Machine-learning the string landscape

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He, Y. (2017). Machine-learning the string landscape. Physics Letters B, 774,

pp. 564-568. doi: 10.1016/j.physletb.2017.10.024

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Physics

Letters

B

www.elsevier.com/locate/physletb

Machine-learning

the

string

landscape

Yang-Hui He

a

,

b

,

c

,

∗

a_{MertonCollege,UniversityofOxford,UK}

b_{DepartmentofMathematics,City,UniversityofLondon,UK} c_{SchoolofPhysics,NanKaiUniversity,China}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received5September2017

Receivedinrevisedform4October2017 Accepted10October2017

Availableonline16October2017 Editor:N.Lambert

Weproposeaparadigmtoapplymachinelearningvariousdatabaseswhichhaveemergedinthestudy

of the stringlandscape. Inparticular, we establishneural networks as bothclassifiers and predictors

and trainthemwithahostofavailabledatarangingfromCalabi–Yaumanifoldsandvectorbundles,to

quiverrepresentationsforgaugetheories,usinganovelframeworkofrecastinggeometricalandphysical

dataaspixelatedimages.Wefindthatevenarelativelysimpleneuralnetworkcanlearnmanysignificant

quantitiestoastoundingaccuracyinamatterofminutesandcanalsopredicthithertoforeunencountered

results,wherebyrenderingtheparadigmavaluabletoolinphysicsaswellaspuremathematics.

©2017TheAuthor.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense

(http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.

1. Introduction

Whereastheoretical physics now inevitably residesin an Age where new physics, new mathematics and new data coexist in a symbiosis transcending disciplines, string theory has spear-headed this vision. That it engenders the cross-fertilization be-tween physics and pure mathematics is without dispute, that it also has been a testing ground for computational mathematics and “big data” is perhaps less known. With the advent of in-creasinglypowerfulcomputers,fromthisfruitfuldialoguehasalso arisena plethora ofdata,ripe formathematical experimentation. Thisemergence ofdata beganwith theincipience ofstring phe-nomenology [1] where compactification of the heterotic string on Calabi–Yau threefolds (CY3) was widely believed to hold the ultimategeometricunification.A race,spanning the1990s,to ex-plicitly construct examples of Calabi–Yau (CY) manifolds ensued, beginning with the so-calledcomplete intersection CY manifolds (CICYs) [2], proceeding to the hypersurfacesin weighted projec-tivespace[3],toellipticfibrations[4]andultimatelyculminating inthe impressive(at leastsome 1010)list ofCY3sfromreflexive polytopes[5].

Withtherealizationthat thelandscapeofstringy vacuamight in fact exceedthe number of inequivalent CY3s[6] by hundreds oforders ofmagnitude, therewas a veringof directiontowarda moremulti-verseoranthropicphilosophy.Nevertheless,hintshave emergedthat thevastnessofthelandscapemightwellbe mostly

*

Correspondenceto:MertonCollege,UniversityofOxford,UK. E-mailaddress:hey@maths.ox.ac.uk.

infertile (cf. the swamp-land of [7]) andthat we could live in a very special universe [8–10], a “des res” corner within a barren vista.

Thus, undaunted by the seeming over-abundance of possi-ble vacua, fortified by the rapid growth of computing power and inspirited by the omnipresence of big data, the first two decades ofthe newmillennium saw areturn tothe earlier prin-ciple ofcreating andmining geometrical data; the notable fruits of this combined effort between pure and computational alge-braic geometers aswell as formally and phenomenologically in-clinedphysicistshaveincluded(q.v.[11]fora reviewofthe vari-ousdatabases):(1) ContinuingwithKreuzer–Skarke(KS)database

[14–22];(2) GeneralizingtheCICYconstruction[23–28];(3)

Find-ing ellipticandK3fibredCYforF-theoryandstringdualities[13,

14,28–32];(4) D-braneworld-volume theoriesassupersymmetric

quivergaugetheories[33–41].

All oftheabove casesareaccompanied bytypically accessible dataofconsiderablesize,representingaconcreteglimpseontothe string landscape, to which we shall refer aslandscape data. For instance, theheterotic linebundleson CICYs areon the orderof 1010_{,thespectral-coverbundlesontheellipticallyfibredCY3,10}6_, thebrane-configurationsintheCYvolumestudies,105_,typeII in-tersectingbrane models,109_{,etc. Evenbytoday’smeasure,these} constitute a fertileplayground ofdata,the likes ofwhichGoogle andIBMareconstantlyanalysing.A naturalcourseofaction, there-fore,istodountothislandscapedata,whatGoogleetal. doeach secondofourlives:tomachine-learn.

