Uni e sal class o ype-IIB lux acua wi h analy ic mass spec um
Jose J. Blanco-Pillado ,1,2,* Kepa Sousa ,3,†Mikel A. U kiola ,1,‡and Je emy M. Wach e 4,§
1Depa men o Physics, Uni e si y o he Basque Coun y UPV/EHU, 48080 Bilbao, Spain
2IKERBASQUE, Basque Founda ion o Science, 48011 Bilbao, Spain
3Ins i u e o Theo e ical Physics, Cha les Uni e si y, V Holesoˇ ičkách 2, P ague, Czech Republic
4Skidmo e College Physics Depa men , 815 No h B oadway Sa a oga Sp ings, New Yo k 12866, USA
(Recei ed 7 Feb ua y 2021; accep ed 8 Ap il 2021; published 4 May 2021)
We epo on a new class o lux acua gene ically p esen in Calabi-Yau compac i ica ions o ype-IIB
s ing heo y. A hese acua, he mass spec um o he comple e axiodila on/complex s uc u e sec o is
gi en, o leading o de in α0and gs, by a simple analy ic o mula independen o he choice o Calabi-Yau.
We p o ide a me hod o ind hese acua and cons uc an ensemble o 17,054 solu ions o he Calabi-Yau
hype su ace WP4
½1;1;1;6;9, whe e he masses o he axiodila on and he 272 complex s uc u e ields can be
explici ly compu ed.
DOI: 10.1103/PhysRe D.103.106006
I. INTRODUCTION
The s udy o he phenomenological implica ions o s ing
heo y demands he cons uc ion o low-ene gy e ec i e
ield heo ies (EFTs) desc ibing i s compac i ica ion o ou
dimensions. Howe e , de i ing hese EFTs is ema kably
challenging and in ol es, in pa icula , in eg a ing ou a la ge
numbe o scala ields ( ypically hund eds) desc ibing he
geome y o he compac i ied dimensions, i.e., he moduli.
Ac ually, inding a mechanism o gene a e he moduli
masses is a c ucial s ep in he bes s udied p oposals o
cons uc de Si e acua in ype-IIB s ing heo y, i.e., he
Kach u-Kallosh-Linde-T i edi (KKLT) [1] and la ge olume
scena ios (LVS) [2,3]. Bo h cons uc ions ely on he
iden i ica ion o lux acua: minima o he e ec i e po en ial
induced by highe -dimensional o m ields. Al hough he
gene al ea u es o he lux po en ial a e well cha ac e ized, a
de ailed compu a ion o he moduli mass spec a in gene ic
scena ios is s ill ex emely di icul due o he complexi y o
he heo y and he la ge numbe o ields in ol ed. Mo e
speci ically, he so-called no-scale s uc u e o he lux
po en ial ensu es ha a subse o he moduli, namely he
axiodila on and complex s uc u e ields, can be ixed a a
pe u ba i ely s able con igu a ion p o ided hey p ese e
supe symme y.
Howe e , he s abiliza ion o he emaining moduli
ields, i.e., he Kähle moduli, equi es including α0
pe u ba i e co ec ions and nonpe u ba i e con ibu ions,
which spoil he no-scale s uc u e [4,5] (see also [6–8]).
The e o e, he unco ec ed lux acua may become
achyonic, o e en cease being c i ical poin s o he
po en ial. Indeed, while he s abili y o he axiodila on/
complex s uc u e sec o in he ully s abilized acuum has
been a gued in KKLT and LVS using scaling a gumen s
[3,9–12] and by he di ec examina ion o explici examples
(see, e.g., [13,14]), i has a ely been s udied in de ail.
Ac ually, as discussed in [15–17], in bo h KKLT and LVS
scena ios he p esence o ligh (o massless) ields in he
spec um o leading o de in α0and quan um co ec ions
may s ill lead o he appea ance o ins abili ies in he inal
acuum. In e es ingly, he p esence o such dange ously
ligh modes has been epo ed o a ise in explici con-
s uc ions o de Si e acua, whe e he moduli a e
s abilized nea special poin s o he moduli space [18–
20]. Fu he mo e, ecen analyses indica e an exis ing
ension be ween he D3- adpole cancella ion condi ion
and he need o s abilize all he complex s uc u e moduli
[21–24], wha could ha e impo an implica ions o he
consis ency o KKLTand LVS p oposals. While signi ican
p og ess has been made in c ucial aspec s o moduli
s abiliza ion in he las couple o yea s [14,20,22,25],a
p ecise cha ac e iza ion o he mass spec um in he
axiodila on/complex s uc u e sec o has emained elusi e,
which calls o u he s udies in his di ec ion.
Ad ances on his ma e we e ecen ly made in [26]
( ollowing [17,27,28]) o compac i ica ions on he o ien i-
old o Calabi-Yau mani olds, which allow he consis en
unca ion o all he complex s uc u e ields excep one.
The analysis o [26] assumed a Calabi-Yau geome y
*[email p o ec ed]
†[email p o ec ed]
‡[email p o ec ed]
§jwach e @skidmo e.edu
Published by he Ame ican Physical Socie y unde he e ms o
he C ea i e Commons A ibu ion 4.0 In e na ional license.
Fu he dis ibu ion o his wo k mus main ain a ibu ion o
he au ho (s) and he published a icle’s i le, jou nal ci a ion,
and DOI. Funded by SCOAP3.
