2+˜
V−ρeK/2|qϕ|δ(χ−R),(5.5)
which in he case o a canonical kine ic e m o he ield ϕ, ha is, K(ϕ, ¯
ϕ) = ϕ¯
ϕ, educe o
ϕ′′ +3ρ′
ρϕ′=∂˜
V
∂¯
ϕ+e|ϕ|2/2hϕ|qϕ|−qeiηiδ(χ−R)(5.6)
ρ′′ =−1
3ρ2|ϕ′|2+˜
V−ρe|ϕ|2/2|qϕ|δ(χ−R),(5.7)
13We ha e omi ed he supe symme y-b eaking cubic e m o simplici y, as i s e ec (in his model, a
leas ) was iden ical o u ning on he quad a ic e m.
– 21 –
JHEP10(2023)061
which a e e y simila o he se o equa ions o be sol ed in he case o a pu ely scala
ield bounce solu ion in cu ed space i s ound by Coleman and de Luccia in [2] wi h
he impo an di e ence ha now bo h i s de i a i es o he unc ions ρand ϕp esen a
jump a he posi ion o he memb ane. This is o be expec ed since locally he beha iou
o he unc ions should be he same as he la memb ane case in hese models.
Jus like in he non-g a i a ional case o sec ion 3.3, one may in eg a e eqs. (5.4)–(5.5)
a ound a small neighbo hood o χ=Rin o de o explici ly ind he jump o ϕ′and ρ′
ac oss he memb ane:
[∂χϕ]|χ=R=eK/2hK¯
ϕ|qϕ|−qeiηi,[∂χρ]|χ=R=−ρeK/2|qϕ|.(5.8)
Finally, in o de o ind he adius o he memb ane R, we will di e en ia e be ween
ansi ions s a ing om AdS and Minkowski, which we will co e in sec ion 5.1.1, and
ansi ions s a ing om dS, which we will analyze in sec ion 5.1.2. Fo he o me cases,
we will ake a simila app oach as he one we ook in he non-g a i a ional case, namely,
we will sol e eqs. (5.4)–(5.4) o se e al choices o Rand ind which one ex emizes he
Euclidean ac ion (5.1). On he o he hand, decays in ol ing a dS alse acuum will equi e
special a en ion due o he compac ness o he ins an on.
One should no e, howe e , ha ano he possibili y exis s in o de o ind he adius
o he memb ane o all o hese cases, which elies on he use o he χχ-componen o he
Eins ein equa ions associa ed o ou sys em:
1−(ρ′)2+ρ2
3Kϕ¯
ϕ|ϕ′|2−˜
V= 0 (5.9)
Indeed, his equa ion will be sa is ied ac oss any p o ile only o he pa icula memb ane
adius Rwhich ex emizes he ac ion. The e o e, an al e na i e me hod o ind such Ris
o ine- une i in o de o he l.h.s. o (5.9) o be as close o ze o as possible ac oss he
ull p o ile.
In wha ollows, we ha e elied on he explici e alua ion o he Euclidean ac ion in
o de o ind R. Fu he mo e, eq. (5.9) has been used o es he accu acy o he nume ical
p o iles.
5.1 Example: qua ic supe po en ial
In his subsec ion, we will apply he machine y desc ibed abo e o he simple model o
eq. (4.23). Taking in o accoun he di e en alues o he supe symme y b eaking pa am-
e e µone can encoun e se e al si ua ions depending on he na u e o he alse acuum. In
pa icula , he geome y o he Euclidean space will depend on he sign o he po en ial a
he alse acuum, i.e., while o V ≤0, he backg ound will be a non-compac space, he
case wi h V >0will yield a compac de Si e space o he backg ound. This di e en
p ope ies o he backg ound will become impo an o he way we handle ou nume ical
solu ions.
All he examples below we e sol ed nume ically using a simple o e shoo -unde shoo
algo i hm o ind he co ec ini ial condi ion o he scala ield. No e ha , as a as he
bounda y condi ions a e conce ned, equi ing he space ime o be egula a he cen e o
he bubble implies ha ρ(χ) = χ+O(χ3) o χ→0, see [2] o u he de ails.
– 22 –
JHEP10(2023)061
5.1.1 AdS/Minkowski o AdS ansi ions
As we men ioned abo e, his unnelling e en occu s wi hin a non-compac space. The e-
o e, he in eg al o he Euclidean ac ion co esponding o he ins an on wi h a memb ane
ixed a a pa icula alue o he coo dina e adius Ris gi en by,
SE= 2π2Z∞
0
dχ hρ3|ϕ′|2+˜
V(ϕ, ¯
ϕ)+ 3 ρ2ρ′′ +ρ(ρ′)2−ρi+ 4π2hρ3e|ϕ|2/2|qϕ|iχ=R+SGH
= 2π2Z∞
0
dχ hρ3|ϕ′|2+˜
V(ϕ, ¯
ϕ)−3ρ(ρ′)2+ρi+ 4π2hρ3e|ϕ|2/2|qϕ|iχ=R(5.10)
whe e, in he las s ep, he e m we ha e in eg a ed ou cancels he con ibu ion om he
GH e m (see [41,42]).