Letusbepreciseaboutwhatwemeanby

deep machine-learning

this landscape. Much of the aforementioned data have been the

https://doi.org/10.1016/j.physletb.2017.10.024

brain-child of the marriage between physicists and mathemati-cians,especiallyincarnatedbyapplicationsofcomputational alge-braic geometry, numerical algebraic geometry and combinatorial geometry to problems which arise fromthe classification in the physicsandrecastintoafinite,algorithmic probleminthe math-ematics (cf. [12]). Obviously, computing power is a crucial lim-itation. Unfortunately, in computational algebraic geometry – on whichmostofthedataheavily rely,ranging frombundles stabil-ityin heterotic compactification to Hilbert series in brane gauge theories–adecisivestepisfindingaGroebnerbasis,whichis no-toriouslyknownto be unparallelizableanddouble-exponentialin runningtime.Thus,muchofthechallengeinestablishingthe land-scapedatahadbeentoeithercircumventthedirectcalculationof theGroebner basesby harnessingof thegeometric configuration –e.g., usingthe combinatorics whendealing with toricvarieties. Still,manyofthecombinatorialcalculations,betheytriangulation ofpolytopes orfinding dual cones, are still exponentially expen-sive.

The good news for our present purpose is that, much of the data have already been collected. Oftentimes, as we shall find out inour forthcoming case-studies, tremendous effort is neededfor deceptivelysimple questions.Hence, todraw inferencesfrom

ac-tual theoretical data by deep-learning therefrom would not only helpidentify undiscoveredpatternsbutalso aidinpredicting re-sultswhichwouldotherwisecostformidablecomputations. Subse-quently,weproposeour

Paradigm: To set-up neural networks (NN) to deep-learn the landscapedata,to recognize unforeseeablepatterns (as classi-fiers)andtoextrapolatetonewresults(aspredictors).

Of course, this paradigm is useful not only to physicists but to alsotomathematicians;forinstance,couldourNNbetrainedwell enoughtoapproximatebundlecohomologycalculations?This,and ahostofotherexamples,wewillnowexamine.

Methodology Neuralnetworksareknownfortheircomplexity, in-volvingusually a complicateddirectedgrapheach node ofwhich isa “perceptron”(an activation function imitatinga neuron)and amongstthemultitudeofwhichthereare manyarrowsencoding input/output.Throughout thisletter, we will usea rather simple multi-layerperceptron(MLP)consistingof5layers,threeofwhich are hidden, with activation functions typically of the form of a logistic sigmoidora hyperbolic tangent.The input layer isa lin-ear layerof 100to 1000nodes,recognizing a tensor(as we will soonsee,algebro-geometric objectssuch asCalabi–Yaumanifolds orpolytopesaregenericallyconfigurationsofintegertensors)and theoutputlayerisasummationlayergivinganumber correspond-ing to a Hodge number, or to rank of a cohomology group, etc. SuchanMLPcan beimplemented,forinstance,onthelatest ver-sions of Wolfram Mathematica. With 500–1000 training rounds, the running time is merely about 5–20minutes on an ordinary laptop.Itisreassuringandpleasantly surprisingthatevensuch a relatively simpleNN can achieve thelevel of accuracy shortlyto bepresented.

This letter is a companion summary of the longer paper [42]where the interested reader can find more details of the computations and the data.

2. Results

WithsimpleNNs,weproceedtoanalyseourlandscapedata,a fertilegroundconstituting morethan2decades ofmany interna-tionalcollaborations between physics andmathematicians. Using

4 concretecasestudies, we first “learn” fromtheinherent struc-tureandthen “predict”unseenproperties; consideringhow diffi-cultsomeofthecalculationsinvolvedhadbeeninestablishingthe databases,theusefulnessofourparadigmisevident.

2.1. Case study 1: CY hypersurfaces in W

P

4

One of the first datasets [3] to experimentally illustrate mir-ror symmetry was that of hypersurfaces in weighted projective space W

P

4_{. The ambientspace W}

_P

4

[w0:w1:w2:w3:w4] withweights

wi=0,...,4

∈

Z

+ isingeneralsingular,butagenericenough

homo-geneouspolynomialofdegree 4

i=0

wiwhichmissesthesingularities

definesahypersurfacethereinwhichisasmoothCY3

X

.Thereare 7555inequivalentsuchconfigurations,eachspecifiedbya5-vector

wi=0,...,4.TheEuler characteristic

χ

of X iseasily giveninterms ofthevector.However,asisusuallythecase,theindividualHodge numbers

(

h1,1

_,

_h2,1

₎

_{are lessamenableto a simple combinatorial} formula. The original computation resorted to Landau–Ginzberg techniquesto obtainthelist ofHodgenumbers [3]. Onecould in principleuseadjunctionandEulersequences, andsingularity res-olution,butthisisnotaneasytasktoautomate.