PHYSICAL REVIEW D 103, 106006 (2021)
2470-0010=2021=103(10)=106006(12) 106006-1 Published by he Ame ican Physical Socie y
admi ing a la ge disc e e isome y g oup which, p o ided
he lux con igu a ion is also in a ian unde hese sym-
me ies, allows he e ec i e educ ion o he complex
s uc u e sec o . In his se ing i was shown ha he
comple e mass spec um o he axiodila on and complex
s uc u e sec o (including he unca ed ields) can be
explici ly compu ed in he la ge complex s uc u e (LCS)/
weak s ing coupling egime. Mo e speci ically, o he
class o acua, which can be ound pa ame ically close o
he LCS poin [29,30] and up o exponen ially small
co ec ions, he scala moduli masses a e gi en by he
simple analy ic o mula
μ2
λ
m2
3=2¼
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
1ffiffiffiffiffiffiffiffiffiffi
ð1−2ξÞ
p
ffiffi3
pˆ
mðξÞ2
λ¼0;
1ffiffiffiffiffiffiffiffiffiffi
ð1−2ξÞ
p
ffiffi3
pˆ
mðξÞ2
λ¼1;
11þξ
32λ¼2;…;h
2;1
−:
ð1Þ
He e, he quan i y ξpa ame izes he complex s uc u e
moduli space, anging in ξ∈½0;1=2Þ o h2;1
−>h
1;1
þo in
ξ∈ð−1;0 o h2;1
−<h
1;1
þ, wi h he LCS poin loca ed a
ξ¼0, and wi h h2;1
−and h1;1
þdeno ing he numbe o
complex s uc u e and Kähle moduli ields o he Calabi-
Yau o ien i old, espec i ely. The quan i y m3=2is he
g a i ino mass, and
ˆ
mðξÞ≡1
ffiffiffi
2
p2þκðξÞ2−κðξÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4þκðξÞ2
q1=2
ð2Þ
wi h κðξÞ¼2ð1þξÞ2=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3ð1−2ξÞ3
p.
In he p esen pape we p o e ha , in he LCS egime
and p o ided he luxes a e con enien ly cons ained, he
EFT o gene ic Calabi-Yau compac i ica ions always
admi s a consis en unca ion o all complex s uc u e
ields bu one. He e, in con as wi h [26], we do no equi e
he p esence o a disc e e isome y g oup, and ou esul
elies ins ead on he exis ence o monod omy ans o ma-
ions a ound he LCS poin . Tha is, we will equi e only he
in a iance o he EFT unde disc e e shi s o he complex
s uc u e ields zi
zi→ziþ i;
i∈Zh2;1;ð3Þ
combined wi h an app op ia e ans o ma ion o he luxes
o he o m ields. This in a iance is a common ea u e o
all Calabi-Yau compac i ica ions in he LCS egime.
Mo eo e , he choice o ield su i ing he unca ion is
highly nonunique, wi h each possibili y associa ed o a
di e en monod omy di ec ion i.
This simple, and ye powe ul, obse a ion allows us
o ex end he esul s o [26] o gene ic Calabi-Yau com-
pac i ica ions and, as a consequence, opens he doo o
gene a ing a la ge landscape o acua wi h an unp ec-
eden ed analy ic con ol o e he mass spec um o he
axiodila on and complex s uc u e moduli. I is impo an o
no e ha solu ions discussed he e a e no he dominan
class in he landscape. In ac , hey only cons i u e a small
ac ion o he o al numbe o acua o la ge alues o
h2;1
−≫1. Howe e , as we shall see below, ou app oach
allows us o sea ch o hese solu ions in a e y e icien
way, which makes i pa icula ly a ac i e o explici
cons uc ions o de Si e acua.
II. CONSISTENT TRUNCATION OF THE EFT
We begin by in oducing he e ec i e supe g a i y
heo y desc ibing he low-ene gy egime o ype-IIB
s ing heo y compac i ied on a Calabi-Yau o ien i old
˜
M3. The couplings o he heo y a e con enien ly exp essed
by speci ying an in eg al and symplec ic homology basis
AI;B
Igo H3ðX3;ZÞ, sa is ying AI∩BJ¼δI
Jand
AI∩AJ¼BI∩BJ¼0, wi h I¼0;…;h
2;1
−. In pa icula ,
he componen s in his basis o he Calabi-Yau (3,0) o m
ΩðziÞcan be encoded in he pe iod ec o
ΠT≡ðFI;XIÞ¼RBI
Ω;RAIΩ:ð4Þ
He e XIa e p ojec i e coo dina es in he complex s uc u e
moduli space, and he co esponding moduli ields can be
de ined o be zi≡Xi=X0,i¼1;…;h
2;1
−. Then, o leading
o de in α0and he s ing coupling gs, he Kähle po en ial
o he co esponding ou -dimensional e ec i e supe g a -
i y heo y eads [31,32]
K¼−2log V−logð−iðτ−¯τÞÞ−log ð−iΠ†·Σ·ΠÞ;ð5Þ
whe e Vdeno es he Kähle moduli-dependen olume o
˜
M3in uni s o 2πffiffiffiffi
α0
pand measu ed in Eins ein ame, and
Σ¼01
−10is he symplec ic ma ix. The quan i ies FI
can be exp essed as he de i a i es o a holomo phic
unc ion o he XI, he p epo en ial FðXIÞ, which in he
LCS egime admi s he expansion
F¼−
1
3! κijkzizjzk−
1
2! κijzizjþκiziþ1
2κ0þ…;ð6Þ
whe e we ha e chosen he gauge X0¼1. The e ms κijk,
κij, and κia e nume ical cons an s which can be compu ed
om he opological da a o he mi o mani old o M3(see
[33]). In pa icula , he quan i ies κijk a e in ege s, he
coe icien s κij and κia e a ional, and he cons an κ0¼
ζð3ÞχðM3Þ=ð2πiÞ3is de e mined by he Eule numbe
χðM3Þo he Calabi-Yau. The p epo en ial also ecei es
con ibu ions om wo ld-shee ins an ons, which a e sub-
leading in he LCS egime, and hus hey will be neglec ed
BLANCO-PILLADO, SOUSA, URKIOLA, and WACHTER PHYS. REV. D 103, 106006 (2021)
106006-2
in he ollowing calcula ions. The p esence o he Ramond-
Ramond and Ne eu-Schwa z-Ne eu-Schwa z h ee- o m
luxes, espec i ely Fð3Þand Hð3Þ, induces he Guko -Va a-
Wi en supe po en ial W o he dila on and complex
s uc u e moduli [34]
ffiffiffiffiffiffiffiffi
π=2
pW¼NT·Σ·Π;ð7Þ
whe e we ha e in oduced he lux ec o
N≡ −τh; ¼RBIFð3Þ
RAIFð3Þ;h¼RBIHð3Þ
RAIHð3Þ;
ð8Þ
wi h I
A; B
I;h
I
A;h
B
Ig∈Z. Then, he con igu a ions o he
axiodila on and complex s uc u e ields τc;z
i
cg, which
minimize he scala po en ial while p ese ing supe sym-
me y, a e hose sa is ying he F- la ness condi ions
ð∂τþ∂τKÞWjτc;zi
c¼ð∂ziþ∂ziKÞWjτc;zi
c¼0:ð9Þ
The p esen desc ip ion o he EFT has an inhe en
edundancy associa ed o he choice o homology basis.