On he o he hand, he in eg al co esponding o he backg ound is gi en by,
SE,bg = 2π2Z∞
0
dχ hρ3
V −3ρ (ρ′
)2+ρ i,(5.11)
whe e
ρ (χ) = H−1sinh (Hχ)(5.12)
is he exp ession o he scale ac o o Euclidean an i-deSi e space in he pa icula slicing
gi en by eq. (5.3) and whe e we ha e in oduced he Hubble pa ame e H=q|V |
3. We
can ake he Minkowski limi o his exp ession o ind ha in he case o V = 0, he scale
ac o becomes, ρ (χ) = χ.
I is easy o see ha hese backg ound Euclidean ac ions as well as he ones ob ained
om he ins an on solu ions a e, in ac , di e gen . Howe e , he physically ele an quan-
i y is he di e ence be ween he ins an on ac ion and he backg ound one. This ac ion
is ini e.
The key o compu ing his di e ence co ec ly lies in pe o ming he in eg a ions up
o a ce ain ρmax, such ha i s co esponding adial coo dina e χmax sa is ies χmax ≫R.
See [42] o mo e de ail and an explici p oo o he con e gence o his di e ence.
In igu e 8we show he esul o compu ing his di e ence o se e al memb ane adii
and supe symme y-b eaking pa ame e alues. We can clea ly see ha he di e ence
be ween ac ions is ini e and eaches a maximum a a ce ain R, depending on µ. Fu -
he mo e, as he po en ial ends owa ds i s o iginal and supe symme ic o m, he adius
o he memb ane in e pola ing be ween bo h b anches o he scala po en ial ge s bigge ,
which is consis en wi h he ac ha a he supe symme ic limi he unneling ansi ions
becomes comple ely supp essed.
The p o iles o bo h he scala ield and he scale ac o co esponding o he adii
wi h maximum Euclidean ac ion di e ence o each µa e shown in igu e 9. The scala ield
p o iles a e all qui e simila in shape, wi h he only di e ences es ing on he posi ions o
he ue and alse acuum, and in he adius whe e he jump happens. On he o he hand,
he scale ac o shows e y clea ly whe e he jump happens as well, and he exponen ial
beha iou seems o pick up qui e as once he memb ane has been c ossed.
– 23 –
JHEP10(2023)061
0.7 0.8 0.9 1.0 1.1 1.2
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Figu e 8. (a) Scala po en ial o n= 2 (dashed) and n= 3 (solid). (b) Euclidean ac ion o
di e en ixed memb ane adii R o a lis o supe symme y b eaking pa ame e s µ. The colo
co esponding o each µis he same o bo h igu es. No e ha he adius Rwhich ex emizes he
ac ion inc eases as we diminish he supe symme y-b eaking pa ame e µ.
0 5 10 15 20 25 30
0.90
0.95
1.00
1.05
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0 5 10 15 20 25 30
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Figu e 9. E olu ion o (a) scala ield and (b) scale ac o , o di e en supe symme y pa ame e s
µ. Each case co esponds o he adius o maximum Euclidean ac ion ob ained in igu e 8.
In o de o compa e all hese p o iles wi h he limi ing BPS case, we show in igu e 10
a close-up plo a ound he memb ane o all o hem. Essen ially, we see ha he BPS
p o ile is ac ually a limi ing case o he scala ield, which concu s wi h ou esul s in he
non-g a i a ional case.
Tuning he supe symme y-b eaking pa ame e µ, we can also analyze an almos
Minkowskian alse acuum (µ= 0.33), see igu e 11(a). No e ha Minkowski alse acua
a e s ill non-compac spaces, and hus hey should be analyzed in exac ly he same ashion
as AdS acua.
A e compu ing he Euclidean ac ion di e ence o se e al adii, we ound ha he
adius wi h maximum Euclidean ac ion was R= 5.2in he Minkowskian case. The p o iles
co esponding o a se ing wi h such a memb ane a e shown in igu e 12.
Finally, as explici ly shown in [10,42], he equi emen o ene gy conse a ion can be
gene alized o unnelings whe e g a i a ional e ec s a e conside ed and he alse acuum
is Minkowskian, by equi ing he ADM mass anishes. A e swi ching o Lo en zian
– 24 –
JHEP10(2023)061
uncos ained eal supe ield V=¯
V. I is con enien o de ine i s θ-expansion as ollows:
V(x, θ, ¯
θ) = C+iθχ −i¯
θ¯χ+iθθ ¯
ϕ−i¯
θ¯
θϕ −θσµ¯
θ µ
+iθθ¯
θ¯
λ+i
2¯σµ∂µχ−i¯
θ¯
θθ λ+i
2σµ∂µ¯χ+1
2θθ¯
θ¯
θD−1
2□C.(A.12)
He e u(x)and D(x)a e eal scala ields, ϕ(x)is a complex scala ield, µ(x)is a eal
ec o ield and χ(x)and λ(x)a e Weyl spino s.