Supposewehaveasimplequestion:

how many such CY3s have a

relatively large number of complex deformations?Wecan,forinstance, consider

h

2,1

_>

_{50 tobe“large”andlettrainingdatabeoftheform}

wi

→

1or0dependingonwhether

h

2,1

(

X

)

>

50.TrainingtheNN, withsay500rounds,takesunderaminuteonanordinarylaptop. Theresultisanoptimised continuousrealoutputbetween0and1, theroundingofwhichcanthenbecomparedwiththeactualdata. Anaccuracyof96.2%isachievedalmosteffortlessly!Toappreciate the predictive power of the network, suppose that we only had partial data. This is particularly relevant when for instance, due to computational limitations, a classification is not yet complete, orwhena quantityin questionhasnot beenorcouldnot be yet computed.

Therefore,let uspretend that we have onlydata available for thefirst3000out ofthe7555

(

X

,

h2,1

₎

_{pairs.We repeatthe} pro-cedure onthe 3000,andthen test againstthefull 7555. Wefind that6078caseswereactuallycorrect.Thus,withratherincomplete training data,theNNhaslearnt, inunderaminute, ourquestion andpredictednewresultsto80%accuracy.

Emboldened,letusmoveontoanotherquestion,ofimportance tostringphenomenology:

Given a configuration, can one tell whether

χ

is a multiple of 3?Intheearlydaysofheteroticstring compactifi-cation,thisquestionwasdecisiveonwhetherthemodeladmitted 3generationofparticlesinthelow-energyeffectivegaugetheory. Again, we can define a binary function taking the value of 1 if

χ

mod 3

≡

0 and 0 otherwise.Training withthe NN, we achieve 82% accuracywith1000training rounds, takingabout2minutes; these figures are certainly expected to improve with increasing number oftraining rounds and withmore layers or more nodes intheNN.

The astute readermight question at this stage why we have adheredto

binary queries

.WhynottraintheNNtoansweradirect query, i.e., to try for instance to learn and predict the value of

[image:4.612.53.262.64.135.2]

Fig. 1.Werealizethesetof7890CICYs(Calabi–Yauthreefoldsascomplete inter-sectionsinproductsofprojectivespaces)as12×15 matrices,paddingwithzeros wherenecessary.ThenallCICYconfigurationsaresuch matriceswithentries in {0,1,2,3,4,5}.Weconsidertheseaspixelcolours anddrawatypicalCICYin(a), with0beingpurple.In(b),weaverageoverallsuchmatricescomponent-wise,and drawthe“average”CICYasapixelatedimage.(Forinterpretationofthereferences tocolourinthisfigurelegend,thereader isreferredtothewebversion ofthis article.)

2.2. Case study 2: CICYs

Having warmed up, let us move onto complete intersection Calabi–Yauthreefolds(CICYs)inproductsofprojectivespaces.This isboththe firstCalabi–Yau database(or, forthatmatter, thefirst databaseinalgebraic geometry)[2]andthe mostheavily studied recentlyforstring phenomenology [23–26,28].It hasthe obvious advantagethattheambientspaceissmoothbychoice.

Briefly,CICYsembedas K homogeneous polynomialsin

P

n1

×

. . .

×

P

nm_{,ofmulti-degree}

_q

r

j,withcompleteintersectionmeaning

that

K

=

m r=1

nr

−

3 andCYconditionimplying K

j=1

qr

j

=

nr

+

1

∀

r

=

1

,

. . . ,

m.TheconstructionofCICYsisthusreducedtoa combina-torialproblemofclassifyingtheintegermatrices.Themostfamous CICY is, of course,

[

4

|

5

]

or simply the matrix

[

5

]

, denoting the quintichypersurface in

P

4_{.Itwas shownthat suchconfigurations} arefinite innumberandthebestavailable computer atthetime (1990’s), viz., the super-computer at CERN [2], was employed. A totalof7890inequivalentmanifoldswerefound,correspondingto matriceswithentries

q

r

j

∈ [

0

,

5

]

,ofsizerangingfrom1

×

1 to max-imumnumberofrowsandcolumnsbeing12and15,respectively. Thisrepresentation is muchin the standard wayto represent an image:to pixelateit intoblocks of

m

×

n, each ofwhich car-rying acolour info, for example,a 3-vector encapturing the RGB data.Therefore,we canrepresentallthe7890CICYsinto12

×

15 matricesover

Z

/

6

Z

, embedded starting fromthe upper-left cor-ner,say,andpaddingwithzeroseverywhereelse,asillustratedin

Fig. 1.ToviewaCICYasapixelatedimage,andindeed,touse

im-ageprocessingtoaddressproblemsingeometryandmathematical physics,isanentirelynewideaworthyofextensiveexploration.