Mo e speci ically, a change o basis induces a ans o ma-
ion o he pe iod and lux ec o s
Π→S·Π;N→S·N; ð10Þ
wi h S∈Spð2h2;1
−þ2;ZÞ, leading o di e en desc ip-
ions o he same heo y. Finally, he equi emen ha he
pe iod ec o ans o ms by symplec ic ans o ma ions
unde he monod omies (3) leads o he ollowing condi ion
on he couplings [35,36]:
κij jþ1
2κijk j k¼0mod Z:ð11Þ
We will now p o e he main esul o his pape :
Theo em: Le us conside a h2;1
−-dimensional ec o io
cop ime in ege s, which lies in he Kähle cone o he mi o
Calabi-Yau. Then, he Ansa z zi¼ˆz iwi h ˆz∈Cde ines a
consis en supe symme ic unca ion o he EFT gi en by
(5),(6),and(7) when he lux con igu a ion is o he o m
N0
A¼0;N
i
A¼ iˆ
NA;
NB
i¼qκijk j kˆ
NB−κij þ1
2κijk kNj
A;ð12Þ
and NB
0a bi a y. He e ˆ
NA≡
ˆ
A−τ
ˆ
hA,ˆ
NB≡
ˆ
B−τ
ˆ
hB
wi h ˆ
A;
ˆ
hA;
ˆ
B;
ˆ
hBg∈Zand q−1≡gcdðκijk j kÞ.
P oo .—Fi s , no e ha he cons ain (11) ensu es ha
he ec o s and hde ined in (8) ha e in ege componen s,
as equi ed by he lux quan iza ion condi ion. To p o e ha
he Ansa z zi¼ˆz iwi h ˆz∈Cde ines a consis en supe -
symme ic unca ion o he EFT wi h he luxes (12),we
need o check ha he F- la ness condi ion wið∂ziþ
∂ziKÞWjˆz i¼0is sa is ied along all di ec ions wio hogo-
nal o he educed ield space de ined by he unca ion
Ansa z, i.e., o hogonal o i, ega dless o he alue o ˆz
and τ[17,37–39]. Subs i u ing he lux con igu a ion (12)
in o (7) we ind ha he F- la ness condi ion eads
κijkwi j kˆzþi
2NA−iqNBþwi½∂ziKWˆz i¼0:
ð13Þ
Ac ually he wo e ms in his exp ession anish
independen ly, as hey a e bo h p opo ional o
κijkwiImðzjÞImðzkÞ¼0, which is ze o in he LCS egime.
Indeed, his quan i y anishes a any con igu a ion ziwhe e
he holomo phic ec o wiis o hogonal o ImðziÞ[26]
(see also Appendix Aand [28,40,41]). ▪
The p e ious esul gua an ees ha he Ansa z zi¼ˆz i
can be consis en ly subs i u ed in o he ac ion, ob aining a
educed heo y wi h an e ec i ely one-dimensional com-
plex s uc u e moduli space pa ame ized by ˆz. The cou-
plings o he educed ac ion a e s ill cha ac e ized by (5)
and (7), bu wi h an e ec i e p epo en ial gi en by
ˆ
F≡−
1
3! κ
ˆz3þ1
2·2! κ
ˆz2þκ
ˆzþ1
2κ0ð14Þ
and an e ec i e ou -dimensional lux ec o
ˆ
N≡ðNB
0;qκ
ˆ
NB;0;
ˆ
NAÞT;ð15Þ
whe e we in oduced he sho hands1κ ≡κijk i j kand
κ ≡κi i. Any solu ion o his educed heo y is also a
solu ion o he ull ac ion in he LCS egime and o leading
o de in α0and gs. Fu he mo e, i he ields su i ing he
unca ion sa is y he F- la ness condi ions (9), hen he
axiodila on/complex s uc u e sec o o he comple e heo y
will sa is y hem as well [17,42].
The e o e, gi en an EFT o some Calabi-Yau compac-
i ica ion, we can immedia ely gene a e la ge amilies o
lux acua in he LCS egime (one amily o each choice o
i), whe e we can compu e he mass spec um o he
comple e axiodila on/complex s uc u e sec o . Indeed, we
jus need sol e he F- la ness condi ions (9) o he educed
model de ined by (14) and (15). Then, he mass spec um a
he esul ing acua can be ob ained using he esul s in [26],
which apply whene e he complex s uc u e sec o can be
consis en ly unca ed o a single ield. Mo e speci ically,
he o mula (1) gi es he squa ed masses o all he 2h2;1
−þ2
1The eedom (10) allows o shi κij i jby an a bi a y
in ege , wha we use o se κij i j¼−1
2κijk i j kin (14) [35].
UNIVERSAL CLASS OF TYPE-IIB FLUX VACUA WITH …PHYS. REV. D 103, 106006 (2021)
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scala modes in he axiodila on/complex s uc u e sec o ,
including he unca ed ones, in e ms o a single pa ame e
ξ≡−3Imκ0
2κ ImðˆzÞ3, and no malized by he g a i ino mass
m2
3=2≡eKjWj2¼3QD3=ðπð2−ξÞV2Þ;ð16Þ
whe e QD3≡ T·Σ·h≥1is he lux induced D3cha ge.
In Eq. (1), he masses wi h λ¼0, 1 a e hose associa ed o
he ields su i ing he unca ion τ;ˆzg, while hose wi h
λ¼2;…;h
2;1
−a e he masses o he emaining ields in he
unca ed sec o . I is also wo h men ioning ha solu ions
o (9) wi h N0
A¼0a e o pa icula in e es , as hey a e he
only ones ha can be ound pa ame ically close o he LCS
poin [26–30], which is whe e we ha e he bes pe u ba i e
con ol o he EFT.