A special chi al supe ield Ycan be ob ained om he gene ic chi al supe ield Φ =
¯
D¯
DP by exp essing he complex p epo en ial by P=−i
4V he eal supe ield V, so ha 17
Y:= −i
4¯
D2V(A.13)
( he p e ac o is chosen o la e con enience). I is easy o check ha i ul ills he condi ion
¯
D˙αY= 0 om i s de ini ion. I s componen s can be p ojec ed using
Y|=−i
4¯
D2V=ϕ(A.14)
DαY|=−i
4Dα¯
D2V=λα(A.15)
−1
4D2Y=i
16 D2¯
D2V=1
2(∂µ µ+iD)(A.16)
No e ha , componen -wise, Yis almos iden ical o he o iginal chi al ield Φin ha i
con ains a complex scala ield, a complex Weyl e mion and a complex auxilia y ield,
albei in his case we a e mos ly in e es ed in he eal pa o he la e , which is gi en by
he de e gence o a 4- ec o ield µ.
Fo ou pu poses, i will be con enien o conside µas he one- o m associa ed
h ough Hodge duali y o a h ee- o m. Indeed, he Hodge dual o he h ee- o m is gi en
by he ollowing ec o ield18
(∗A3)µ≡Aµ=1
3!ϵµνρσAνρσ (A.17)
whe e he indices ha e been aised using he la space ime me ic. The di e gence o his
ec o ield is ela ed o he Hodge dual o he 4- o m ield s eng h F4, associa ed o Aνρσ
h ough19
∗F4=1
4!ϵµνρσFµνρσ =∂µAµ(A.18)
17No e ha his de ini ion allows o some eedom in choosing V, since (A.13) is in a ian unde he
gauge supe space symme y V→V+L, whe e Lis he so-called eal linea supe ield which sa is ies
¯
D2L= 0 = D2L. Fu he mo e, he de ini ion o Ymay be gene alized o he case whe e he eal supe ield
Vis no independen bu composed om a so-called complex linea supe ield Σ, which obeys D2Σ = 0,
and i s complex conjuga e ¯
Σ, which obys ¯
D2¯
Σ = 0. This las case may be used o cons uc special chi al
supe ields which desc ibe a double 3- o m supe mul iple . See [18,21,23] o u he de ail.
18In ou con en ions, ϵ0123 =−ϵ0123 = 1 o Lo en zian signa u e (which we will use h oughou his
appendix), while ϵ0123 =ϵ0123 = 1 o Euclidean one.
19We ha e omi ed he con ibu ion o he me ic de e minan in hese exp essions o cla i y he de i-
ni ions. In he case o a cu ed space ime, hese exp essions a e (∗A3)µ≡Aµ=1
3! √−gϵµνρσAνρσ and
∗F4=1
√−g∂µ(√−gAµ).
– 31 –

JHEP10(2023)061
Wi h hese exp essions in hand, we ind ha i µis iden i ied wi h Aµ, i.e, wi h he Hodge
dual o he 3- o m ield, he composi e auxilia y ield o he special chi al supe ield Y eads
FY=−1
4D2Y=1
2(∗F4+iD).(A.19)
The ad an age o ea ing he ec o µas dual o 3- o m is ha a memb ane can be
coupled ’elec ically’ (o minially) o he 3- o m ield (see below) and hus o he composi e
auxilia y ield in he special chi al supe ield Y. Since Ddoes no en e he memb ane pa
o he ull ac ion, we will be able o emo e i om he ac ion by sol ing i s algeb aic
equa ions ea ly on, lea ing ∗F4un ouched o i s in e play wi h he scala ield and he
memb ane.
We can easily apply hese exp essions o a single h ee- o m mul iple Y, wi h a ce ain
Kähle po en ial K(Y, ¯
Y)and supe po en ial W(Y). Plugging he componen ields o Y
in o he bosonic Lag angian (A.10), we ind
L|bos. =−Kϕ¯
ϕ∂µϕ∂µ¯
ϕ+1
4Kϕ¯
ϕ(∗F4+iD) (∗F4−iD) + 1
2(∗F4+iD)Wϕ+1
2(∗F4−iD)¯
W¯
ϕ
=−Kϕ¯
ϕ∂µϕ∂µ¯
ϕ+1
4Kϕ¯
ϕ(∗F4)2+1
2(∗F4)Wϕ+¯
W¯
ϕ+1
4Kϕ¯
ϕD2+i
2DWϕ−¯
W¯
ϕ
(A.20)
The equa ion o mo ion o he auxilia y ield D eads
D=−iKϕ¯
ϕWϕ−¯
W¯
ϕ(A.21)
which is comple ely algeb aic, as expec ed. Plugging his in o (A.20) yields
L|bos. =−Kϕ¯
ϕ∂µϕ∂µ¯
ϕ+1
4Kϕ¯
ϕ(∗F4)2+1
2(∗F4)Wϕ+¯
W¯
ϕ+1
4Kϕ¯
ϕWϕ−¯
W¯
ϕ2(A.22)
=−Kϕ¯
ϕ∂µϕ∂µ¯
ϕ−1
4·4!Kϕ¯
ϕFµνσρFµνσρ +1
2·4! Wϕ+¯
W¯
ϕϵµνσρFµνσρ
+1
4Kϕ¯
ϕWϕ−¯
W¯
ϕ2(A.23)
whe e, in he second s ep, we ha e ew i en he Hodge duals in e ms o hei o iginal
enso ields o cla i y.