Can we deep-learn, say the full list of Hodge numbers? As usual, the Eulernumber is relatively easy to obtain andthere is a combinatorial formula in terms of the integers qr_j, whilst the individualHodgenumbers

(

h1,1

_,

_h2,1

₎

_{involvesomenon-trivial} ad-junctionandsequence-chasing,whichluckilyhadbeenperformed forus[2]. Again,we setup a list oftraining rules (padded con-figurationmatrix

→

h1,1_{) and findthattheNNcan betrainedto} an accuracyof 99

.

91% inunder10 minutes! Whataboutthe NN asapredictor,whichisobviouslyamoresalientquestion?Suppose theNNweretrainedwiththefirst5000ofthedata,then,checking againstthefulldatasetcomprisingofconfigurations/imagestheNN hasnever before seen, we achieve 77% accuracy.Considering (1) thatwehaveonlytrainedtheNNforamere6minutes,(2)thatit hasseenonlyalittleoverhalfofthedata,(3)thatitisrather el-ementarywithonly5forwardlayers,and(4)thatthevariationof theoutputisintegralrangingfrom0to19,withnoroomfor con-tinuoustuning,suchaccuracywithsolittleeffortisquiteamazing.

2.3. Case study 3: bundle cohomology

The subject of vector bundle cohomology has, since the so-named “generalized embedding” [1]of heterotic compactification onsmoothCY3 X endowedwitha(poly-)stableholomorphic vec-tor bundle V,become one ofthe mostactive dialoguesbetween algebraicgeometryandtheoreticalphysics.Therealization[9]that the theoretical possibility of [1]can be concretely achieved by a judicious choice of

(

X

,

V

)

to give the exact MSSM spectrum in-ducedmuchactivityinestablishingrelativelylargedatasetstosee howoftenthismightoccurstatistically[11,18,22,24,31],

culminat-ingin[25,26]whichfoundsome200outofascanof1010bundles

whichhaveexactMSSMcontent.

Upon thisvast landscape let ustake an insightfulglimpse by taking the dataset of [31], which are SU

(

n

)

vector bundles V

on elliptically fibred CY3. By virtue ofa spectral-cover

construc-tion [4,30], thesebundlesare guaranteed to be stableandhence

preserves

N

=

1 supersymmetry in the low-effective action, to-getherwithGUTgaugegroups E6, S O

(

10

)

andSU

(

5

)

respectively for n

=

3

,

4

,

5. We take the base of the elliptic fibration – of which there is a finite list[29] – asthe r-th Hirzebruchsurface (r

=

0

,

1

,

. . . ,

12 denoting theinequivalent wayswhich

P

1 can it-self fibre over

P

1 _{to give a complex surface), in which case the} stable SU

(

n

)

bundleisdescribed by5numbers

(

r

,

n

,

a

,

b

,

λ)

,with

(

a

,

b

)

∈

Z

₊ and

λ

∈

Z

/

2 beingcoefficientswhichspecifythe bun-dle via the spectral cover. This ordered 5-vector will constitute our neural input. The database of viable models was set up in [31],viablemeaningthatthebundle-cohomologygroupsofV are such that h0

₍

_X

_,

_V

₎

₌

_h3

₍

_X

_,

_V

₎

₌

_{0 and}

_h1

₍

_X

_,

_V

₎

₋

_h2

₍

_X

_,

_V

₎

_≡

0

(

mod3

)

, where the first is a necessary condition for stability and the second, that the GUT theory has the potential to allow for3netgenerations ofparticlesuponbreakingtoMSSMby Wil-sonlines.OveralltheHirzebruch-basedCY3,14,264modelswere found;a sizeableplay-ground.

Supposetheoutputbea2-vector,indicating(I)whatthegauge group is, asdenoted by n, and(II) whether thereare more gen-erationsthan anti-generations,asdenotedby thesignof the dif-ference

h

1

₍

_X

_,

_V

₎

₋

_h2

₍

_X

_,

_V

₎

_{; thisisclearly a phenomenologically} interesting question. With1000training rounds onaNNwithan output linear layer, and withthe dataset consistingof entries in the form

(

r

,

n

,

a

,

b

,

λ)

→

(

n

,

Sign

(

h1

(

X

,

V

)

−

h2

(

X

,

V

)))

, in about 10minutes,weachieve100%accuracy(i.e.,theneuralnetworkhas completely learnt the data). Training withpartial data, say 8000 points, alittle overhalf,achieves68.9% predicativeaccuracy over theentireset.

