Counting string theory standard models

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Citation

: Constantin, A., He, Y. ORCID: 0000-0002-0787-8380 and Lukas, A. (2019).

Counting string theory standard models. Physics Letters, Section B: Nuclear, Elementary

Particle and High-Energy Physics, 792, pp. 258-262. doi: 10.1016/j.physletb.2019.03.048

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Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Counting

string

theory

standard

models

Andrei Constantin

a

,

b

,

Yang-Hui He

c

,

d

,

e

,

∗

,

Andre Lukas

f

a_{PembrokeCollege,OxfordUniversity,OX11DW,Oxford,UK} b_{MansfieldCollege,OxfordUniversity,OX13TF,Oxford,UK} c_{MertonCollege,UniversityofOxford,OX14JD,UK}

d_{DepartmentofMathematics,City,UniversityofLondon,EC1V0HB,UK} e_{SchoolofPhysics,NanKaiUniversity,Tianjin,300071,China}

f_{RudolfPeierlsCentreforTheoreticalPhysics,OxfordUniversity,1KebleRoad,Oxford,OX13NP,UK}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received13December2018

Receivedinrevisedform22March2019 Accepted25March2019

Availableonline28March2019 Editor:M.Cvetiˇc

We derive an approximate analytic relation between the number of consistent heterotic Calabi-Yau compactifications of string theory with the exact charged matter content of the standard model of particlephysicsandthetopologicaldataoftheinternalmanifold:theformerscalingexponentiallywith the numberofKähler parameters.Thisisdonebyanestimateofthenumber ofsolutionstoasetof Diophantineequationsrepresentingconstraintssatisfiedbyanyconsistentheteroticstringvacuumwith threechiralmasslessfamilies,andhasbeencomputationallychecked toholdforcompleteintersection Calabi-Yauthreefolds(CICYs)withuptosevenKählerparameters.WhenextrapolatedtotheentireCICY list,therelationgives_∼1023 _{stringtheorystandardmodels;fortheclassofCalabi-Yauhypersurfacesin}

toricvarieties,itgives∼10723_{standardmodels.}

©2019TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introductionandsummary

It isgenerallybelievedthat string compactifications that have theexactchargedmatter contentofthestandard modelof parti-clephysics (andnoother chargedmatter exceptmoduli)are few in number. The purpose of this letter is to show that, although

such compactifications maybe rare andhard tofind, their

num-berissubstantial.Admittedly,thisbiashascomeinthepastfrom thedifficultytoconstructphenomenologicallyviable compactifica-tions.However,sincethebirthofstringphenomenologyinRef. [1], fromtheadventofthefirst standard-likestringmodel[2],tothe first exact particle spectrum directly derived from a string com-pactification[3–6],tothefirstresult[7] fromalgorithmicheterotic compactification [8], until the comprehensive computer scan of Refs. [9–11,15], as well asthe various statisticalperspectives on the heterotic landscape [12,38] (cf. [13] in Type II and beyond [14]),therehasbeenmuchprogress.

Whileitcanbespecifiedatdifferentlevelsofsophistication,for thisletter a “string standard model” is a modelwith a massless

spectrum which is exactly that of the minimally

supersymmet-*

Correspondingauthor.

E-mailaddresses:[email protected](A. Constantin),

[email protected](Y.-H. He),[email protected](A. Lukas).

ric standard model(MSSM),plus anynumber ofmassless modes

(modulifields)

uncharged

underthestandardmodelgaugegroup. The general strategy of heterotic string phenomenology is to considerasmooth,compactCalabi-Yauthreefold(CY),say X,with a non-trivial fundamental group

,together witha holomorphic, (poly-)stable vector bundle V over X, typically with structure group SU

(

5

)

or SU(4

)

. Subsequently, a

-Wilson line can break theGrandUnifiedTheory(GUT)group,typically

SU

(

5

)

orS O(10

)

, totheMSSMgroupandthe -equivariantcohomologyof

V

aswell asitsappropriatetensorpowerscorrespondtotheMSSMparticles. From analgorithmic pointofview,onecan (1)takemanifolds

˜

X from existing databases, mostof which are simply connected,

then search for discrete, freely acting symmetry groups

on X

˜

,

and consider the quotient X

˜

X/ with fundamental group

;

(2) construct and classify families of

-equivariant bundles V

on X,ensurestability,andthencomputetherelevantcohomology groups; and(3)scan through theresultsto lookforexact MSSM

particle content. Much of these can be implemented on a

com-puter.