In o de o p oceed, ecall ha he physical ield we wish o ex emise in he ac ion is
no he ield s eng h, bu a he i s an isymme ic enso po en ial Aµνρ. In he ollowing,
i will be mo e con enien o wo k wi h he Hodge-dual Aµ, which is ela ed o ∗F4by
eq. (A.18). The a ia ion o (A.23) wi h espec o Aµis
1
2Kϕ¯
ϕ(∗F4) (∂µδAµ) + 1
2(∂µδAµ)Wϕ+¯
W¯
ϕ
=δAµ1
2∂µ−Kϕ¯
ϕ(∗F4) + Wϕ+¯
W¯
ϕ+∂µδAµ
1
2Kϕ¯
ϕ(∗F4)−Wϕ−¯
W¯
ϕ.(A.24)
We can clea ly see ha he anishing o he i s e m in he .h.s. will gi e us an equa ion
o mo ion o he 3- o m gauge ield, while he second one will p oduce a bounda y e m.
– 32 –
JHEP10(2023)061
As o iginally discussed in [10], in o de o deal wi h he second e m in (A.24) we will
need o add a bounda y e m o ou o iginal Lag angian o con e his con ibu ion in o
an exp ession con aining he a ia ions o gauge in a ian quan i ies (F4and scala ields).
Indeed, o he wise he a ia ional p oblem wi h espec o gauge po en ial would no be well
posed (see [18] o ecen discussion). This bounda y e m no only ensu es he consis ency
o he a ia ional p oblem bu , as we will see sho ly, i will ha e a no iceable e ec on he
inal, on-shell esul .
The equi ed bounda y e m is gi en by
Lbd =−1
2∂µhAµKϕ¯
ϕ(∗F4)−Wϕ−¯
W¯
ϕi
=1
2·3!∂µhAνρσ Kϕ¯
ϕFµνρσ +ϵµνρσ Wϕ+¯
W¯
ϕi (A.25)
while he equa ion o mo ion o he o m ield is
∂µ1
2Kϕ¯
ϕ(∗F4) + Re Wϕ= 0 ⇒ ∗F4=−2Kϕ¯
ϕ(ReWϕ−n)(A.26)
whe e n∈Ris an a bi a y eal (in eg a ion) cons an . As his cons an appea s in he
exp ession o he 4- o m ield s eng h, i can be iden i ied wi h he lux o he 3- o m
gauge ield.
Plugging his esul in o (A.23) and aking in o accoun he non- anishing con ibu ion
o he bounda y e m yields
L|bos.,on-sh. =−Kϕ¯
ϕ∂µϕ∂µ¯
ϕ−Kϕ¯
ϕ(Wϕ−n)¯
W¯
ϕ−n.(A.27)
F om his inal o m o he Lag angian we can conclude ha he con ibu ion o 3- o ms
in a supe symme ic se up esul s in a linea con ibu ion o ou o iginal supe po en ial
in which he ield is mul iplied by a cons an associa ed wi h he lux o he 3- o m gauge
ield. Thus, se ing he 3- o ms on shell educes he o iginal heo y o a model o a scala
ield desc ibed by he o iginal Kähle po en ial and an e ec i e supe po en ial gi en by
ˆ
W(ϕ)≡W(ϕ)−nϕ. (A.28)
B Rigid supe symme y, bosonic memb anes, and BPS equa ions
As i was shown in [36,37], he ac ion o he in e ac ing sys em composed o he supe -
g a i y mul iple and a bosonic memb ane is in a ian unde a hal o local supe symme y,
p o ided he memb ane e m o he in e ac ing ac ion is gi en by he bosonic ‘limi ’ o a
supe memb ane. Fu he mo e, he p ese ed pa o he local supe symme y e lec s he
local e mionic κ-symme y o he o iginal supe memb ane ac ion.
In he case o an in e ac ing sys em composed o supe symme ic ma e and a supe -
memb ane, he e is no igo ous way o ind igid supe symme y in a iance o he in e -
ac ing sys em including he ma e supe mul iple and a bosonic memb ane. Howe e , as
we will show below, he e exis s a ick allowing us o see he κ-symme y o he o iginal
– 33 –
JHEP10(2023)061
supe memb ane pa o he ac ion o he in e ac ing sys em. I implies he manipula ion
o he supe symme y ans o ma ion o he bosonic memb ane ac ion wi h he use o some
ansa ze bo h o he ields and o memb ane con igu a ion. Ac ually, such a possibili y,
when i exis s, e lec s he exis ence o pu ely bosonic supe symme ic solu ions o he com-
ple e supe symme ic sys em.20 In he ollowing we will show how using his a gumen s
we ob ained he simple o m o he BPS equa ions o ou b ane in ou model.