2.4. Case study 4: quiver gauge theories

As a final examplelet ustackleaffinevarieties inthe context of quiver representations. Physically, these correspond to world-volume gaugetheories comingfromD-braneprobesofgeometric singularities in string theory, as well as the space of vacua for classes of supersymmetric gauge theories in various dimension; theyhavebeendata-minedsincetheearlydaysofAdS/CFT(cf.[33, 34]). When the geometryconcerned is an affinetoric CY variety, therealizationofbrane-tiling [35]hasbecomethecorrectwayto understandthegaugetheoryandsincethendatabaseshavebegun tobecompiled[36,37].

(2) F-term matrix

Q

F eachcolumnofwhichdocumentswhereand withwhatexponentthefieldcorrespondingtothearrowappears in

∂

W. Concatenating QD and QF gives the so-call total charge matrix Qt ofthe moduli spaceas a toric variety (q.v. §2of [34] forthepreciseprocedure).Thecombinatoricsandgeometryofthe above isa longstory spanning a lustrum of research to uncover followedbyadecadeofstill-ongoinginvestigations.

Inthefirstdatabaseof[36],ahostofexamplesweretabulated. A totalof375quivertheoriesmuchliketheabovewerecatalogued (a catalogue which has recently been vastly expanded in [37]). Thoughnot verylarge,thisgivesusaplaygroundtotest someof ourideas.The inputdataisthetotal chargematrix Qt,the max-imal ofwhose numberof rowsandcolumns are,respectively 33 and36,andall takingvaluesin

{−

3

,

−

2

,

. . . ,

3

,

4

}

. Now,suppose wewishtoknowthenumberofpointsofthetoricdiagram associ-atedtothemodulispace,whichisclearlyanimportantquantity.In principle,thiscanbecomputed(albeitcomputationallyintensive): theintegerkernelof Qt shouldgive amatrixwhosecolumnsare thecoordinatesofthetoricdiagram,withmultiplicity (associated totheperfectmatchingsofthebipartitetiling).Training withour NNwiththefulllistachieves,inunder5minutes,99

.

5% accuracy.

2.5. A sanity check

Lestthereaders’optimismbeelevatedtounreasonableheights bythe stringofsuccesses withthe NNs,it isimperative that we be awareof deep-learning’s limitations.We therefore finish with asanity check that aNN isnot some omnipotent oracle capable ofpredicting

any

pattern. Anexamplewhich mustbedoomed to failureistheprimes(or,forthatmatter,thezerosoftheRiemann zetafunction).Indeed,ifanNNcouldlearnsomeunexpected pat-ternintheprimes,thiswouldbearatherfrighteningprospectfor mathematics.Wetestthesequenceofprimes(i.e.,dataoftheform

i -> Prime[i]

)withourNN,andachievenobetterthana0.1% accuracy. Our NNis utterly useless against this formidable chal-lenge;we are better off trying a simple regression against some

n log

(

n

)

curve, asdictatedby the primenumber theorem.Thisis asoberingexerciseaswellasafurtherjustificationofthevarious casestudiedabove,thatitisindeedmeaningfultodeep-learnthe landscapedata andthat ourvisual representationof geometrical configurationsisanefficientmethodology.

3. Discussion

Therearemanyquestionsintheoreticalphysics,oreveninpure mathematics, for which one would only desire a qualitative, ap-proximate,orpartialanswer,andwhosefullsolutionwouldoften eitherbebeyondthecurrentscope, conceptualorcomputational, or would have taken considerable effort to attain. Typical such questions could be “what is the likelihood of finding a universe withthree generationsofparticles within thelandscapeofstring vacua or inflationary scenarios”, or “what percentage of known Calabi–Yau manifolds has Hodge numbers within a prescribed range”?Attemptingtoaddresstheseprofoundquestionshave,with theever-increasingpowerofcomputers, engenderedour commu-nity’sversion of“big data”,which though perhaps humble com-paredtosomeotherfields,docomprise,especiallyconsideringthe abstractnatureoftheproblemsathand,ofsignificantinformation oftenresultingfromintensedialoguebetweenteamsofphysicists andmathematiciansformanyyears.

Onthestill-ripeningfruitsofthislabourthephilosophyofthe last decadeor so, particularlyfor thestring phenomenology and computationalgeometrycommunity,has beento(I)createlarger andlarger datasetsand (II) scan through themto test the likeli-hoodofcertain salientfeatures.Nowthat thedataisaugmenting

in size andavailability, it is only natural to follow the standard procedures ofthedata-miningcommunity.Inthisletter,we have proposed theparadigm ofapplyingdeep-learning,vianeural net-works, suchdata.Thepurposeistwofold,theneuralnetworkcan actas

Classifiers: by association ofinput configurationwitharequisite quantity,andpattern-matchoveragivendataset;

Predictors: by extrapolating to hithertofore unencountered con-figurations,havingdeep-learntagiven(partial)dataset.

