JHEP03(2016)045 Springer March 8, 2016 : January 15, 2016 February 29, 2016 : : Published 10.1007/JHEP03(2016)045 Received Accepted doi: Published for SISSA by = 2 vacua by focusing on certain distributions of Calabi-Yau vari- . 3 N ´ Etudes Scientifiques, 1512.01170 The Authors. c Differential and Algebraic Geometry, Superstring Vacua, Flux compactifica-

We use methods from topological data analysis to study the topological fea- , [email protected] Av. Rovisco Pais, 1049-001 Lisboa,Institut Portugal des Hautes Le Bois-Marie, 35 route de Chartres, F-91440E-mail: Bures-sur-Yvette, France Center for Mathematical Analysis,Instituto Geometry and Superior Universidade T´ecnico, Dynamical de Systems, Lisboa, Open Access Article funded by SCOAP Keywords: tions, Superstrings and Heterotic Strings ArXiv ePrint: eties and Landau-Ginzburg models. We thenwe turn can to use flux topological compactifications data and analysistechniques discuss to how to extract physical certain information. phenomenologically Finally realisticbility we heterotic of apply models. characterizing these string We vacua discuss using the the possi- topological properties of their distributions. Abstract: tures of certain distributionsapproach used of to analyze string the topological vacua. featuresical of characteristics a Topological persist dataset over by data identifying a which analysiscontexts. long homolog- range We is analyze of a scales. multi-scale We apply these techniques in several Michele Cirafici Persistent homology and string vacua JHEP03(2016)045 = 1 effective N 8 = 2 models one has to fix a Calabi- 4 6 N 21 6 – 1 – 24 4 11 15 16 16 9 19 10 1 29 = 2 vacua We can represent this collection of choices with a set of points. Each point could 4.1 Flux compactifications 4.2 Rigid Calabi-Yau 4.3 A Calabi-Yau hypersurface example N 3.1 Calabi-Yau compactifications 3.2 Landau-Ginzburg vacua 2.1 Elements of2.2 algebraic topology Point clouds2.3 and the Rips-Vietoris Persistent complex homology2.4 and barcodes Approximation schemes2.5 and witness complexes Topological analysis studying the equations associated with the physical model, as minima of a superpotential; represent a particular compactificationvacuum manifold, for or a the fixedpoints set background of could geometry. parameters be determining For associatedtheir a origin example with could in integral be a more fluxesdistribution flux mysterious, of which manifolds for compactification stabilize with example certain these associated the properties. with physical In a the moduli. (yet former not case understood) these Or points arise from subject to various constraints.all In all of these which cases seemThere we are equivalent are presented before by with a many nowwonder possibilities, if detailed many there study is available a of techniquesset simpler the to setting of in choices. low perform which one energy such can effective an understand qualitative theory. analysis, features of yet this one can When studying string compactifications ina many occasions discrete one set faces of a parameters.Yau set variety, For of labelled instance choices among in for certain examplemodels by are its parametrized Hodge by numbers; the choice similarly of certain a collection of integers, representing flux quanta, 6 Conclusions 1 Introduction 5 A first look at heterotic models 3 4 Persistence in flux vacua Contents 1 Introduction 2 An introduction to topological data analysis JHEP03(2016)045 .  bar- . We ]. We struc- 1 5 – >  3 2  , a set of points topological ]. point cloud 11 and then to die at 1  ]. The basic idea is that those homological 2 . To obtain topologically invariant information, – 2 – ,  ) of all the homology classes: these are called 1 [ . A little thought shows that the only thing that can  persistence persistence intersect pairwise. The main idea of topological data analysis is as a function of  ] for a sample of the literature. Among the strengths of topological data 10 – 6 each representing a vacuum. Then we “fatten” each point into a sphere of radius . At the end, we have associated a collection of barcodes to each distribution of vacua. N One could envision a program to study systematically string vacua from this per- In plain words, the idea behind this note is simple. First of all, we take a set of string Such techniques are becoming standard practice in data analysis, with a long range Topological data analysis studies how homological features of a dataset persist over In this note we pose the following question: is there any particular R finally plot the lifespans (their codes Such a collection captures the topological features of the distributionspective at and every length ask scale. which subsets of physically relevant vacua are characterized by which that a number of pointsciated form spheres an of edge, radius ato face study or such a complexes higherwe dimensional then simplex, pass if to the theof asso- homology homology groups of parametrized this by continuoushappen family is of for complexes, an and homology class obtain to a be family born at some value vacua, for example obtained asof critical compactification points spaces. in Out a ofin flux this collection compactification, we or construct as a To a this collection configurations we associate a continuous family of simplicial complexes by declaring of applications, from biologysyntax, and see neuroscience [ toanalysis complex is networks, its natural robustness