Le us s a by looking a he supe symme y ans o ma ion in ou model. Al hough
we did no w i e hose ans o ma ions o he 3- o m supe mul iple in he main ex ,
hey can be easily es o ed om he associa ion o he space ime ields wi h he supe -
ield componen s in eqs. (A.14), (A.15), (A.16) and he iden i ica ion o supe symme y
as e mionic supe ansla ions in supe space. This leads o he ollowing supe symme ic
ans o ma ions o he ields,
δϵϕ=ϵαDαY|=ϵαλα,(B.1)
δϵλα=ϵβDβDαY|+ ¯ϵ˙
β¯
D˙
βDαY|= 2ϵαFY+ 2i(σµ¯ϵ)α∂µϕ
=ϵα(∂µAµ+iD) + 2i(σµ¯ϵ)α∂µϕ , (B.2)
and
δϵFY=1
2(∂µδϵAµ+iδϵD) = −1
4¯ϵ˙
β¯
D˙
βD2Y|=i(∂µλσµ¯ϵ).(B.3)
These ans o ma ions also imply
δϵD = ∂µλσµ¯ϵ+ϵσµ∂µ¯
λ , δϵAµ=i(∂µλσµ¯ϵ−ϵσµ∂µ¯
λ),(B.4)
as well as
δϵAµνρ =iϵµνρσ(∂µλσσ¯ϵ−ϵσσ∂µ¯
λ),(B.5)
o he dual 3- o m Aµνρ =ϵµνρσAσ.
Le us now pe o m he abo e supe symme y ans o ma ion o he scala and 3- o m
ield in he bosonic memb ane ac ion (3.7):
δϵSmemb .=−2|q|Zd3ξ√−hRe ϵαλα
¯
ϕ
|¯
ϕ|!+ 2qZd3ξIm (λσµ¯ϵ)ϵµνρσ∂0xν∂1xρ∂0xσ.
(B.6)
Gene ically, his las exp ession does no anish. Howe e , i we conside a la memb ane
lying a x3=z= 0 in he s a ic gauge, i.e.,
xa(ξ) = ξa, x3(ξ) := z(ξ)=0,(B.7)
hen √−h= 1 and eq. (B.6) educes o
δϵSmemb .=−2|q|Zd3ξRe ϵαλα
¯
ϕ
|¯
ϕ|−iq
|q|(¯ϵ˜σ3λ)!.(B.8)
20The ela ion o he κ-symme y o a supe -p-b ane wi h supe symme y p ese ed by solu ions o
equa ions was desc ibed o he i s ime in [39].
– 34 –
JHEP10(2023)061
This o mally anishes i we se
ϵα=i(¯ϵ˜σ3)αqϕ
|qϕ|z=0
.(B.9)
This equa ion has non i ial solu ions o a cons an e mionic spino ϵαi qϕ
|qϕ|z=0 is
independen o ξa(i.e., as long as i ep esen s a cons an phase). This condi ion is clea ly
sa is ied i we assume he scala ield o depend on he zcoo dina e only, ha is,
ϕ(xa, z) = ϕ(z).(B.10)
This and he es ic ion o a la memb ane gi en by he las equa ion in (B.7) de ine he
ansa z o he supe symme ic bosonic solu ion o he in e ac ing sys em o he 3- o m
mul iple and supe memb ane.
Le us epea ha hese o mal calcula ions e lec he κ-symme y o he comple e
supe memb ane ac ion, which gua an ees he exis ence o he pu ely bosonic supe sym-
me ic solu ion o he equa ions o he in e ac ing sys em o a supe memb ane and he
single 3- o m ma e supe mul iple .