This is, of course, the archetypal means by which Google deep-learnsthe internet andhand-writing recognition softwareadapts tothereader’sesotericscript.

Itisintriguingthat by goingthrougha wealthofconcrete ex-amplesfromwhatwehavedubbedlandscape data,someofwhose creationthe authorhadbeen apart,thisphilosophyremains en-lightening. Specifically, we havetaken test casesfroma range of problemsin mathematicalphysics,algebraic geometryand repre-sentationtheory,suchasCYdatasets,classificationofstablevector bundles, and cataloguesof quiver varietiesandbrane tilings. We subsequentlysawthatevenrelativelysimpleNNcandeep-learning toextraordinaryaccuracy.

Insomesense,thisisnotsurprising, thereisunderlying struc-tureto anyclassification probleminour context,which maynot bemanifest.Indeed,whatisnovelistolookatthelikesofaCICY oraquivertheory asa pixelated image,nodifferentfroma hand-writtendigit,forwhoseanalysismachine-learninghasbecomethe defactomethodandablossomingindustry.

The landscape data, be

they work of human hands, elements of Nature or conceptions of Mathe-matics, have inherent structure, sometimes more efficiently uncovered by AI via deep-learning.Thereby,onecanrapidlyobtainresults,before embarkingonfindingareductionistframeworkforafundamental theoryexplainingtheresultsorproceedtointensivecomputations fromfirstprinciples.Thisparadigmisespeciallyusefulwhen clas-sification problems become intractable, which is often the case, here a pragmatic approach would be to deep-learn partial clas-sificationresultsandpredictfutureoutcome.

Under thisrubric, the possibilities are endless. Several imme-diate and pertinent directions spring to mind. First, the largest datasetinalgebraic geometry/stringtheory isthe Kreuzer–Skarke

list [5,20,21] of reflexive polytopes in dimension 4 fromeach of

whichmanyCYmanifolds(compactandnon-compact)canbe con-structed.Todiscoverhiddenpatternsisanongoingenterprise[14, 17]andthehelpofdeep-learningwouldbeamostwelcomeone. Next, the issue of bundle stability and cohomology is a central problem in heterotic phenomenology aswell as algebraic geom-etry.Inmanyways,thisisaperfectproblemformachine-learning: the input isusually encodable into an integer matrixor alist of matrices, representingthecoefficientsinan expansioninto effec-tive divisorclasses, the output is simply a vector of integers (in thecaseofcohomology) ora binaryanswer(withrespectivetoa givenKahlerclass, thebundle iseitherstableornot). The brute-force way involvesthe usual spectral sequences and determining allcoboundarymapsorfindingthelatticesofsubsheafs,expensive byanystandards.Inthecaseofstabilitychecking,thisisan enor-mousefforttoarriveatayes/noquery.Withincreasingnumberof explicitlyknownexamplesofstablebundlesconstructedfromfirst principles,to deep-learnthisandthenestimatetheprobability of agivenbundlebeingstablewouldbetremendoustime-saver.

com-puter is powerless), and

100

,

000 manifolds were found from

25

,

000 polytopes. The KS list has h1,1 _{going up to 496, thus} wehavenot eventouchedthetipoftheiceberginansweringthe simplest question of enumerating CY3s. Here, the NN would be extremelyusefulinpredictingan estimate,havinglearntthedata from[21],whichalreadytook

∼

5000 core-hourswithtraditional methodsonthecluster;thisiscurrentlyunderinvestigation.

Wehopethereaderhasbeenpersuadedbynotonlythescope butalso the feasibilityof our proposed paradigm,a paradigm of increasing importance in an Age where even the most abstruse of mathematics or the most theoretical of physics cannot avoid compilationsofandinvestigations onperpetuallygrowingdatasets. Thecasestudiesofdeep-learningsuchlandscapeofdatahere pre-sentedarebutafewnuggets inanunfathomablyvast gold-mine, richwithnewscienceyettobediscovered.

Acknowledgements

WeareindebtedtotheScienceandTechnologyFacilities Coun-cil,UK, forgrantST/J00037X/1,theChineseMinistryofEducation, foraChangJiangChairProfessorshipatNanKaiUniversity,andthe city ofTian-Jin fora Qian-RenAward, aswell asMerton College, UniversityofOxford forcontinuedsupport.

References

[1]P.Candelas,G.T.Horowitz,A.Strominger,E.Witten,Vacuumconfigurationsfor superstrings,Nucl.Phys.B258(1985)46.