imagesup respect and as to very short-lived noisy persistent samples,states homology and since classes. enumerative spurious For problems an features in application typically string to show and the field study theory of BPS see [ will discuss how thisa characterization distribution can of be vacua; usedits for non-trivial to example persistent analyze homology by the classes measuringrequirements physical in on its significance higher the topological degree, of parameters complexity or which if in correspond there terms to are distinctive of certain topological physical features. a long range ofout scales. of This a is datasettopological obtained and spaces by studying is constructing its a called features homologies family which at of are various simplicial more lengthtopological complexes persistent scales. characterization over of This the the approach range original to of dataset, length as scales reviewed for can example be in used [ to give a characterized by certain physicalture, properties or “complex”, is with “simple”, manyfeatures non-trivial with homology of almost generators. distributions no We ofthat will topological such study vacua, fea- topological topological in information could thetechniques be from appropriate of topological physical sense, data relevance. analysis and We to will the consider do problem the so of by counting possibility applying vacua in string theory. classifying higher dimensional manifolds. ture in these setthe of tendency points? to For cluster example inanalog one distinct higher can regions, dimensional ask or structures. if if vacua the Similarly in distribution one a of can given vacua wonder distribution presents if holes have a or distribution of vacua in the latter from some ad hoc mathematical construction, as part of the open problem of JHEP03(2016)045 gut = 2 and N = 1 models which give rise N – 3 – = 2 vacua obtained from Calabi-Yau compactifi- as we vary a certain physical parameter in the N ]. Here we take a topological approach, which is different change 18 ]. Such models are characterized by a Calabi-Yau and an holomorphic 21 – , which then can be higgsed by Wilson lines giving a Standard Model-like 19 ] this is a very promising avenue to apply statistical techniques to distribu- ], and part of these results have been put on firmer ground using tools from gut 17 14 , – 13 15 ], which parametrizes Calabi-Yau varieties obtained from reflexive polyhedra in four- 12 In this note we content ourselves to understand how to use techniques of compu- Finally we end with a very cursory look at a class of promising heterotic models Next we will discuss flux vacua in type IIB compactifications. As originally pointed As a first step we will discuss = 1 supersymmetry. In this context we will learn how to interpret and extract physical to investigate it more thoroughly in the future. and one could obtaining a different collections deeper of intuition vacua.corresponding by Hopefully to enlarging phenomenologically by the working viablephysical datasets with models, collections insights, available one of or such could larger consider- use astopological datasets these if features. methods specific At to physical this gain stage characteristics we are have accompanied no by evidence certain for this intriguing idea, and plan how these topological features spectrum. We will useof this vacua example has to a make higher more degree precise of the topological statement complexitytational that compared topology a to collection to others. extractdiscuss physical which information kind from of distributions facts of one string might vacua hope and to learn. Of course this is just a first step bundle, plus a series ofto an additional SU(5) choices. We willspectrum. consider We will study thethe topological Hodge features numbers of of a the distributionbundle. of underlying vacua Calabi-Yau Then parametrized and we by the will Chern take classes a of somewhat the different holomorphic perspective, and study in a simple example the topological structure ofselves the with whole discussing distribution the ofperspective counterpart vacua. of of In persistence, results this leaving already a note known more we in detailed content the analysis our- literature forconstructed from in the the [ future. tions of string vacua.distribution A of great the amount numberple of of in work [ flux has vacua beenrandom with algebraic dedicated certain geometry to properties, [ understandingfrom as the the reviewed statistical counting for of exam- vacua with certain properties and can in principle address ticular homological feature, andgeometries with if certain distinctive properties, characteristicssimilar for approach appear example to when low Landau-Ginzburg restricting Euler models. characteristics. to We willout take in a [ eties are known andlist have [ been constructed explicitly.dimensions. An Such example lists offer isno only the systematic samples Skarke-Kreuzer general of construction the is fulling known. set these From of vacua a is Calabi-Yau more akin varieties geometricalCalabi-Yau and to varieties. perspective, studying indeed study- the We topological will properties consider of the the question distributions if of such known distributions present any par- the persistent homology associatedN with a fewinformation from distributions the of barcodes. vacua, with cations of the type II string or from Landau-Ginzburg models. Many Calabi-Yau vari- topological features. In this note we take a small step towards this direction by studying JHEP03(2016)045 – ]. f Σ 3 10 ∈ + 1. – 6 σ k . matlab we discuss 3 . A simplicial map 2 mathematica S ) of σ ( f . -simplex if it has cardinality k gives a basic introduction to topologi- ]. It applies homological methods to col- 2 2 , . 