Le us look a he ans o ma ion o he ields in he bulk. Fo a pu ely bosonic solu ion
ob ained wi h he ansa z (B.10), he p ese a ion o supe symme y implies δϵλ= 0 which,
a e using (B.2) and he auxilia y ield’s equa ion
2FY= (∗F4+iD) = −Kϕϕ ˆ
¯
W¯
ϕ(B.11)
educes o
i(σ3¯ϵ)α∂zϕ=ϵαKϕϕ ˆ
¯
W¯
ϕ.(B.12)
This equa ion has a non i ial solu ion wi h he cons an e mionic spino obeying p ojec -
ing condi ion
ϵα=eiηi(σ3¯ϵ)α⇔ϵα=−eiηi(¯ϵ˜σ3)α(B.13)
i he scala ield obeys he BPS equa ion
∂zϕ(z) = eiηKϕ¯
ϕˆ
¯
W¯
ϕ.(B.14)
Fu he mo e, he supe symme y p ese ed by he bosonic con igu a ion will also p e-
se e he supe symme y o he s a ic la bosonic memb ane con i gu a ion gi en by (B.7)
i (B.13) coincides wi h (B.9), i.e. when
eiη =−qϕ
|qϕ|z=0
.(B.15)
Gi en he BPS equa ion (B.14), i is easy o check ha he phase o ∂zˆ
W emains
cons an e en ac oss he memb ane. Indeed, i we mul iply bo h sides o he equa ion by
ˆ
Wϕ, we ind
(∂zϕ)ˆ
Wϕ=eiη ˆ
WϕKϕ¯
ϕˆ
¯
W¯
ϕ⇒∂zˆ
W=eiη hV(ϕ, ¯
ϕ) + |qϕ|δ(z)i(B.16)
– 35 –
JHEP10(2023)061
The e o e, one inds ha eiη may be w i en as
eiη =−qϕ
|qϕ|z=0
=∆ˆ
W
|∆ˆ
W|(B.17)
whe e ∆ˆ
W≡ˆ
Wz=+∞−ˆ
Wz=−∞. No e ha his las esul exac ly ep oduces he phase
ob ained in [3], albei in ou case i includes he e ec o he memb ane localized a z= 0,
which e ec i ely changes he supe po en ial when c ossing i .
One may also use his esul in o de o ind a closed exp ession o he ension o
he domain wall solu ion in he p esence o a supe memb ane. As shown in [23], one may
ew i e he ac ion (3.13) as
S=−Zd3xZdzKϕ¯
ϕh∂zϕ−eiβKϕ¯
ϕˆ
¯
W¯
ϕih∂z¯
ϕ−e−iβKϕ¯
ϕˆ
Wϕi
−Zd3x2|qϕ|z=0 + 2Re he−iβ(∆ ˆ
W+qϕ|z=0)i.(B.18)
whe e he phase eiβ is, a his s age, a bi a y. Howe e , upon choosing eiβ =eiη one
can see ha he exp ession o he ension o any ield con igu a ion is maximized by he
solu ion o he BPS equa ion (B.14). Taking his in o accoun he ension becomes
TDW+memb. =|2∆ ˆ
W|.(B.19)
No e ha e en hough his exp ession is o mally simila o he one in he scala ield model
in [3] he esul is now w i en in e ms o he e ec i e supe po en ial ˆ
Wwhich includes
he jump due o he di e en luxes on bo h sides o he memb ane.
As desc ibed in [21] and e e ences he ein, simila a gumen s can be applied o ob ain
he coun e pa o (B.14) o dynamical sys ems including supe g a i y, which lead o
eqs. (4.13)–(4.14).
C Th ee- o m mul iple s in supe g a i y
In his sec ion we will ollow a simila easoning as he one abo e, wi h g a i y included.
We will i s p esen he ac ion o scala mul iple s desc ibed by gene ic chi al supe ields,
which we will la e on gene alize o include special chi al supe ields which ha e ec o
o 3- o m componen s among hei bosonic ing edien s. All o he esul s we p esen he e
ha e also been de i ed in [18,23] using a supe -Weyl in a ian app oach o ma e -coupled
supe g a i y, eaching he same conclusions.
We ecall ha supe g a i y can be desc ibed in e ms o he supe space supe ielbein
EA
M(z)and he spin connec ion ωab
M(z) = −ωba
M(z)subjec o a se o o sion cons ain s.
A e ixing he so-called Wess-Zumino gauge, he ield con en educes o
•eµ
a, he ielbein,
•ψµ
α, he g a i ino,
•bµ, a eal ec o auxilia y ield,
•M, a complex scala auxilia y ield.