To end his sec ion, le us b ie ly commen on he
D3- adpole cons ain . In a gi en compac i ica ion, he
numbe o solu ions a LCS compa ible wi h he spec um
(1) Ncan be es ima ed using he con inuous lux app oxi-
ma ion o [43]. We ind
NðQD3≤Q
D3;g
s≤g
sÞ∝X
i∈CK
g
sjImκ0jðQ
D3Þ3
q2κ2
;ð17Þ
whe e Q
D3is he a ailable D3cha ge in he compac i i-
ca ion, gs¼ðImτÞ−1is he s ing coupling, and he p o-
po ionali y cons an is o o de one (see Appendix C). The
sum in he p e ious o mula ex ends o e all ec o s iin
he Kähle cone (CK) o he mi o dual o M3. Al hough
he ac ual numbe o acua depends on he choice o
compac i ica ion, his esul shows ha Nonly ep esen s a
e y small ac ion o he o al numbe o lux acua N≪
N o al ∝ðQ
D3Þ2ðh2;1
−þ1Þwhen h2;1
−≫1[43]. Ne e heless,
he me hod desc ibed abo e allows us o e y e icien ly
sea ch o hese solu ions, as we demons a e nex wi h an
explici example.
III. EXAMPLE: THE HYPERSURFACE WP4
½1;1;1;6;9
We will now illus a e ou esul s by cons uc ing an
ensemble o he class o acua p esen ed abo e. Fo his
pu pose we will conside he compac i ica ion o ype-IIB
s ing heo y in an o ien i old o he Calabi-Yau hype su -
ace WP4
½1;1;1;6;9, which has h1;1
þ¼2Kähle moduli and
h2;1
−¼272 complex s uc u e ields. Fo geome ies admi -
ing a G¼Z18 ×Z6isome y g oup, and p o ided only
G-in a ian luxes a e u ned on, he complex s uc u e
sec o can be consis en ly unca ed, lea ing only wo
su i ing complex s uc u e ields which also ans o m
i ially unde G. In he LCS egime, he couplings o he
wo G-in a ian complex s uc u e ields a e de e mined by
a p epo en ial wi h coe icien s [44]
κ111 ¼9;κ112 ¼3;κ122 ¼1;
κ11 ¼−
9
2;κ22 ¼0;κ12 ¼−
3
2;ð18Þ
κi¼ð
17
4;3
2Þ, and κ0¼−540ζð3Þ=ð2πiÞ3. Recall ha he
p esence o he g oup Gis no necessa y o ou esul s o
apply, howe e , such isome ies a e o en equi ed o make
he compu a ion o he EFT couplings ac able (see [45]).
We will also assume he same con igu a ion o o ien i old
planes and D7-b anes as in [20], which allows o lux
acua wi h D3cha ge sa is ying QD3≤138.
The p ocedu e desc ibed abo e allows us o u he
educe he complex s uc u e sec o o a single ield.
Conside o de ini eness he unca ion Ansa z de ined
by he monod omy di ec ion i¼ð1;1Þ. The esul ing
e ec i e p epo en ial (14) is gi en by he couplings
κ ¼21 and κ ¼23
4. In o de o cons uc he acua
ensemble, we i s gene a ed he collec ion o all lux uples
B
0;h
B
0;
ˆ
A;B;
ˆ
hA;Bgwi h en ies in he in e al ½−15;15
sa is ying he adpole cons ain (∼2×107in o al). Fo
each o hese, we nume ically sol ed he F- la ness con-
di ions (9) o he educed model gi en by (14) and (15),
employing he so wa e Pa amo opy [46–48]. The
esul ing se o 17,054 solu ions is displayed in Fig. 1
(blue do s), which shows he dis ibu ion o acua on a
FIG. 1. Nume ically gene a ed dis ibu ion o lux acua on
he undamen al domain o he educed ield space, wi h
Reτ∈½−1=2;1=2Þ,Imτ>1and Reˆz∈½−1=2;1=2Þ o he
WP4
½1;1;1;6;9model. The plo ep esen s a o al o 28,683 acua
ob ained by educing he EFT along he monod omy di ec ion
i¼ð1;1Þ. We indica ed in ed acua wi h la ge (>5%) ins an on
co ec ions o he Kähle me ic and m3=2, and in blue (17,054
solu ions) hose wi h co ec ions <5%.
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undamen al domain o τ;ˆzg. This ensemble includes
only solu ions a he weak s ing coupling/LCS egime,
i.e., whe e gs<1and wi h small ins an on co ec ions
o he p epo en ial (using he simila c i e ia o [26]).
I is in e es ing o no e ha o his pa icula b anch o
acua, he con inuous lux app oxima ion (17) p edic s
Nj i¼ð1;1Þ∼105, which ep esen s a e y small ac ion o
he o al numbe o acua, N o al ∼1013. This esul shows
he e iciency o ou me hod, which led us o ind a
signi ican ac ion o his b anch o solu ions, ega dless
o hem a ising wi h low equency in he landscape.
As we de ailed be o e, he unca ion Ansa z zi¼ iˆz
oge he wi h (12) allows us o li each o hese solu ions o
a acuum o he comple e WP4
½1;1;1;6;9model. A e he li ,
we compu ed he scala mass spec um a each acuum o
he axiodila on and he G-in a ian zimodes (λ¼0,1,2)
by di ec diagonaliza ion o he Hessian o he lux
po en ial o he WP4
½1;1;1;6;9model. The esul pe ec ly
ma ched he o mula (1) in all cases. I is impo an o
emphasize ha , a each o he ob ained solu ions, Eq. (1)
also gi es he masses o he 270 unca ed complex ields,
which ans o m non i ially unde G, i.e., he modes wi h
λ¼3;…;272. This is a ema kable esul , gi en ha we
only used he EFT couplings o he G-in a ian moduli
compu ed in [44]. A s a is ical analysis o he lux acua
ob ained by his p ocedu e can be ound in Appendix B.
IV. DISCUSSION
In his pape , we p esen ed a me hod o cons uc
ensembles o lux acua o gene ic Calabi-Yau compac i-
ica ions a LCS, whe e he masses o he axiodila on and
complex s uc u e moduli a e gi en by he uni e sal
o mula (1). This esul p o ides ull analy ic con ol, o
leading o de in α0and gs, o e he masses o hose ields
and, he e o e, he acua we conside a e an excellen
s epping s one owa ds he comple e s abiliza ion o he
compac i ica ion, i.e., including he Kähle moduli.