The most extensively used databases of manifolds are the

Complete IntersectionCalabi-Yau three-folds(CICYs)embedded in products of projective spaces of around 8000 manifolds [16,17] as well as the Kreuzer-Skarke (KS) dataset of Calabi-Yau hyper-surfaces embedded in around half-billion four-dimensional toric varieties[18–20] (the actualnumberofCalabi-Yau willvastly

ex-https://doi.org/10.1016/j.physletb.2019.03.048

ceedthisnumberbecauseoftriangulationsanda recentestimate wasmadein[21]).Thecomprehensivescanof[11] wasperformed

onCICYmanifoldswitha numberofKählerparameterslessthan

h1,1

_(X

₎

₌

_{6 andvectorbundlesconstructedfromsumofline}

bun-dles.A totalofabout35

,

000 SU

(

5

)

heterotic linebundle models werefound, allwiththerightfield contentto inducelow-energy standard-likemodels.

It is obviously important to have a count of MSSM models

expectedwithin string theory.While oursis still a relatively un-refinednotion ofthestandard model,even countingatthislevel isnoteasysinceitrequiresinformationonallvectorbundlesand theircohomologyonCYswhichisnot availableinanysystematic form.An exceptiontothisruleistheclassofvectorbundlesthat splitintoasumoflinebundles.Holomorphiclinebundlesare clas-sifiedbytheir first Chernclass,whichcan be expressedinterms of

h

1,1

₍

_{X)integers.Assuch,linebundlescanbeenumerated.}

Moreover,thereisenoughevidencethatindicatestheexistence ofanalytic formulaefortheranksoflinebundle valued cohomol-ogygroups in terms ofthe line bundle integers [23–26]. Finally, linebundlesumsofferanaccessiblewindowintothemodulispace ofnon-abelianbundles[26–28]:ifalinebundlesumcorresponds toastandard-likemodel,thenusuallyitcanbedeformedinto non-abelianbundlesthatalsoleadtostandard-likemodels.

Ourmodelbuildingexperienceforsuch(rankfive)linebundle modelsonCICYs suggeststhata significantnumber ofconsistent

models with the correct chiral asymmetry will descend to

stan-dardmodelsafterdividingby thefreelyactingdiscretesymmetry

group

. Moreover, by far the most frequent symmetry is

Z

2.

Thissuggeststhatanindicationofthenumberofstandardmodels shouldbeprovidedbycountingtheconsistentupstairslinebundle modelswithchiralasymmetry6,relevantfor

Z

2symmetries.

Westartouranalysisby outliningtheconstraintsonthe

com-pactificationdatathat guarantee an exactMSSM spectrum. Fora

fixed manifold, most of these constraints take the form of

Dio-phantineequationsandinequalities,wheretheunknownvariables arethelinebundleintegers. FortheclassofCICYmanifoldswith lessthansixKählerparametersthissystemwassolvedinRef. [11] byexplicitlycheckingeverypossiblelinebundlesum.Weaugment thisdatasetoflinebundlemodelswithresultsfor7 newmanifolds with

h

1,1

_(X

₎

₌

₆

_,

_7.

Thesescanssuggest asimple rule: thenumberofline bundle

models increases roughly by an order of magnitude with every

incrementof h1,1

_(X

₎

_{by one. However, it is difficult to test this}

relationforlarger values of

h

1,1

_(X)

_{due tocomputer limitations.}

Instead,we estimate thenumber ofsolutions tothe Diophantine systemofconstraintsusingaresultfromthemathematical litera-ture[29].Forthistohold, we definea boundon thelinebundle integersintermsoftopologicaldataofthemanifold.

Finally,we comeback totheempiricaldatasetoflinebundles andcorrelatethenumberofsolutions not onlywith

h

1,1

_(X)

_{, but}

alsowithanumberoftopologicalinvariants,constructedfromthe intersectionformandthesecond Chernclass oftheCYmanifold, that displaylittle variation withincreasing h1,1

_(X)

_{. Extrapolating}

themulti-linearregressiontothemaximalvalueof

h

1,1

₍

_X)found

intheCICYdataset,weestimateatotalof

N

CICY

1023linebundle

models(weowetheexpression“amoleofmodels”toTristan Hüb-sch)while forthe manifoldsintheKreuzer-Skarkelistwe expect NKS

10723 linebundlemodels.