– 36 –

JHEP10(2023)061
The mos gene al supe space ac ion o in e ac ing supe g a i y and scala supe mul iple s
in e ms o EA
M(z)and gene ic chi al supe ields (de ined in cu ed supe g a i y supe -
space) can be ound e.g. in [25]. The space ime ac ion o he componen ields is hen
ob ained upon ixing he Wess-Zumino gauge and in eg a ing o e he e mionic coo di-
na es o supe space. In he con en ions o [25], he space ime Lag angian o he bosonic
sec o o such an ac ion eads
1
√−gL=1
2Re−1
3K+ Ωa¯
b∂µϕa∂µ¯
ϕ¯
b−1
3e−1
3K˜
M˜
¯
M−˜
M¯
W−˜
¯
MW
+e−1
3KKa¯
bFa¯
F¯
b+Fa(Wa+KaW) + ¯
F¯
b(¯
W¯
b+K¯
b¯
W)
−1
9Ωbµbµ−i
3bµ(∂µϕaΩa−∂µ¯
ϕ¯
bΩ¯
b),(C.1)
whe e
Ω(Φ,¯
Φ) = −3e−1
3K(Φ,¯
Φ),˜
M=M+K¯a¯
F¯a,˜
¯
M=¯
M+KaFa.(C.2)
This ac ion is no w i en in Eins ein ame. The e o e, i is cus oma y o escale he
ielbein as ollows:
ea
µ7→ ea
µe1
6K,(C.3)
and o supplemen his wi h a sui able ans o ma ion o he spin connec ion. Fu he mo e,
escaling he auxilia y ields as
Fi7→ Fie−1
6K, M 7→ M e−1
6K,(C.4)
we a i e a he ollowing ac ion in Eins ein ame
1
√−gL=1
2R−Ka¯
b∂µϕa∂µ¯
ϕ¯
b−1
3˜
M˜
¯
M−e1
2K˜
M¯
W−e1
2K˜
¯
MW
+Ka¯
bFa¯
F¯
b+e1
2KFa(Wa+KaW) + e1
2K¯
F¯
b(¯
W¯
b+K¯
b¯
W).(C.5)
No ice ha in his las exp ession he auxilia y ields bµha e been in eg a ed ou using hei
equa ions o mo ion. I we a e dealing wi h minimal supe g a i y and scala mul iple s, ˜
M
and all Fia e independen , and he auxilia y ield equa ions ead
˜
M=−3e1
2KW , FaKa¯
b=−e1
2K(¯
W¯
b+K¯
b¯
W).(C.6)
Subs i u ing his in o (C.5) we ind he well known ma e -coupled N=1, D= 4 supe g a -
i y Lag angian
1
√−gL=1
2R−Ka¯
b∂µϕa∂µ¯
ϕ¯
b−V(ϕ, ¯
ϕ)(C.7)
whe e he po en ial is gi en by
V(ϕ, ¯
ϕ) = eKDaWKa¯
bD¯
b¯
W−3|W|2,(C.8)
and Da=∂a+Kaa e he so-called Kähle -co a ian de i a i es.
– 37 –
JHEP10(2023)061
C.1 Supe g a i y in e ac ing wi h 3- o m mul iple s
Jus as in he non-g a i a ional case, we will implici ly in oduce h ee- o ms by passing
om gene ic o special chi al supe ields. In his case, special chi al supe ields desc ibing
single h ee- o m mul iple s a e gi en by
S=−i
4(¯
D2−8R)P,P= (P)∗(C.9)
whe e Ris he so-called main chi al supe ield o minimal supe g a i y, whose leading
componen is p opo ional o he complex scala auxilia y ield, R| =−1
6M, and Pis
an uncons ained eal supe ield. The la e is de ined up o shi by a eal linea supe -
ield (C.9)
P 7→ P +L, (¯
D2−8R)L= 0 = (D2−8¯
R)L. (C.10)
This eedom is he mani es a ion o a gauge symme y which can be used o ix he Wess-
Zumino gauge, whe e
P| = 0 , DαP| = 0 ,¯
D˙αP| = 0 .(C.11)
The emaining pa o he Lsymme y, p ese ing his gauge, coincides wi h he 2- o m
gauge symme y o he 3- o m dual o ec o componen o he p epo en ial supe ield,
σa
α˙α[Dα,¯
D˙α]P| = 4Aa, Aa=∗(A3)a.(C.12)
In he gauge (C.11), he highes componen o he special chi al supe ield o Ssimpli ies
o (in he no a ion o [25])
F ≡ FS=−1
4D2S|=i
16 D2¯
D2P|− i
2RD2P| =1
2(DµAµ+id) + 1
3(s¯
M+ 2¯sM).(C.13)
whe e s=S|,Mand ¯
Ma e he scala auxilia y ields o minimal supe g a i y, dis a eal
auxilia y scala ield, and
∗F4=DµAµ=1
e∂µ(eAµ), e = de ea
µ=√−g . (C.14)
Fixing he WZ gauge and in eg a ing o e he e mionic coo dina es o supe space in
he supe ield ac ion we a i e a he ollowing Lag angian in he bosonic limi :
1
√−gL=1
2Re−1
3K+ Ωs¯s∂µs∂µ¯s−1
3e−1
3K(M+K¯s¯
F) ( ¯
M+KsF)−M¯
W−¯
MW
+e−1
3KKs¯sF¯
F+FWs+¯
F¯
W¯s−1
9Ωbµbµ−i
3bµ(∂µsΩs−∂µ¯sΩ¯s) + 1
√−gLbd
(C.15)
This Lag angian o mally coincides wi h ha o scala mul iple s in e ac ing wi h supe -
g a i y (C.5) up o bounda y e m, and up o he composi e na u e o he F-componen
Fo he special chi al supe ields (C.13).
– 38 –
JHEP10(2023)061
Be o e subs i u ing he exp ession o he F-componen s o he special chi al supe -
ields, (C.13), i is con enien o pe o m a Weyl escaling o he ields wi h (C.3), which
will b ing ou ac ion o he Eins ein ame. In he ollwoing, will ca e ully conside he
case o supe g a i y in e ac ing wi h special chi al supe ields desc ibing 3- o m mul iple s,
as his case has some peculia i ies wi h espec o he case o gene ic chi al supe ields
desc ibing scala supe mul iple s.