In e es ingly, up o an o e all scale, he masses gi en in
(1) a e comple ely de e mined by he acuum alues o he
complex s uc u e ields. As a consequence, knowing he
magni ude o he α0and nonpe u ba i e co ec ions which
gene a e he Kähle moduli po en ial, i is possible o
gua an ee he s abili y o he axiodila on and all he
complex s uc u e ields by es ic ing he sea ch o acua
o app op ia e egions o moduli space. In pa icula , he
spec um (1) in ol es a single asymp o ically massless
mode in he neighbo hood o he LCS poin μ2
−1jξ→0¼0
[26], wi h all he emaining masses being a leas o he
o de o he g a i ino mass m3=2. In o he wo ds, in he LCS
limi he e is only one po en ially dange ous mode which
migh h ea en he s abili y o he compac i ica ion. On he
con a y, away om he LCS poin he mass o he ligh es
mode in (1) becomes o he o de o m3=2, and hus as long
as he pe u ba i e and nonpe u ba i e con ibu ions o he
EFT a e unde con ol, he inal acuum wi h he Kähle
moduli ixed will no de elop an ins abili y. Ne e heless, i
is expec ed ha such con ibu ions will induce small
co ec ions in he spec um (1), which, in pa icula , will
li he degene acy o he modes λ¼2;…;h
2;1
−.
As a inal ema k, no e ha he class o acua we
discussed is only app op ia e o he cons uc ion o
LVS solu ions, bu no o he KKLT scena io. Fo he
solu ions p esen ed he e, he lux supe po en ial sa is ies
W0≡Vm3=2≥1=ffiffiffi
π
p[see Eq. (16)], while he KKLT
acua equi e W0 o be exponen ially small. The e o e, a
logical u u e di ec ion would be o conside he s abiliza-
ion o he Kälhe moduli a he class o acua p esen ed
he e wi hin he LVS amewo k. Ano he in e es ing
con inua ion o his wo k would be o s udy o he unca-
ion schemes compa ible wi h he mo e gene al acua
discussed in [26], whe e he spec um can also be explici ly
compu ed, and W0could be a bi a ily small.
ACKNOWLEDGMENTS
This wo k is suppo ed by he Spanish Minis y
MCIU/AEI/FEDER G an No. PGC2018-094626-B-C21,
he Basque Go e nmen G an No. IT-979-16, and he
Basque Founda ion o Science (IKERBASQUE). K. S.
is suppo ed by he Czech science ounda ion GAČR
G an No. 19-01850S. M. A. U. is also suppo ed by he
Uni e si y o he Basque Coun y G an No. PIF17/74. Fo
he nume ical wo k, we used he compu ing in as uc u e
o he ARINA clus e a he Uni e si y o he Basque
Coun y (UPV/EHU).
APPENDIX A: F-FLATNESS CONDITION FOR
THE TRUNCATED MODULI
The p oo o Eq. (13) elies on he ollowing p ope y
sa is ied by he couplings o he p epo en ial (6) in he LCS
egime (neglec ing ins an on co ec ions)
κijkwiImðzjÞImðzkÞ¼0;ðA1Þ
whe e wiis any holomo phic ec o o hogonal o ImðziÞ
wi h espec o he moduli space me ic. This esul was
de i ed in [26] (see Eqs. (4.2) and (4.5) he e), howe e , as
he con en ions used he e a e sligh ly di e en o hose
o [26], o comple eness we will b ie ly ou line he
p oo in his Appendix. We will assume he Eule numbe
o he Calabi-Yau o be non anishing, bu i is s aigh o -
wa d o ex end he a gumen o he case χðM3Þ¼0,
whe e ξ¼Imκ0¼0.
The Kähle po en ial o he complex s uc u e sec o
de i ed om (5) and (6) eads
Kcs ¼−log 4
3κijkImðziÞImðzjÞImðzkÞ−2Imðκ0Þ;
ðA2Þ
UNIVERSAL CLASS OF TYPE-IIB FLUX VACUA WITH …PHYS. REV. D 103, 106006 (2021)
106006-5
and he co esponding moduli space me ic on his sec o
Ki¯
j≡∂
i∂¯
jKcs is
Ki¯
j¼−2eKcs κijkImðzkÞ
þ4e2Kcs κilmκjnpImðzlÞImðzmÞImðznÞImðzpÞ:ðA3Þ
Fo he ins an on con ibu ions o he p epo en ial (6) o be
supp essed, i.e., in he LCS egime, he ield con igu a ion
mus be such ha ImðziÞlies in he Kähle cone o he
mi o Calabi-Yau (see [33]), wha , in pa icula , implies
ha κijkImðziÞImðzjÞImðzkÞmus be non anishing and
posi i e.
Le us conside now he p oduc Ki¯
juiImðzjÞ, whe e uiis
a gene ic holomo phic ec o no necessa ily o hogonal
o ImðziÞ. Using ha he enso κijk is o ally symme ic,
and he de ini ion ξ≡−2eKcs Imκ0
1þ2eKcs Imκ0,2 his p oduc can be
exp essed as
Ki¯
juiImðzjÞ¼−ξð1−2ξÞ
ð1þξÞ2Imκ0
κijkuiImðzjÞImðzkÞ:ðA4Þ
Se ing ui¼ImðziÞin he p e ious equa ion, we ob ain
Ki¯
jImðziÞImðziÞ¼3ð1−2ξÞ
4ð1þξÞ2;ðA5Þ
implying ha a ield con igu a ions whe e ξ¼1=2, he
moduli space me ic becomes degene a e, and hus he EFT
is no well de ined. Mo eo e , he case jξj→∞co e-
sponds o con igu a ions ou side he LCS egime whe e
κijkImðziÞImðzjÞImðzkÞ¼0, and hus he EFT de ined by
he polynomial p epo en ial (6) canno be us ed.3Finally,
o p o e Eq. (A1) we subs i u e ui¼wiin (A4), wi h wi
o hogonal o ImðziÞ, leading o
−ξð1−2ξÞ
ð1þξÞ2Imκ0
κijkwiImðzjÞImðzkÞ¼0;ðA6Þ
which can only be anishing a physical con igu a ions
away om he LCS poin (ξ¼0) p o ided (A1) is sa is ied.