1 Σ is a p , is a map between the corresponding vertex 2 ∈ ≤ – 4 – S The accompanying software and dataset can be k σ 1 and ]. 1 22 S is mapped into a simplex we summarize our findings. 1 6 S of -simplex, with σ k Σ. We say that is a collection of non-empty sets Σ, its simplices, such that if library from [ http://appliedtopology.github.io/javaplex/ ∈ S τ ]. After reviewing some basic elements of algebraic topology, we introduce contain applications of persistent homology to the study of flux vacua in 11 5 ]. Manipulations of datasets have been carried out with , then javaplex and 23 σ -simplex into a 4 p ⊆ τ In this section we will survey some basic ideas and techniques of topological data This note is organized as follows. Section All the persistent homology computations in this paper have been done with Software available at = 2 vacua obtained from Calabi-Yau compactifications and Landau-Ginzburg models. 1 ], which we will follow. For a more detailed discussion aimed at physicists and various One can define maps betweenbetween simplicial two complexes simplicial in complexes thesets natural way. such A that simplicial a map takes simplex a a triangle, and sosimplicial on. complex A is face basically of ais collection a part of simplex of simplices is it, with thesimplicial the then convex complex property so hull that of are if aand all a subset of simplex of its its faces. vertices. We A can state this more abstractly as follows: a We start with athe quick following. review of In sometriangulation. many notions applications This can of it by algebraic is donehull by topology useful of means that to a of a we approximate series simpliciala a shall of complex. 0-simplex topological use points, A is in its space simplex a vertices. is by single the a vertex, Its convex a dimension 1-simplex is is the an number edge of between vertices two minus vertices, one: a 2-simplex is 5 examples, see [ the Vietoris-Rips complex andschemes persistent in homology. the computation We of also barcodes discuss and some discuss approximation 2.1 how to set up the Elements topological of analysis. algebraic topology torial by construction, by studyingmodel the are relations varied. between The objects analysisfields as such of as the data biology, parameters based computer on of vision, topology the neuroscience, is languages rapidly or gaining complexanalysis. momentum systems in The [ reader interested in a more in depth discussion should consult the reviews [ Topological data analysis is a relatively recenttechniques approach based at managing on large computational sets of topologylections data [ of using data arranged as pointonly clouds sensitive to to extract topological qualitative information information. andof not a The to results metric geometrical are or quantities, of such as a a system choice of coordinates. Furthermore it has the advantage of being func- Sections type IIB compactifications andrespectively. to Finally in certain section phenomenologically realistic heterotic models, 2 An introduction to topological data analysis using the found at [ cal data analysis, presentingN all the elements that we will need. In section JHEP03(2016)045 , I i is ). = σ I i ( k U , for (2.2) (2.3) (2.4) (2.1) 2 a } ⊆ 2  H p i S 0 balls, , i P Nerv gives the ˇ Cech I = and . We define ···  > ∈ ⊆ , = dim i c 1 0 +1 } ) for a subset ) i k S ) i k I { i ( b ∂ ( (  1   = B increases, the balls B I σ {  , ∈ . Consider a covering i ] = Im ˇ Cech U k S k , v B = . One defines the boundary Nerv ··· ··· ··· / / , σ X , , i U a small prime). The boundary ˆ v ⊆ ) o , −→ · · · f p τ ( 1 1 ) ) . Note that as = 0 ··· , with an ordering of the vertex set. − − 1 1 and where a set  we have that p , k S S S 0 I 2 C -boundaries ( ( S 6 C  is then defined in terms of non-empty v  1 1 and its Betti numbers k [ . This construction does not depend on i − − ∅ ≤ U k p p \ ) between chain complexes, such that the is derived from an underlying topological 1) −→ 1 2 C C and to 6= / /  and − /B S k S ( applied to the covering 2 1 p I k ( k i S S p p C measures the number of independent holes of • Z ∂ k ∂ ∂ (typically for U =0 – 5 – i C ) k X construction, the Betti numbers give information by considering linear combinations p f b 1)-subsimplices S ⊆ U | ( between two simplicial complexes ) ) k Nerv −→ Z p 1 1 n − 2 C C S S −→  ]) = ∈ = ker S ( ( ˇ k . +1 Cech = k ) p p i k ∩ · · · ∩ k 1 X a U is the nerve of a covering C C , v C 0 or Z / / S i . The nerve of ( −→ U • I X ··· -chains 1 C and , Nerv k Nerv S 1 as S : consider for example the covering given by radius ): , v · · · −→ -cycles 1 0 ··· ··· f . Heuristically k ) as the quotient f as − ( v X p k • X ([ S Z ) associated to the set . In particular assume we can write C C k ; I • X ∂ (  ∈ C x ( −→ is the union of its ( . If the simplicial complex } k ) k k -simplex in σ H ˇ x Cech C B k ( labelled by a set  : . I B k k ∈ { dim i is a ∂ } i , for example via the i − . Then the nerve construction σ U X -simplex k ) = { X k Perhaps the most important feature of this construction is that it is functorial. Con- From a simplicial