C.1.1 Supe -Weyl ans o ma ions
A he beginning o his appendix, when conside ing he in e ac ion o supe g a i y an
chi al supe ields Φ, we assumed ha each supe ield Φand i s componen s ϕand Fa e
ine unde Weyl ans o ma ions. This is a consis en assump ion in he case o a gene ic
chi al supe ield.21 Howe e , since he chi al supe ield Shas been w i en in e ms o a
eal supe ield P, i is impo an o check how supe -Weyl ans o ma ions, ac on hem.
These a e de ined ia he ollowing ans o ma ions o supe ielbein [25,68]:
Ea7→ ˜
Ea=eΥ+¯
ΥEa,(C.16)
Eα7→ ˜
Eα=e2¯
Υ−ΥEα−i
4Ea¯
D˙α¯
Υ˜σ˙αα
a,(C.17)
¯
E˙α7→ ˜
¯
E˙α=e2Υ−¯
Υ¯
E˙α+i
4Ea˜σ˙αα
aDαΥ,(C.18)
whe e Υis a chi al supe ield
¯
D˙αΥ=0, Dα¯
Υ=0.(C.19)
These ans o ma ions mus be supplemen ed by a sui able ans o ma ions o he spin
connec ion; howe e , hei explici o m is no needed o ou pu poses (see [25,68] o
mo e de ail).
I is impo an o no e ha he supe g a i y chi al p ojec o ans o ms in an inho-
mogeneous way unde supe -Weyl ans o ma ions:
(¯
D¯
D−8R)7→ e−4Υ(¯
D¯
D−R)e2¯
Υ,(DD −8¯
R)7→ e−4¯
Υ(DD −8¯
R)e2Υ.(C.20)
In he case o a gene ic chi al supe ield Φ=(¯
D¯
D−8R)P, his p ojec o ac s on he
gene ic complex supe ield po en ial P. Choosing he ans o ma ion o his supe ield o
be P7→ e+4 ¯
Υe−2ΥPwe can ac ually make Φine unde he supe -Weyl an o ma ions.
On he o he hand, his is no possible in he case o a special chi al supe ield S(C.9)
cons uc ed om he eal supe ield p epo en ial P= (P)∗. In his case, in he ligh
o (C.20), he only way o ob ain a co a ian supe -Weyl ans o ma ion o he special
chi al supe ields (C.9) is o a ibu e o i s eal p epo en ial he ans o ma ion ule
P 7→ P e−2Υ−2¯
Υ(C.21)
21This can be checked by w i ing he chi al ield in e ms o an uncons ained complex supe ield (wi h a
simila de ini ion as (C.9)). Choosing he Weyl weigh s o he ans o ma ion acco dingly, i can be shown
ha a gene al chi al supe ield Φis in a ian unde hese escalings.
– 39 –
JHEP10(2023)061
which esul s in
S7→ S e−6Υ (C.22)
and in he ollowing ans o ma ions o i s leading componen 22
s7→ s e−6Υ|.(C.23)
We will be in e es ed in he pu ely bosonic pa o he supe -Weyl ans o ma ions
wi h
Υ|=1
12K=¯
Υ|, DαΥ|= 0 , D2Υ|= 0 ,(C.24)
since, in ha case,
ea
µ7→ ea
µe1
6K(C.25)
as needed o w i e he Lag angian in Eins ein ame, jus as in he case o supe g a i y
in e ac ing wi h chi al mul iple s. Howe e , in his scena io, bo h Mand Fwill be a ec ed
by his ans o ma ion. This can be seen om
(D2−8¯
R)S|=−4F −8RS|=−4F+4
3s¯
M(C.26)
whose ans o ma ion wi h he use o (C.23) and (C.24) esul s in
s7→ e−1
2Ks(C.27)
F 7→ e−2
3KF(C.28)
M7→ e−1
6KM . (C.29)
As a as he scala ield is conce ned, i is con enien o combine he Weyl escaling wi h
he ield ede ini ion
ϕ:= e−1
2Ks(C.30)
so ha he kine ic e m o ϕ ield emains in a canonical o m.
Taking all o he abo e in o accoun and in eg a ing ou he auxilia y ield bµusing i s
algeb aic equa ions o mo ion, we ind ha
1
√−gL=1
2R−Kϕ¯
ϕ∂µϕ∂µ¯
ϕJ−1
3(M+e−1
2KK¯
ϕ¯
F) ( ¯
M+e−1
2KKϕF)
−e1
2KM¯
W−e1
2K¯
MW +e−KKϕ¯
ϕF¯
F+FWϕ+¯
F¯
W¯
ϕ+1
√−gLbd.(C.31)
I is con enien o u he ede ine he supe g a i y auxilia y ield Mas
ˇ
M≡Me1
2K(C.32)
22We do no w i e ans o ma ion o P|explici ly because i anishes in he Wess-Zumino gauge (C.11).
– 40 –