[2]P.Candelas,A.M.Dale,C.A.Lutken,R.Schimmrigk,CompleteintersectionCY manifolds,Nucl.Phys.B298(1988)493;

P.Candelas,C.A.Lutken,R.Schimmrigk,CompleteintersectionCalabi–Yau man-ifolds.2.Threegenerationmanifolds,Nucl.Phys.B306(1988)113; M.Gagnon,Q.Ho-Kim,AnexhaustivelistofcompleteintersectionCalabi–Yau manifolds,Mod.Phys.Lett.A9(1994)2235;

T. Hubsch,Calabi–Yau Manifolds: aBestiaryfor Physicists,World Scientific, ISBN 981021927X,1992.

[3]P.Candelas,M.Lynker,R.Schimmrigk,CYmanifoldsinweightedP(4),Nucl. Phys.B341(1990)383.

[4]A.Grassi,D.R.Morrison,GrouprepresentationsandtheEulercharacteristicof ellipticallyfiberedCalabi–Yauthreefolds,arXiv:math/0005196[math-ag]; R.Y.Donagi,Principalbundlesonellipticfibrations,AsianJ.Math.1(1997)214, arXiv:alg-geom/9702002.

[5]A.C. Avram,M. Kreuzer,M. Mandelberg, H. Skarke, Thewebof Calabi–Yau hypersurfacesintoricvarieties,Nucl.Phys. B505(1997)625,arXiv:hep-th/ 9703003;

VictorV.Batyrev,LevA.Borisov,OnCalabi–Yaucompleteintersectionsintoric varieties,arXiv:alg-geom/9412017;

M.Kreuzer,H.Skarke,Reflexivepolyhedra,weightsandtoricCalabi–Yau fibra-tions,Rev.Math.Phys.14(2002)343,arXiv:math/0001106[math-ag].

[6]S.Kachru,R.Kallosh,A.D.Linde,S.P.Trivedi,DeSittervacuainstringtheory, Phys.Rev.D68(2003)046005,arXiv:hep-th/0301240.

[7]C.Vafa,Thestringlandscapeandtheswampland,arXiv:hep-th/0509212.

[8]F.Gmeiner,R.Blumenhagen,G.Honecker,D.Lust,T.Weigand,Oneinabillion: MSSM-likeD-branestatistics,J.HighEnergyPhys.0601(2006)004, arXiv:hep-th/0510170.

[9]V.Braun,Y.-H.He,B.A.Ovrut,T.Pantev,TheexactMSSMspectrumfromstring theory,J.HighEnergyPhys.0605(2006)043,arXiv:hep-th/0512177.

[10]P.Candelas,X.delaOssa,Y.-H.He,B.Szendroi,Triadophilia:aspecialcorner inthelandscape,Adv.Theor.Math.Phys.12(2008)429,arXiv:0706.3134.

[11]Y.H.He,Calabi–Yaugeometries:algorithms,databases,andphysics,Int.J.Mod. Phys.A28(2013)1330032,arXiv:1308.0186[hep-th].

[12] Y.-H.He,P.Candelas,A.Hanany,A.Lukas,B.Ovrut,Computationalalgebraic geometryinstring,Gaugetheory.Specialissue,Adv.HighEnergyPhys.2012 (2012)431898,http://dx.doi.org/10.1155/2012/431898,Hindawipublishing.

[13]V.Braun,ToricellipticfibrationsandF-theorycompactifications,J.HighEnergy Phys.1301(2013)016,arXiv:1110.4883.

[14]W.Taylor,OntheHodgestructureofellipticallyfiberedCalabi–Yauthreefolds, J.HighEnergyPhys.1208(2012)032,arXiv:1205.0952[hep-th].

[15]A.P.Braun,J.Knapp,E.Scheidegger,H. Skarke,N.O.Walliser,PALP –auser manual,arXiv:1205.4147[math.AG].

[16]P.Candelas,A.Constantin,H.Skarke,AnabundanceofK3fibrationsfrom poly-hedrawithinterchangeableparts,arXiv:1207.4792[hep-th].

[17]Y.H.He, V. Jejjala,L. Pontiggia, Patterns in Calabi–Yau distributions, arXiv: 1512.01579[hep-th].

[18]P.Candelas,R.Davies,NewCalabi–YaumanifoldswithsmallHodgenumbers, Fortschr.Phys.58(2010)383,arXiv:0809.4681[hep-th].

[19] W.A.Stein,etal.,http://www.sagemath.org. [20] CalabiYaudata,hep.itp.tuwien.ac.at/~kreuzer/CY/.

[21] R. Altman, J. Gray, Y.H. He, V.Jejjala, B.D. Nelson, A Calabi–Yau database: threefoldsconstructedfromtheKreuzer–Skarkelist,J.HighEnergyPhys.1502 (2015)158,arXiv:1411.1418,CYDatabase:www.rossealtman.com.