APPENDIX B: VACUA STATISTICS FOR THE
WP4
½1;1;1;6;9HYPERSURFACE
In his Appendix, we will discuss he s a is ical p ope -
ies o he class o solu ions p esen ed in he main body. In
pa icula , we will analyze he dis ibu ion o hese acua on
he educed moduli space τ;ˆzgand he p obabili y densi y
unc ions o he scala masses in (1). Fo his pu pose,
ollowing he me hod p esen ed be o e, we nume ically
cons uc ed an ensemble o lux acua on an o ien i old o
he Calabi-Yau hype su ace WP4
½1;1;1;6;9and ex ac ed he
co esponding p obabili y dis ibu ions by he di ec exami-
na ion o his se o solu ions. As shown below, hese
nume ical esul s a e in good ag eemen wi h he analy ical
p obabili y dis ibu ions de i ed in [26], which desc ibe he
s a is ics o compac i ica ions wi h an e ec i ely one-
dimensional complex s uc u e sec o (see also [43]). I
is impo an o emphasize ha ou analy ical desc ip ion o
he p obabili y dis ibu ions is independen o he choice o
Calabi-Yau and he unca ion Ansa z. The e o e, he
s a is ical ea u es obse ed he e o he WP4
½1;1;1;6;9ensem-
ble a e expec ed o be p esen in gene ic compac i ica ions
as well.
In o de o ha e a su icien ly la ge sample o acua o
he s a is ical analysis, we conside ed he e ec i e educ-
ion o he complex s uc u e sec o along he monod omy
di ec ions i¼ ð1;1Þ;ð1;2Þ;ð1;3Þg, and we combined in
a single ensemble he solu ions o he F- la ness condi ions
(9) ound o each o he h ee cases. O he amilies could
also ha e been conside ed; howe e , he s udy o any o
hem is e y compu a ionally demanding and, due o he
uni e sal ea u es o hese acua, we do no expec o gain
any new in o ma ion om s udying a di e en amily.
Fu he mo e, o elax he adpole cons ain on he
luxes, we conside ed he se ing adop ed in [49], whe e
he ype-IIB compac i ica ion on WP4
½1;1;1;6;9was ega ded
as he o ien i old limi o F- heo y on an ellip ically ibe ed
Calabi-Yau ou old, M4. In he F- heo y amewo k, he
maximum allowed D3cha ge induced by he luxes is
de e mined by he Eule numbe o he ou old, leading in
he p esen case o4QD3≤χðM4Þ=24 ¼273 [49].
Fo each choice o i, we p oceeded in a simila way as
desc ibed in he main body o he pape . Fi s , we gene a ed
a collec ion o 107 lux uples B
0;h
B
0;
ˆ
A;B;
ˆ
hA;Bgd awn
om a uni o m dis ibu ion wi h suppo in ½−25;25and
subjec o he adpole cons ain QD3≤273. Then we
sea ched o solu ions o he co esponding F- la ness
Eqs. (9) wi h he aid o he so wa e Pa amo opy. The
esul ing ensemble con ains 206,479 acua in he weak
s ing-coupling egime, i.e., wi h ðImτÞ−1¼gs<1, ou o
which 95,626 a e in he LCS egime. He e we de ined he
LCS egime by he condi ion ha he leading ins an on
con ibu ions o he p epo en ial (6), gi en by (see [44])
Fins ¼−
135
2π3iei2πz1−
3
8π3iei2πz2þ…;ðB1Þ
2This de ini ion educes o he one in he main ex when
zi¼ˆz i.
3Ac ually, all ield con igu a ions wi h ξ<−1o ξ>1=2a e
unphysical since he ield space me ic always has a leas one
nega i e eigen alue he e [26].
4The ca ea on his app oach is ha i in oduces addi ional
D7-b ane moduli ields. Fo simplici y, he e we will igno e hose
addi ional moduli, and we e e he eade o [23,50–54] o
discussions on hei s abiliza ion.
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induce small ela i e co ec ions (<5%) o moduli space
geome y (i.e., o he ield space me ic and he canonically
no malized couplings κijk) and o he g a i ino mass m3=2
[26]. I is impo an o men ion ha ou de ini ion o he
LCS egime is mo e es ic i e han jus equi ing Fins o
be small (in absolu e alue) wi h espec o he pe u ba i e
pa o he p epo en ial (6) (see, e.g., [55]). Indeed, he
moduli space me ic becomes degene a e a om he LCS
poin (ξ→−1 o χðM3Þ>0and ξ→1=2 o χðM3Þ<0),
and hus, in ha egime, he me ic eigen alues a e small
and e y sensi i e o he ins an on co ec ions, e en o
small a ios jFins =Fj∼0.01.
The me hod used he e o a oiding duplici ies in he
coun ing o acua is essen ially he same as he one used in
[26] (see also [56]). Howe e , he case a hand equi es
ce ain speci ic conside a ions ela ed o he unca ion o
he moduli space so, o comple eness, we will b ie ly
summa ize his me hod in he nex sec ion.
1. Redundancies and solu ion duplica es
The desc ip ion o he EFT p esen ed in he main ex has
wo inhe en edundancies, namely hose associa ed o he
choice o holonomy basis (10) and he well-known
SLð2;ZÞmodula ans o ma ions ac ing on τ. Those
acua, which can be ela ed o each o he by hese gauge
ans o ma ions, should be ega ded as physically equi -
alen , and hus when cons uc ing he ensemble one mus
ensu e ha each dis inc solu ion is only coun ed once.
Rega ding he choice o holonomy basis, he coe icien s
κij,κi, and κ0a e only de ined modulo in ege s wi h di -
e en ep esen a i es associa ed o di e en choices o his
basis. The e o e, by selec ing a pa icula exp ession o
he p epo en ial he symplec ic gauge is pa ially ixed,
wi h he esidual gauge gi en by he monod omy ans-
o ma ions a ound he LCS poin , i.e., zi→ziþδi
pwi h
p∈1;…;h
2;1
−. As a esul o imposing he unca ion
Ansa z zi¼ˆz i, he gauge eedom is u he educed,
lea ing as he only sou ce o gauge edundancy he
monod omy ans o ma ions zi→ziþ i, which amoun s
o he shi
ˆz→ˆzþ1ðB2Þ
on he ield su i ing he unca ion. The co esponding
symplec ic ans o ma ion Sð Þ∈Spð2h2;1
−þ2;ZÞac s on
he pe iod ec o as (see [35])
Πðziþ iÞ¼Sð Þ·ΠðziÞ;wi h Sð Þ≡AB
0C:
ðB3Þ
The ma ices Aand Ba e gi en by
A¼1− i
01;B¼2κ þ1
6κ −κj þ1
2κj
−κi −1
2κi −κij ;
ðB4Þ
and C¼ðATÞ−1. No e ha he condi ion (11) is necessa y
o Sð Þ o ha e in ege en ies, which also equi es he
addi ional cons ain 2κ þ1
6κ ¼0mod Z[36].