Acknowledgments.WearegratefultoK. DienesandT. Hübschfor

valuable commentson the draft. AC andAL wouldlike to thank

the Mainz Institute for Theoretical Physics for hospitality during partofthecompletionofthisproject.ALispartiallysupportedby theEPSRCnetworkgrantEP/N007158/1.YHHthanksSTFCforgrant ST/J00037X/1.

2. CountinglinebundleMSSMs

The models of interest for our count have an exact MSSM

particle contentandare constructed fromheterotic compactifica-tionsonasmooth,compactCalabi-YauthreefoldsX endowedwith slope-zero, poly-stable direct sums of linebundles. Leth denote thePicardnumberof X,

h

:=

h1,1

_(X)

_{,andchooseanintegralbasis}

of H2

_(X

₎

_denotedby

_{

_J

i

}

,where

i

=

1

,

. . . ,

h.Inthisbasis,letthe

second Chern classof X be

c

2,i and thetriple intersection

num-bersbe

d

i jk

=

X Ji

∧

Jj

∧

Jk.LinebundlesL

→

X withfirstChern

class

c

1

(L)

=

kiJiaredenotedby

L

=

O

X

(

k

)

.Thereaderisreferred

toSec. 4ofRef. [11] forfurtherdetailsontheconstraints. First, we focus on SU(5

)

bundles V for thefollowing reason. From a group theoretic point ofview [22], thereare many ways

tobreaktheGUTgroup totheMSSMgroup usinganappropriate

discreteWilsonline,forexample,theexactMSSMspectrumof[4] was achieved witha

Z

3

×

Z

3 Wilson linefroman S O(10

)

GUT

group.However,CYmanifolds X

˜

withalargefreelyactingdiscrete symmetry group

are quite rare.This canbe seen, forinstance, fromthecomplete classificationoffreely-acting[33] and residual [34,35] symmetriesonallCICYs,orfromtheKSdatasetof hyper-surfacesintoricFanofourfolds[30,32].

Therefore, generically, it is expectedthat Calabi-Yau manifolds withasmallfundamentalgroup

π

1

(X

)

,shouldfarexceedin

num-ber thosewithalarge

π

1

(

X)(thisshould becontrasted withthe

relative paucity ofCalabi-Yau manifoldsof smallHodge numbers [36,37,39]). The smallest possible

π

1

(X

)

that breaks the SU

(

5

)

GUT group to the Standard Model gauge group is

=

Z

2 and

this setup is expected to dominate. Now, in SU

(

5

)

(commutant of the SU

(

5

)

of the bundle in E8) GUTs, the 10 representation

corresponds entirely to anti-families which we desire to be ab-sent. Under the branching of E8 to SU

(

5

)

, this corresponds to

the condition that h2

(

X, V

)

=

0, so that stability (implying that h0

_(X,

_V

₎

₌

_h3

_(X

_{, V}

₎

₌

_{0) in conjunctionwiththe index theorem}

gives

−

3

|

| =

ind

(

V

)

=

3

i=0

hi

(

X

,

V

)

= −

h1

(

X

,

V

) .

(1)

Thus,weimposeind

(V

)

= −

h1

(X,

V

)

= −

6.

Insummary,the countingproblemcan thenbe formulated as

follows:

PROBLEM:Whatisthenumber

N

=

N

(h,

c2,i

,

di jk

)

ofrankfiveline

bundle sums V

= ⊕

5_a=1La, where La

=

O

X

(

ka

)

satisfying the

fol-lowingconstraints:

E8embedding: c1

(V

)

=

5

a=1

ki

a

=

! 0 forall

i

=

1

,

. . . ,

h;

Anomaly cancellation:

c2,i(V

)

= −

1 2di jk

a

kajkka

!

≤

c2,ifor alli

=

. . . ,

h

;

Supersymmetry/Zero Slope: thereisacommonsolution

t

i _tothe

vanishingslopes

μ(

La

)

=

di jkkiat j

atka

=

! 0 fora

=

1

, . . . ,

5

suchthat J

=

tiJi

∈

interioroftheKahlercone; Particle generations: thechiralasymmetryissix,i.e.

ind

(

V

)

=

1 6di jk

a

[image:4.612.321.527.70.209.2]

Table 1

NumberofmodelsfoundinacomputerscanforfavourableCICYswith4≤h≤7 foreachh.

h=h1,1_{(X) 4 5 6 7}

number of CYs 9 10 8 5

[image:4.612.34.284.97.295.2]

number of models 75 2949 36692 1428856 number,N¯, of models per CY 8.3 294.9 4586.5 285771.0

Fig. 1.ThelogarithmofN¯,theaveragenumberofmodelsperCY,asafunctionof

h=h1,1_{(X),takenfromthedatainTable1.Theredlineisalinearfittothedata.}

Weemphasizethat

N

isafunctionoftheprescribedHodge num-ber

h

,thesecondChernclass

c

2,i,aswellasthetripleintersection

numbers

d

i jk of X.