[22]Y.H.He, S.J. Lee, A.Lukas, Heterotic models from vector bundles on toric Calabi–Yaumanifolds,J.HighEnergyPhys.1005(2010)071,arXiv:0911.0865 [hep-th];

Y.H.He,M.Kreuzer,S.J.Lee,A.Lukas,HeteroticbundlesonCalabi–Yau man-ifolds with small Picard number, J. High Energy Phys. 1112 (2011) 039, arXiv:1108.1031.

[23]L.B. Anderson, F. Apruzzi, X. Gao, J. Gray, S.J. Lee, A new construction of Calabi–Yaumanifolds:generalizedCICYs,Nucl.Phys.B906(2016)441,arXiv: 1507.03235[hep-th].

[24]L.B.Anderson,Y.H.He,A.Lukas,Heteroticcompactification,analgorithmic ap-proach,J.HighEnergyPhys.0707(2007)049,arXiv:hep-th/0702210[hep-th].

[25]L.B.Anderson,J.Gray,A.Lukas,E.Palti,Heteroticlinebundlestandardmodels, J.HighEnergyPhys.1206(2012)113,arXiv:1202.1757[hep-th].

[26]L.B. Anderson, A. Constantin, J. Gray, A. Lukas, E. Palti, A comprehensive scanforheteroticSU(5) GUTmodels,J.HighEnergyPhys.1401(2014)047, arXiv:1307.4787.

[27]J.Gray,A.S.Haupt,A.Lukas,AllcompleteintersectionCalabi–Yaufour-folds, J. HighEnergyPhys.1307(2013)070,arXiv:1303.1832[hep-th].

[28]P.Gao,Y.H.He,S.T.Yau,ExtremalbundlesonCalabi–Yauthreefolds,Commun. Math.Phys.336 (3)(2015)1167,arXiv:1403.1268[hep-th].

[29]D.R.Morrison,C.Vafa,CompactificationsofFtheoryonCalabi–Yauthreefolds. 1&2,Nucl.Phys.B473(1996)74,arXiv:hep-th/9602114,Nucl.Phys.B476 (1996)437,arXiv:hep-th/9603161.

[30]R.Friedman,J.Morgan,E.Witten,VectorbundlesandFtheory,Commun.Math. Phys.187(1997)679,arXiv:hep-th/9701162.

[31]M.Gabella,Y.H.He,A.Lukas,Anabundanceofheteroticvacua,J.HighEnergy Phys.0812(2008)027,arXiv:0808.2142[hep-th].

[32]M.Cvetiˇc,J.Halverson,D.Klevers,P.Song,OnfinitenessoftypeIIB compact-ifications:magnetizedbranesonellipticCalabi–Yauthreefolds,J.HighEnergy Phys.1406(2014)138,arXiv:1403.4943[hep-th].

[33]A.Hanany,Y.H.He,Nonabelianfinitegaugetheories,J.HighEnergyPhys.9902 (1999)013,arXiv:hep-th/9811183;

A.Hanany,Y.H.He,Amonographontheclassificationofthediscretesubgroups ofSU(4),J.HighEnergyPhys.0102(2001)027,arXiv:hep-th/9905212.

[34]B.Feng,A.Hanany,Y.H.He,D-branegaugetheoriesfromtoricsingularitiesand toricduality,Nucl.Phys.B595(2001)165,arXiv:hep-th/0003085.

[35]S.Franco,A.Hanany,D.Martelli,J.Sparks,D.Vegh,B.Wecht,Gaugetheories fromtoricgeometryandbranetilings,J.HighEnergyPhys.0601(2006)128, arXiv:hep-th/0505211.

[36]J.Davey,A.Hanany,J.Pasukonis,Ontheclassificationofbranetilings,J.High EnergyPhys.1001(2010)078,arXiv:0909.2868[hep-th].

[37]S.Franco,Y.H.He,C.Sun,Y. Xiao,A comprehensivesurveyofbranetilings, arXiv:1702.03958[hep-th].

[38]J.Davey,A.Hanany,N.Mekareeya,G.Torri,M2-branesandFano3-folds,J.Phys. A44(2011)405401,arXiv:1103.0553[hep-th].

[39]A.Hanany,R.K.Seong,Branetilingsandreflexivepolygons,Fortschr.Phys.60 (2012)695,arXiv:1201.2614[hep-th].

[40]Y.H. He, R.K. Seong, S.T. Yau, Calabi–Yau volumes and reflexive polytopes, arXiv:1704.03462.

[41]S.Franco,S.Lee,R.K.Seong,C.Vafa,Branebrickmodelsinthemirror,J.High EnergyPhys.1702(2017)106,arXiv:1609.01723.