complex we can get interesting topological information by passing to For a metric space Let us consider a couple of examples. A simple simplicial complex which can be Z X = ( ⊆  ˇ diagram dimension sider a continuous map example induced by aduces map the between chain the map two underlying topological spaces. This map in- and define the spacesthe of homology dim space about the topology of where the hatted variables are omitted. We can now form the chain complex We form the vector spacewhere of of a operator B I Cech complex get bigger and bigger and therefore whenever its homology. Assume we have a simplicial complex that is, as the simplicialdefines complex whose a vertex simplex set if is the and particular only if details ofhomotopy the equivalent covering. to In the space particular, under general assumptions, attached to a topological space U sub-collections of sets JHEP03(2016)045 . ) ) X X X ( (   such Nerv (2.5) (2.6) ) as a } p VR VR k ) at the i Z ), again p x p , that is );  Z Z will be the ; X ) of a point 2 ( ); 2 X  S ( X ,..., S (  ( 0 • i  VR x ( H i VR { . We will call such ) whose vertex set and VR -simplex in n X ( H k = ( i 1 R  −→ S H σ in , a )  p I we associate the Vietoris- ˇ Cech ∈ Z i ; X } 1 i . x S ) (  . { ), by fattening the points of i • X ( x ≤ ˇ is the radius of the balls, while in Cech complex with a simpler vari- H  ), is that in defining the Vietoris- ( 2  : 0, also called the proximity param-  X ( B ). ? i, j  p f X  > ˇ ≤ Z Cech ∈ i ˇ Cech complex x ) is that it is computationally lengthy to -simplices we use a version of the ]. We will use these facts extensively in ); ˇ ⊆ Cech k X σ X S ) ( (  ◦  for 0 the X = (  f   whose pairwise distance is less than 2 VR X – 6 – X ( ˇ Cech } i k 2 to each point, and we connect two points by an VR i H ] = [  , x / σ ⊆ ). To the set of points in [ ≤ ? ) X ( f ) . To define X  ˇ j Cech complex ( ,..., X x -simplices are collections of points  0 i , k VR i x x { ( ˇ d Cech it is natural to define a simplicial complex whose vertices cor- ˇ Cech complex . In most practical applications a point cloud is constructed out ). This idea leads to persistent homology. p X and its induces the homomorphism Z ; complex we define the space . 2 X f is replaced by a ball of radius ∅ ] changes sign under an odd permutation. ) we only have to compute the distance for a pair of points at a time. S X k ( X X + 1 points , which leads to a family of homology spaces • 6= (  , v  k ) H point cloud n i VR x a ··· ( , . Instead of a simplicial complex, now we have a collection of them,  ) in 0 X  p v B Z ; Vietoris-Rips =0 1 , we can compute its homology k n S X each point of ( T •  The main difference with the The problem with the Given a point cloud X H -simplex [ 2.3 Persistent homology andThe idea barcodes behind persistent homologyfunction is of to study theparametrized homology spaces by the latter is the distance between the centers, leads to the inclusions Finally now that wecloud know how to construct the Vietoris-Rips complex edge anytime their ballsp intersect. The natural orientation is given byRips declaring complex that the Furthermore the fact that in the former the parameter Rips simplicial complex as follows.is a Given collection a of proximity parameter Equivalently we assign balls of radius is the set of pointsthat of check all the intersections. It isant, useful the to approximate the respond to theconstruction. set From of pointsIn in eter. For example now we can associate to We will be interested inanymore a a topological version space, of but the aa finite previous collection collection constructions. of points Theof starting point a is multidimensional not dataassociates set. topological Topological information to data a analysis point is cloud, basically via the a homology framework of which a certain complex. level of the homologies,the such following; that in ourinclusion. case the Then map functoriality can betweenof be two different used simplices to understand the fate2.2 of the homology Point classes clouds and the Rips-Vietoris complex commutes. The map JHEP03(2016)045 ) p Z (we (2.7) (2.8) (2.9) , the ); j ’s. We X  ( is a real ≤ b  a < b  persistence k - ) are the lift VR N ( ≤ . i 2.9 i H , with b ). Since homology  ’s. Here “large” can → · · · , 2.8  ) and ) p ). Persistence modules p Z Z i,j . ); ρ be a field. A ); X X ( K = ( , together with a collections a 2   K ). The maps in ( k,j → · · · , , ρ VR VR 2.9 ) -persistence module, and that it is ( · ( i i X N over ( H -persistence module, since i,k H 2 , such that whenever  ρ N R j → · · · ∈ , → i , VR 2 } ≤ )  i p V i X → , Z { ) ); we can construct the associated Vietoris-Rips → , X is a finite set. – 7 – X ( a 1 (   1 J 1   X X VR for every VR → , ( i j 0 →  , where V H X ) J and then to die at a subsequent “time” X ∈ → ( , a a 0 −→  we construct the sequence of inclusions of spaces } ; any choice will do, but a choice has to be made. )  . For each a p ). Barcodes are a simple tool to visualize homological features i R  X V Z -persistence module. To do this we have to chose an ordering { VR ∞ ··· : N ); as −→ X < ( i,j  = + 0 homology class is present both in . Therefore the only thing that can happen is for an homology class to 2 N ρ  b b :  . While these in principle are continuous families, there will be only a  VR <  is a family of vector spaces f ( ). This leads to the filtration of simplicial complexes i and 1 same X K H a ( -th