2.1. Preliminary count

Forthe subset offavourable CICYswith h

≤

6,the number N wasdeterminedbythecomputerscan[11] andwehaveextended thisscantoincludemanifoldsfor

h

1,1

_(X

₎

₌

_{7 aswellassome}

non-favourablemanifold for

h

1,1

_≤

_{7 whichwerepreviouslydiscarded.}

Thecumulativeandaveragenumberofmodelsfoundforeach

Pi-cardnumber

h

issummarisedinTable1.Notetherearenoviable modelsfor

h

=

1

,

2

,

3 sincethesupersymmetry(slopezero) condi-tionsaretooconstrainingforthosecases.

Asimpleapproachistoassumethatthemaindependenceof

N

ison

h

,andtoneglectthepossibleeffectsof

c

2,i and

d

i jk.The

av-eragenumber,N

¯

= ¯

N

(h)

ofmodelsperCYasafunctionof

h

,taken fromthelastrowofTable1,hasbeenplotted(logarithmically)in Fig.1.

Alinear fitto thisdata(which corresponds to thered linein Fig.1)leadsto

log

(

N

¯

(

h

))

−

5

.

0

+

1

.

5h

.

(2)

The largest known Picard number ofany CYthreefold is

h

max

=

491,whichappears within theKS dataset,andthe largestvalue within theCICYlist is

h

CICY

=

19. Using(2) toboldly extrapolate

tothosevalueswefind

¯

N

(

hCICY

)

1023

,

N

¯

(

hmax

)

10721

.

(3)

Clearly,thesenumbersarequitedramatic,evenifwerestrict our-selvestotheCICYs.Thepredictednumberofstandardmodelseven withinthissetissignificantlylargerthancanbecurrentlystored,

let alone found by a scan. However, the method so far is quite

crude andtheextrapolation tolarge

h

adventurous.Tosee some oftheproblems,considerFig.2whichshowsthenumberof mod-elsasafunction of

h

foreach CICY,ratherthan theaverageover allCICYswiththesame

h

,asinFig.1.Thevariation withingiven hcanbeseentobeconsiderable- clearindicationthatthereisa strongdependenceof

N

on

c

2,i and

d

i jk,inadditiontoh.

Fig. 2.ThelogarithmofN,thenumberofmodelsforeachoftheCICYmanifolds,as afunctionofh=h1,1_(X).

2.2. Some theoretical considerations

While the computer scan gives a finite number of models in

eachcase,itisactuallynoteasytoprovethat

N

isfinite.A succinct argument was presented in Ref. [15] based on the moduli space metric

Gi j

= −

3

κi j

κ

−

3 2

κiκ

j

κ

2

,

(4)

where

κ

=

di jktitjtk,

κi

=

di jktjtkand

κi j

=

di jktk,with

t

ibeingthe

Kählermoduli.We notethattheslopezeroconditionscanbe ex-pressedas

κi

ki

a

=

0 sothat

0

≤

a

kT_aGka

= −

3

κ

di jk

a

ki_akajtk

=

6

κ

t

i_c

2,i(V

)

≤

6

κ

t

i_c

2,i

(

T X

)

≤

6

κ

|

t

| |

c2,i

(

T X

)

|

.

(5)

Then, introducing the scale-invariant modified metric G

˜

=

₆κ_|t|G,

wegetthebound

a

k_aTG

˜

ka

≤ |

c2,i(T X

)

|

.

(6)

This by itself does unfortunatelynot bound the vectorska since

themetricG

˜

mightbecomesingularattheboundaryoftheKähler cone.However,ifwerequirethatwestayina“physical”regionof theKählerconewhereallcurve volumestobegreaterthan1(so that thesupergravityapproximationisvalid)andthevolume

κ

is

bounded fromabove (sothat we are not de-compactifying)then

theeigenvaluesofG

˜

areboundedfrombelowbyastrictlypositive number.

Eq. (6) then impliesthat thelength

_a

|

ka

|

2 isboundedfrom

aboveand,hence,thatthereisonlyafinitenumberofpossible in-tegervectorska.Morequantitative statementsdepend verymuch

onthespecificexamplebutwhatcanbesaidisthatthelengthof thekvectorsisboundedbyaradius Rwhichroughlyscalesas

R

∼

_c

2

d

1/2

,

(7)

where

c

2and

d

aretypicalvaluesof

c

2,iand

d

i jk.