Finally, o ob ain he ac ion o he esidual monod omy
ans o ma ion (B3) on he luxes o he educed heo y, we
jus need o impose he Ansa z (12) oge he wi h (10).We
ind he ans o ma ion ules
ˆ
NA→NA;
ˆ
NB→
ˆ
NB−q−1ˆ
NA;
NB
0→NB
0þκ ðˆ
NA−q
ˆ
NBÞ:ðB5Þ
The condi ion N0
A¼0is p ese ed.
In addi ion o hese ans o ma ions, one mus also ake
in o accoun he modula ans o ma ions SLð2;ZÞ, which
ac on he axiodila on and he luxes as
τ→
aτþb
cτþd;
h→ab
cd
·
h;ðB6Þ
wi h a; b; c; d ∈Zand ad −bc ¼1.
In o de o elimina e equi alen solu ions ela ed by
he ans o ma ions (B3) and (B6), all he acua in he
ensemble we e anspo ed o a undamen al domain
de ined by Reτ∈½−1=2;1=2Þ,jτj>1, and Reˆz∈
½−1=2;1=2Þusing (B2),(B5), and (B6). Once in he
undamen al domain duplica e solu ions a e easily iden i-
ied and disca ded, as hey co espond o hose wi h he
same con igu a ion o he ields and he luxes. The esul
o his p ocedu e o he ensemble o acua discussed in he
main ex was displayed in Fig. 1. The co esponding
dis ibu ion o acua on he undamen al domain o he
ensemble analyzed in his Appendix shows no signi ican
di e ences wi h espec o Fig. 1, and hus we ha e no
displayed i he e.
2. Analy ic o mulae and nume ical esul s
We now u n o he analysis o he s a is ical p ope ies
o he ensemble. As shown in [26], o compac i ica ions
wi h an e ec i ely one-dimensional complex s uc u e
sec o and a la ge D3-cha ge adpole QD3jmax ≫1, he
s a is ics o he lux ensemble can be accu a ely desc ibed
using he con inuous lux app oxima ion o [43].This
app oxima ion consis s in neglec ing he quan iza ion o
he luxes, which a e hen ea ed as con inuous andom
a iables wi h a uni o m dis ibu ion, only subjec o he
adpole cons ain T·Σ·h≤QD3jmax.Using hissim-
pli ica ion as he s a ing poin , i is possible o de i e he
UNIVERSAL CLASS OF TYPE-IIB FLUX VACUA WITH …PHYS. REV. D 103, 106006 (2021)
106006-7
ollowing exp ession o he dis ibu ion o acua in he
educed complex s uc u e space [26]
ρðξÞdξ¼N·ð1þξÞ
ð2−ξÞ2ξ2=3dξ;ξ≡−3Imðκ0Þ
2κ ImðˆzÞ3;ðB7Þ
whe e, o con enience, we ha e gi en he dis ibu ion o
ImðˆzÞin e ms o he pa ame e ξ. In he p e ious exp ession
and he ollowing ones, N ep esen s a no maliza ion
cons an , which should be de e mined o each pa icula
dis ibu ion. I is ema kable ha his dis ibu ion is inde-
penden o he de ails o he Calabi-Yau o ien i old, o he
choice o he su i ing ield in he educed heo y, i.e., o i.
As a consequence, his exp ession can be used o desc ibe
mixed ensembles con aining acua om di e en compac-
i ica ions and/o ob ained om di e en unca ion
Ansä ze.
The dis ibu ion o alues o he pa ame e ξob ained
nume ically o ou ensemble o acua, combining he
cases i¼ ð1;1Þ;ð1;2Þ;ð1;3Þg, is displayed in Fig. 2.
The igu e shows a s acked his og am wi h he 95,626
acua a he LCS egime indica ed in (ligh and da k) blue
and in o ange hose solu ions wi h a la ge con ibu ion om
ins an ons (>5%). Since he o mula (B7) was ob ained
while comple ely igno ing he con ibu ion om ins an-
ons, i is expec ed o wo k only in he egime o ξspace
whe e ew acua, o none, a e disca ded due o ha ing la ge
co ec ions ( ha is, o ξ≲0.12). Fu he mo e, due o he
limi a ions o ou nume ical me hod, he lux in ege s in he
ensemble ange only in he in e al ½−25;25, leading o an
a i icial bound o how close he acua in he ensemble can
be o he LCS poin , ξ≳0.001 [26]. As a consequence, he
dis ibu ion (B7) is expec ed o desc ibe co ec ly he
s a is ics o acua in he ange ξ∈½0.001;0.12, which
we ha e indica ed in Fig. 2in da k blue.
The dis ibu ion (B7), no malized in i s ange o alidi y,
is also indica ed in he igu e wi h a dashed line and, as i
can be obse ed, i p o ides a e y good desc ip ion o he
densi y o lux acua. I is also in e es ing o no e ha ,
despi e he di e gence o he dis ibu ion (B7) a ξ¼0, his
unc ion is no malizable in ξ∈½0;1=2Þ, and hus i p edic s
a ini e numbe o acua in any neighbo hood o he
LCS poin .
Rega ding he axiodila on, i can also be shown ha ,
acco ding o he con inuous lux app oxima ion, he s ing
coupling cons an gs¼ðImτÞ−1has a uni o m p obabili y
dis ibu ion in his class o acua o , equi alen ly, he
p obabili y densi y unc ion o he imagina y pa o τis o
he o m ρðImτÞ∝ðImτÞ−2. This is also consis en wi h he
dis ibu ion, which we ob ained nume ically, as i can be
seen in Fig. 3.