homology gives  i <  0 VR  over The usefulness of persistence modules is the existence of a classification result. This Let us make the above discussion a bit more precise. Let Secondly, it can be useful to think a bit more about ( From the point cloud module can be decomposed in simpler modules. is a generalization offinite the dimensional similar vector result spaces fromthe are elementary classified same linear by fashion, algebra, their certain which dimension, classes states up of that to isomorphisms. persistence modules In are classified by their barcodes. on the particular approximations one is using when constructingmodule the dataset. of homomorphisms homomorphisms are compatible (incan the be sense added, that to create a new persistence module. Viceversa we can ask if a persistence of a data set. Theypersistence are is precisely to what look keepshave for tracks different features of these meaning, which births persist depending andis over on deaths. a asking. the large The Persistent point range idea features of of clouddataset. are or On a on the measure the other of hand, particular the short-lived underlying questions signatures topological one are structure interpreted of as the noise, which depend is functorial, these maps keepwe track know of if the the correspondingfor homology arbitrary classes. In other words be born at aallow certain for “time” the case variable. However only finitelywe many can simplicial consider complexes thispreserving will map a be distinct and therefore to homology of the inclusions between the Vietoris-Rips complexes in ( We will see momentarily thatcompletely this is characterized an by example its ofFirst barcode. an of Let all us what however we begin have with defined a is couple technically of an remarks. for 0 = complex Taking the finite number of inequivalent simpliciallabel these complexes values which of appear at finitely many parametrized by JHEP03(2016)045 i n ) so ≤ (2.10) (2.11) to the i consists m, n m K i where we L ), obtained ≤ N 2.9 . Each landmark X ), given by and we represent it -persistence module. N ), where 0 i from . Then the classification m, n ( is finite dimensional and L , n ∞ i i K V barcode m = + . Given two integers ( i > n n n can be decomposed as -persistence modules which arise at random. This is quite useful, or , N -persistence module ( K ) assigns the vector space ) if each L j N ) is clearly an , n -persistence modules can become quite ) is the distance between the landmark j -persistence module is equivalent to the m, n z i < m 2.9 ( N N , tame m i K ( if otherwise L is ( + 1-th landmark point is chosen in order to K as vertices of the simplicial complexes. d i N X ∈ -persistence module =0 – 8 – N , i j M 0 K N } of i ' V L i { selector. The idea is to select points by induction, ( } . In words i ), where n V z , { ) = i ≤ L , is then uniquely determined by its barcode. In the rest of ( i -persistence module over d m, n maxmin N . We call such an assignment a ( i < j ∞ K . Note that with a maxmin landmark selection, the landmark set −→ X is finite, it is also tame. Each ≤ z consists of a large number of points, the computation of the Vietoris- ∈ is an isomorphism for large enough X m z X +1 to be + n for V i K , and the decomposition is unique up to the ordering of the factors. In plain n ]. Note that we can extend this definition for N −→ , we introduce the “interval” = id n n m, n V i,j ≤ : ρ and the point m i chosen landmark points, then the next A landmark selector is an operator which chooses the subset Each collection of maps and vector spaces ( +1 L i n,n maximize the function set will consist of points spreading apartthe from each set other will as much cover as theis possible. dataset, that As the in a consequence maxmin principle algorithm better will than generically a choose random outlier selection. points. The drawback An obvious landmark selectoralthough depending picks on the how elementswill scattered of is mostly the use dataset, ain it so-called order may to miss important maximizefrom features. the a randomly distance We chosen from point.of the The already remaining chosen ones set. are chosen by More induction: precisely if we start Rips simplicial complexes andintractable. of the We will associated nowone discuss could select certain only approximation a schemes, limited based subset on theselector has idea its that own advantages and disadvantages and it is important to be aware of them. this paper we willfrom compute certain the point clouds barcodes and associated discuss the to physical interpretation of2.4 their persistent features. Approximation schemesWhen and a witness point complexes cloud and we allow graphically by a collection of bars, each one associatedSince with the point the cloud aforementioned intervals. by taking homology in degree for a certain words a tame persistence moduleassign is non completely zero determined vector by spaces.assignment the Therefore of intervals a in a tame collection of pairs of non-negative integers ( where interval [ result states that any tame ρ that We say that the persistence module JHEP03(2016)045 - ) . ν ,  L is in- , (2.13) (2.12) X so that and its ( N -simplex x X k ∈ and the LW . To define ∈ so that x ν L x ) holds when- ). Again this X 2 mathematica . Therefore we . We define the ,  2 ∈ ,  X L form a L x , , i l ] for more examples <  X X ( 5 ( 1 –  3  . LW LW ) is given by ⊂ ,  ) + ) L x ]. We will use 