Apartfromtheslopezeroconditionstheconstraintsontheline bundlescanbewrittenasasystemofDiophantineequations

5

a=1

k_ai

=

! 0

,

−

1 2di jk

a

kajkka

=

! c2,i

,

1

6di jk

a

[image:5.612.335.535.67.214.2]

Fig. 3.ThelogarithmofNfromthecomputerscanofCICYsversusthe“theoretical” upperboundxth,asdefinedinEq. (9).

for i

=

1

,

. . . ,

h. Here we have replaced the inequality in the anomalyconditionwithanequality,assumingthatthebulkofthe contributioncomesfromlinebundleswiththelargestallowed

in-tegers. We can homogenise these equations (by introducing one

additionalcoordinate)andthink ofthemasaset ofequationsin

P

n_,where

_n

₌

_{5h.Assumingtheyprovideacompleteintersection,} Z,itsdimensionisgivenby

m

=

n

−

2h

−

1

=

3h

−

1.

Ref. [29] providesan upper bound forthe numberofrational points NZ

(B)

within aboxofsize B (wherethe sizeismeasured

bythemaximumnorm)on Z whichisgivenby NZ

(B)

Bm.

Us-ingtheaboveradius

R

asanupperboundfor Bwefind

N

R3h−1

∼

_c

2

d

(3h−1)/2

=:

xth

.

(9)

The comparisonbetween thisupper bound and the resultsfrom

thecomputerscanonCICYmanifoldsisshowninFig.3.Withall thedatapointswell belowtheredlineitis clearthat Eq. (9) in-deedprovides anupper boundbutit isequallyobviousthat this upperboundisratherweak. Therearea numberofpossible rea-sonsforthis. First,the resultof Ref. [29] isonlyan upperbound (which, in addition, counts rational rather than integer points). Second, theradius R fromEq. (7) is a crude estimate andis, by itself,onlyanupperboundonthesize

B

oftheboxconsideredin Ref. [29].

2.3. A more sophisticated count

Ourcount in §2.1,based on considering only thedependence of

N

onthePicardnumber

h

isclearlysomewhatunrefined,while the theoretical upper bound xth from §2.2 is clearly too weak

to allow for a meaningful extrapolation to larger values of h. In thissubsection, we seeka moresophisticated equation for N as afunction of

h

, c2,i anddi jk,drawing inspirationfromthe above

discussions.

Thereis an obviousdifficulty ofwritingdown even an ansatz

for N asa function of c2,i and di jk: both of thesequantities are

basis-dependent (ona choice ofintegral basis Ji for H2

(X

)

), but

N clearly cannot depend on such a choice of basis. This means

weshouldthinkaboutbasis-independentquantitieswhichcanbe constructedfrom

c

2,iand

d

i jk.

Unfortunately,bothquantitieshave“allindicesdown”sothere isnoinvariant whichcan be obtainedby a simplecontractionof indices, given that the only available metric, G from Eq. (4), is

moduli-dependent.Thisproblemhasbeenencountered before,in

[image:5.612.64.269.68.212.2]

thecontextofpracticalapplicationsofWall’stheorem,anda solu-tionhasbeenproposedinRef. [17],p. 174.

Fig. 4.ThelogarithmofNfromthecomputerscanofCICYsversusthequantityxin Eq. (12),wheretheconstantsAiandBihavebeendeterminedtoprovidethebest

fit.

From the intersection form

λ(

α

,

β,

γ

)

:=

X

α

∧

β

∧

γ

∈

Z

≥0,

completely symmetric in

α

,

β,

γ

,the following invariantscan be constructed:

1

=

gcd

{

λ(α, β,

γ

)

|

α, β,

γ

∈

H2

(

X

,

Z)

}

2

=

gcd

{

λ(α, β, β)

|

α, β

∈

H2

(

X

,

Z)

}

3

=

gcd

{

λ(α,α,α)

|

α

∈

H2

(

X

,

Z)

}

.