In o de o w i e he exp ession o he mass dis ibu-
ions, i is con enien o de ine he ollowing unc ions o
he pa ame e ξ:
˜
mλðξÞ¼8
>
>
>
<
>
>
>
:
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1−2ξÞ=3
pˆ
mðξÞλ¼0;
ffiffiffiffiffiffiffiffiffiffi
ð1−2ξÞ
p
ffiffi3
pˆ
mðξÞλ¼1;
1þξ
3λ¼2;…;h
2;1
−;
ðB8Þ
which gi e he e mion masses a no-scale acua mλ,
no malized by he g a i ino mass ˜
mλ≡mλ=m3=2[26].
Then, combining he p e ious exp essions wi h (B7) and
using ha he unc ions ˜
mλðξÞa e mono onic, i is imme-
dia e o ob ain h2;1
−þ1sepa a e p obabili y dis ibu ions,
one o each o he escaled e mion masses
FIG. 2. Densi y o lux acua in he educed complex s uc u e
space in e ms o he pa ame e ξ. The plo shows he nume ical
dis ibu ion ob ained di ec ly o m he ensemble o 206,479 lux
acua. We ha e indica ed in (da k and ligh ) blue he 95,626
acua wi h small (<5%) ins an on co ec ions and in o ange
hose whe e he ins an on con ibu ion is la ge (>5%). The
dashed line ep esen s he analy ic dis ibu ion (B7) no malized
in he ange ξ∈½0.001;0.12, whe e mos acua a e in he LCS
egime and he con inuous lux app oxima ion holds (da k blue).
FIG. 3. Dis ibu ion o he s ing coupling gs, in e ms o
g−1
s¼Imτ. The his og am ep esen s he no malized dis ibu ion
o he da a poin s lying in he in e al ξ∈½0.001;0.12, while he
dashed cu e is he expec ed esul om he con inuous lux
app oxima ion.
BLANCO-PILLADO, SOUSA, URKIOLA, and WACHTER PHYS. REV. D 103, 106006 (2021)
106006-8
ρ
λð˜
mλÞd˜
mλ¼N·ð1þξÞ
ð2−ξÞ2ξ2=3ðd˜
mλðξÞ=dξÞξð˜
mλÞ
d˜
mλ;
ðB9Þ
whe e λ¼0;…;h
2;1
−. Finally, om he ela ion
μ2
λ¼ðm2
3=2mλÞ2ðB10Þ
be ween he scala and e mion masses [17], we can ob ain
h2;1
−þ1sepa a e p obabili y dis ibu ions, one o each pai
o no malized scala masses ˜μ2
λ≡μ2
λ=m2
3=2
ρs
λð˜μ2
λÞd˜μ2
λ¼N·˜μ−1
λ½ρ
λð1þ˜μλÞþρ
λðj1−˜μλjÞd˜μ2
λ:
ðB11Þ
In o de o gene a e he nume ical mass dis ibu ions
o ou ensemble o acua, a each solu ion o (9) we
diagonalized he Hessian o he scala po en ial induced by
he luxes, i.e., he po en ial in he heo y de ined by (5),
(6), and (7) wi h he couplings (18), which desc ibe he
G-in a ian sec o o he moduli space in he WP4
½1;1;1;6;9
model. In all cases, he esul ing masses o he h ee
G-in a ian modes (including he axiodila on) we e in
ag eemen wi h Eq. (1) wi h λ¼0, 1, 2. The nume ical
dis ibu ions o he scala μ2
0,μ2
1, and μ2
2a e displayed
in Fig. 4, along wi h he heo e ical dis ibu ion (B11)
no malized in he ange ξ∈½0.001;0.12. As expec ed
om he analysis o he dis ibu ion ρðξÞ(B7), in Fig. 4we
can see ha he heo e ical p obabili y densi ies o he
masses a e in good ag eemen wi h he ob ained nume ical
esul s. The mos signi ican ea u e o hese plo s is ha he
densi y dis ibu ion o μ2
1is peaked a ound ze o, indica -
ing ha a la ge ac ion o acua in ol e a ligh ield in he
spec um. This can be unde s ood ecalling ha , on he one
hand, acua wi h N0
A¼0(as hose discussed he e) can be
ound pa ame ically close o he LCS poin [28–30], and
hus a la ge ac ion is expec ed o be ound nea ξ¼0(see
Fig. 2). On he o he hand, as we men ioned in he main
ex , om (1) i ollows ha he spec um o hese acua
con ains an asymp o ically massless mode in he limi
ξ→0, which explains he peak o ρs
1ðμ2
1Þa μ2
1¼0
obse ed in Fig. 4(b). This ea u e is expec ed o be gene ic
o he class o acua discussed he e, ega dless o he
choice o Calabi-Yau compac i ica ion o he unca ion
Ansa z, as bo h he mass spec um (1) and he p obabili y
dis ibu ions (B10) and (B11) a e comple ely uni e sal.
No e also ha hal o he masses in he spec um a e smalle
han he g a i ino mass m2
3=2.
The sha p edges o he mass spec a shown in Fig. 4
co espond o he cu o s we ha e se on he pa ame e
ξ∈½0.001;0.12, wi h he peaks o he p obabili y dis i-
bu ions co esponding o he minimum alue o ξ. The
e ec o changing he bounds o ξcan be seen in Fig. 5.
The plo in Fig. 5(a) ep esen s he combined dis ibu ion
o he masses o he h ee G-in a ian modes wi h
ξ∈½0.001;0.02. As is can be seen he dis ibu ion
becomes e y peaked, wi h he maxima a he alues
(a)
(b)
(c)
FIG. 4. Nume ical dis ibu ions o he no malized scala
masses ˜μ2
λ¼μ2
λ=m2
3=2o he G-in a ian modes, λ¼ 0;1;2g
in he ensemble o 95,626 acua a LCS. In each igu e, he
dashed line ep esen s he analy ic o mula (B11) o each alue
o λ, no malized in he same ange ξ∈½0.001;0.12. The da ke
egions ep esen he solu ions o which he con inuous lux
app oxima ion applies.
UNIVERSAL CLASS OF TYPE-IIB FLUX VACUA WITH …PHYS. REV. D 103, 106006 (2021)
106006-9