1 , ( 0 vertices ) when  . k X 22 2 ,  ( m L ) the distance between ,  , x W ) + L ( k > X x . To define simplexes we need to in- , k ( ( from a point cloud programs and datasets are available L } ≤ X ) = 0 otherwise. Then, given two ver- ) as the distance between ) m ( m x L x x ( LW ( , W k m l m ( } ≤ = 1. ⊆ ) ) if we can find a witness point x ) ν 1 , ,  matlab 2 L l ,  , available from [ , – 9 – ( L , . . . , d X ) , , d ( x ) X we define to perform all the homology computations and our , x ( ) and there exists a witness point 1 , l X LW 1 ( W ,  l ) is again given by ( ∈ L -persistence modules and we can study their persistent , d and we denote by , d ) x N ,  javaplex { X x L X ( ] is in , , 2 0 matlab is non zero, and set l l X ∈ W ( 1 ( max l ν d x { ) as follows. The vertex set of LW ,  L max , X ( , the edge [ W L in and again we can construct filtrations by inclusion. The usefulness of the lazy 2 l 2 <  ] if all its faces are in and k tically it determines theconnected topological components, properties or of the a presencefaces. dataset, of loops such Being topological, as or in this its generalany information clustering higher metric is in dimensional used independent sur- for on the the set analysis. of For coordinates example or by regarding a dataset as a statistical ]. 1 l 1 l  Topological data analysis provides qualitative information about a dataset. Heuris- In this paper we will use Another approximation scheme is the lazy witness complex A landmark selector can greatly simplify the computation of the homology. We will 23 1. ... + 1-th closest landmark point. Then a collection of 0 l 2.5 Topological analysis Finally we collect some qualitative ideashomology on will how the be topological used analysis in basedof our on the context. persistent practical Again uses we refer of the these reader techniques. to [ of a dataset. In this paper we willprograms always will set employ the library for the manipulations ofat the [ datasets. Our Finally a higher dimensional simplexif is all an of element its of edgesever the are. lazy Note witness that complex againwitness the inclusion complex is that1-skeleton. it The is lazy less witness computationally complex involved is since the it simplest way is to determined study from the persistent its homology troduce a notion of distance.th For closest landmark point if tices Passing to the homologyfeatures defines by looking at the barcodes. complex depends on avolved. landmark The selection, vertex set but of this time an extra parameter [ In particular we havecan the construct inclusion a filtration of witness complexes as function of the proximity parameter use landmark selection to definecomplexes. approximations Assume to we the have Vietoris-Rips chosenwitness complex, a complex the landmark witness set simplexes, we pick ak point JHEP03(2016)045 .  -cycle. n = 2 superconformal field theories. -cycles as well as their characteristic N n – 10 – = 2 vacua of the type II string. Examples of such vacua include Calabi-Yau N -cycle, it is possible that there exists relations between themselves, in the form n = 2 vacua This perspective is commonlythe used barcodes in of different fields datasets suchour can as case reveal biology one if a or cansay certain neuroscience, for a drug example where was certain select effective particle or stringqualitative present not. differences, vacua in In and with the of or low which without energy type. a spectrum, certain and feature, ask if this comports and depends sensitively oninteresting the but physical short-lived problem. persistent homology Wewhen classes. will the In see existence particular example this of of canvarious a happen physically length symmetry scales. or modular property forces the same behavior at lived classes pointclear out and towards follows from homologically theTherefore robust definition one features. of is the lead Vietoris-Ripsclasses. This complex to Of and is look course, its what for intuitively does variations. long-lived it mean bars, to be encoding short- the or long-lived persistent is somewhat homology subjective length scale. This canFor be example, seen if as in evidencean a of existing certain correlations region between ofof the the a data. point series cloud of the algebraic equations points which are describe disposed the along computes the Betti numbersformation of about such the a homologically manifold. non-trivial length From scale the as barcodes measured one by obtains the in- proximity parameter forming higher dimensional homologically non-trivial cycles, at least at a certain approximation to an underlying topological manifold at a certain length scale, one The topological analysis can be most effective when comparing different datasets. Short-lived persistent homology classes are generically regarded as noise, while long- The presence of barcodes in higher degree indicates that the data are organized In particular we will study the following (incomplete) set of string vacua: N 4. 3. 