(10)

Furthermore,combiningtheintersectionformand

c

2

=

c2

(T X)

we

can define the form

(

α

,

β,

γ

,

δ)

=

(λ(

α

,

β,

γ

)c

2

(δ)

+

3permuta

tions

)

whichgivesrisetotheinvariants

4

=

gcd

{

(α, β,

γ

, δ)

|

α, β,

γ

, δ

∈

H2

(

X

,

Z)

}

5

=

gcd

{

(α, β,

γ

,

γ

)

|

α, β,

γ

∈

H2

(

X

,

Z)

}

6

=

gcd

{

(α, β, β, β)

|

α, β

∈

H2

(

X

,Z)

}

7

=

gcd

{

(α,

α,

α,

α)

|

α

∈

H2

(

X

,Z)

}

.

(11)

Ref. [17] alsoprovides apractical wayofcomputingthese invari-ants whichinvolvesa scanover onlyafinite subsetof H2

(X

,

Z

)

, sothattheycanbeworkedoutfrom

d

i jkand

c

2,i.

Combiningtheapproachesof§2.1and§2.2andusinginvariants

i=1,...,7 aplausibleansatzfor

x

:=

logN is

x

=

7

i=1

(

Ai

+

Bih

)

log i

,

(12)

where

A

iand

B

iareconstantstobedeterminedbyregression.

Us-ingthedatafromtheCICYscan,togetherwiththeHodgenumbers handtheinvariants

i foreach CICYinvolved,the bestfitvalues

of Aiand

B

iare

(

Ai)

(

12

.

2

,

0

,

11

.

6

,

−

3

.

5

,

−

11

.

7

,

1

.

6

,

1

.

8

)

(

Bi)

(

−

2

.

3

,

0

,

−

1

.

9

,

0

.

9

,

2

.

0

,

−

0

.

7

,

0

.

2

) .

(13)

The comparisonofthisfitwiththe dataisprovided inFig. 4.

Each point corresponds to a CICY with x is the value computed

fromtheRHSof Eq. (12),using thevaluesin(13) for Ai, Bi and

log

(N

)

isthevalueofstandardmodels onthisCICYfoundbythe computerscan.Theredlineisthediagonal,log

(N)

=

x,which rep-resentsa perfect fit.It seems from Eq(13) that x dependsmost heavily on 1,3,5 as the corresponding coefficients dominate in

magnitude. Examining these, we see that they essentially come

Itturnsoutthattheinvariants

ishowrelativelylittlevariation

with increasing Picard number h. If we insert typical values for thesequantities,together with

h

CICY

=

19 and

h

max

=

491 intothe

abovefitresulttheextrapolatednumbersbecome

N

(

hCICY

)

1023

,

N

(

hmax

)

10723

,

(14)

whichisnottoofarawayfromtheearlierresult (3).

3. Conclusions

ThefitillustratedinFig.4looksratherconvincingandwe be-lievethat an extrapolationto large h

=

h1,1 _{istrustworthy (since}

the invariants i show little variation relative to h), as long as

the underlyingmodel-building assumptions continue to be satis-fiedforlarge

h

.WebelievethatthisisthecasefortheCICYdataset and,hence,thenumberof

1023standardmodelswithinthisset shouldbetakenseriously.

The extrapolation to hmax

=

491, the maximal known Picard

numberofanyCY,ismorequestionable.Someofthemodel

build-ingassumptionsmadeherehavenotyetbeencheckedfortheKS

set,andthereareevenindicationsthat theymaynotbe satisfied. First, it is not clear that the KS set contains manifolds that

ad-mit freely-acting symmetries with the same frequency as CICYs.

Theonlysystematiccheckscarriedoutforlow

h

1,1 _wherethe

fre-quencyofsymmetriesislowerthantheCICYs [30].Noinformation isavailableforlarge

h

1,1 _yet.

AnothergenericfeatureofCICYmodelsisthefrequentabsence oflarge numbers of vector-like pairs, so that checking the index was sufficient toguarantee thecorrect spectrumfora significant fractionofthemodels.Itisnot clearthat thisfeaturepersists for constructionsbasedon other CYmanifolds.Infact, theresultsof Ref. [31] suggest that the presence of phenomenologically prob-lematicnumbersofvector-likepairsmightbeagenericfeatureof someother CYconstructions. Again,nodefinitestatement onthis isavailablefortheKSset.

In summary,the number of

10723 standard models forthe

extrapolation to h1,1

₌

_{491 should be viewed with considerable}

caution.However, theextrapolationto

h

1,1

₌

_{19,the maximal}

Pi-cardnumberintheCICYsethastobetakenseriouslyandleadsto

1023_{standardmodels.Eventhisnumber,almostcertainlya}

con-servative lower bound, is frighteningly large and beyond current computerstorageandsystematicsearch.

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