2. eties still rather mysterious. Itcan is be natural applied to to wonder the if known the distributionstion techniques of can we Calabi-Yau varieties, we have and exposed hope what so kind towhich far of gain. play informa- an Similar arguments important hold part in in the the case classification of of Landau-Ginzburg models, Having set up ourexamples of main computational tools,manifolds and we Landau-Ginzburg proceed models. toown The and use construction while them of hundreds Calabi-Yaus of inexhaustive. is thousands a an The of classification art few examples problem on are specific is its known still explicitly, wide the open, list and is the far origin from of Calabi-Yau vari- In the following we will apply these ideas on certain distributions3 of string vacua. JHEP03(2016)045 ) is C ; X ) (for X ( n 1 , ). More 2 + ¯ Y where ( H 1 , 1 X χ, n h × 1 , 3 , where a Calabi- ) = ) = dim X R X X ( ( 1 1 , , ]. From this list we take 2 2 h h 26 . The Calabi-Yau theorem )), and study its persistent 1 , ) K¨ahlermoduli, which cor- )). Of course this is a rather 3 X C ( R ; X ( ) and X , χ ( ) Y , χ 1 ( , ) 1 X 1 , ( 2 X 2 ( H , h 1 2 ]. , h 1 h 11 ) = ) there exists a unique Ricci flat K¨ahlermet- ) + X C ) + ( ) = dim X ; 1 , ( X 1 – 11 – 1 X X ( , ( ( h 1 1 , 1 1 , , 1 h 1 1 ]. We parametrize these models by ( h ], containing Calabi-Yaus which can be realized as a h = 2 supersymmetry in H 27 = ( 24 . Gromov-Witten theory produces symplectic invariants ∈ x N X ω )). X ( 1 ) and they represent the same physical vacuum. The Hodge , 2 = 0). h χ − X,Y ) X ( 1 , 1 ], which contains the 7890 CICYs constructed in [ h 25 corresponds to the point ) = 2( the 266 pairs of distinct Hodge numbersA and list add of their Landau-Ginzburg mirrors. discrete models, torsion, their taken abelian fromsimplicity orbifolds we [ and remove certain by hand modelscharacterized all by with those models which need an extra label, which are hypersurface in a toricdra. variety, and Of correspond this to list, four 30108 dimensional varieties reflexive haveComplete polyhe- distinct Intersection Hodge Calabi-Yaus numbers. (CICYs),intersection which of are polynomials constructed within viafrom a [ a product complete of projective spaces. We take the list The Skarke-Keuzer list from [ X . In this note we will only consider the cruder topological information contained X ( Therefore as a first approximation we label a Calabi-Yau compactification by its Hodge The superconformal theories which describe the propagation of strings on Calabi-Yaus • • • χ X numbers and Euler characteristic ascrude ( approximation, since differentwould Calabi-Yaus can like have to the consider sameYau a Hodge distribution numbers. of We these values as a point cloud corrections due to worldsheetare associated instantons, with which an extremely modifycounting interesting holomorphic enumerative the curves problem, on low Gromov-Witten theory, energyof effective action, in the Hodgeenumerative numbers problems and of Calabi-Yau Euler appear characteristic; in [ applications of persistent homology to form a mirrornumbers pair for ( mirror pairsdeeply are complexified related K¨ahlermoduli as andcertain complex quantities structure associated moduli withto are a interesting exchanged. Calabi-Yau mathematical can Often predictions be for computed the exactly mirror and manifold. this leads For example quantum vector multiplets and hypermultiplets.by The Euler characteristic of a Calabi-Yau iscome given in pairs,established a for phenomenon many known pairs as mirror of symmetry. Calabi-Yaus. Mirror In symmetry this has case been we say that two Calabi-Yaus states that for each K¨ahlerclass ric. The moduli spacecomplex of Calabi-Yau structure metrics deformations is and parametrizedrespond by to scalar fieldscharacterize the in low the energy effective effective four action dimensional by theory. determining These geometrically the Hodge number numbers of 3.1 Calabi-Yau compactifications We begin by considering vacua of thea type Calabi-Yau threefold, II a string complex which three have dimensionalThese the manifold form compactifications with preserve trivial canonical bundle. JHEP03(2016)045 . 1 ) for 2 ). The Z )). The ,  ] denote ); L 5 X , ( ,  X L ( , χ , ) X LW X , . . . , z ( ( 1 2 ] and their Hodge z , 1 LW h ( 28 • , H 26 ) + ], and the classification ], by classifying reflexive X 30 ( 12 1 , 1 h of the quintic descends from the ) = 0, where [ = ( 5 ω x ] (to which we add the mirror Hodge . The results are shown in figure ). To do so we wrote a program in The Kreuzer-Skarke list of Calabi-Yaus. 25 obtained in this way from collections ,  , . . . , z L X 1 , ) = 0. More general constructions can be z Left. X 5 ( ( 5 f the vector javaplex LW – 12 – X , . . . , z 1 z ( 5 f . The K¨ahlerform 4 P ], using the library ]. Such a list is available at [ 23 29 , while the complex structure moduli correspond to the independent de- 4 P . Barcode computation for the distribution of Calabi-Yau manifolds. The point cloud , available at [ Complete Intersection Calabi-Yaus, with their mirrors. The point cloud has 532 elements, Out of each one of these two lists we construct a point cloud and study its persistent An example of such a list is the class of complete intersection Calabi-Yaus (CICYs), The simplest construction of a Calabi-Yau manifold is as a hypersurface in a complex is constructed assigning to a Calabi-Yau homology using the lazymatlab witness complex On the left wethe Kreuzer-Skarke have list the of barcodes Calabi-Yaus corresponding corresponding to to reflexive the polyhedra, modules on the right product of projective spaces. Theirnumbers classification computed in was [ completed in [ numbers). Another list was obtainedpolyhedra by in Kreuzer four-dimensions. and Skarke Reflexive in polyhedrathreefolds [ in by four-dimensions realizing describe them Calabi-Yau proceeds as using hypersurfaces the in powerful toric combinatorial varieties techniques [ of toric geometry. formations of the definingthought equation of as more elaboratefamilies versions are of available this in simple the example literature. and many lists ofthat Calabi-Yau is those which can be constructed via the complete intersection of polynomials in a projective space. Forof example a the homogenous quintic polynomialthe threefold homogeneous of can coordinates degree be of five K¨ahlerform of seen as the vanishing locus homology. A relatedof question varieties is arising if from different pointfrom constructions clouds their have barcodes. different topologicalnumbers We features, of will as Calabi-Yaus. address seen these questions with certain known lists of Hodge X computation of the persistentlandmark homology set is consists done ofThe using 200 point the points cloud lazy in consists witnessRight. both of complex cases. 30108 points;and the the homology homology computations computation involves runs 251926 over simplices. 49744 simplices. Figure 1 JHEP03(2016)045 nearby  ], on the From this 31 3 ). Note that right ( 30 of the proximity for which the points 1 -persistence modules  ∼ N  in the list we are considering. The empty 2 increases before showing any mirror structure, (at this value we see the birth of an homology  – 13 –  ). This region was identified as special in [ ) the vast majority of connected components associated X greater than the characteristic length of the empty region, (  1 , left 2 h ( 1 ) + X on the left, measured by the barcode in degree one, means that ( 1 , 1 1 -persistence modules in degree zero (we will informally call these “Betti in figure h N  . 1 is that certain bars come in pairs: this is a consequence of mirror symmetry, or 1 Now we focus on a particular zone, namely the “tip” of the distribution, that is the A similar discussion holds for the distribution of CICYs, in figure Of course all these features could equivalently be “seen” by plotting the Hodge number Most bars in the barcode in degree zero are relatively short-lived. This is a sign that Of course one should also take into account that the HodgeWe numbers have are confirmed integers this while conclusion the by proximity repeating the computation for different landmark selections. 2 3 sions of the Standard Modelbe with rather few rare generations and of the fermions. corresponding area Such quite Calabi-Yaus unpopulated. appear to We wish toparameter examine is this area a realnoticed variable. in figure This will induce a few spurious 1-cycles, which are however too small to be which can be regardedperspective as the “noise” set and of excluded CICYs is from topologically the simpler persistence than analysis. theregion set of with varieties small in theground KS list. that heterotic models with small Hodge numbers engineer supersymmetric exten- all sign of non-trivialparameter, homological much structures smaller disappear than atshows in values the even case less of structure,formation the of a KS list. 1-cycles sign is that Now the accidental. the barcode CICYs in This are degree is one more an evenly example distributed of and short-lived the homology classes becoming a single connected component. distribution. The purpose ofstatement this discussion concerning is its to topologicalprovides turn the them features. necessary into tools. a The mathematically precise formalism of figure more down to earth thetwo areas symmetry of of the the distribution Hodgeany with bars: numbers. exactly the The areas same mirror which behavior.will bars are start Of correspond close to course to to “interact” this with the is each not symmetry other true axis as for in the Hodge numbers distribution the distribution of manifoldspoints is become generically part rather of dense, thevisible same since values simplex, at of ceasing small to values bewith of isolated individual connected points components. has For to already the disappeared. existence of isolated This points is or not clusters true in for the distribution. every bar, An interesting pointing feature of there are zones where noregions Calabi-Yau in is present, the distributionsurrounding of Calabi-Yaus these are areas detected are atclass) closer and values disappear than of at values of which is now filled up (and the homology class dies because it becomes a boundary). corresponding to number 0”) are a measureSimilarly of a the barcode number in ofvarieties degree under connected one consideration components, does is at not a everyof have signature length any 1-cycles scale. barcode of in in non-trivial higher figure 1-cycles. degree. The The appearance collection of the list of CICYs to which we have added the mirror Hodge numbers. Recall that bars JHEP03(2016)045 ) + 22, X < ( 1 ) , 1 X ). Let us h ( 1 , 2 h right 22. The filtered )+ and run overs 827 < 22 ( X ) 2 ( < ], according to which 1 Z X , ( ) 1 31 1 , h X 2 ( h 1 , 2 ) over h ) + X . ( X   ( ) + 1 , VR 1 X h ( 1 , 1 h of 170 points, and run over 118362 simplices. L ) and – 14 – left we show the barcodes for said distributions, with 40 ( The point cloud parametrizes Calabi-Yaus with 2 < ) Left. X . ( 1 , 1 2 h with a landmark selection 2 ) + Z X ( on the right. The two odd features at Betti number one appear clearly, 1 , 1 2 ) over h ,  L , X ( 40 and consists 479 points. The persistent homology is computed via the lazy witness . Barcode computation for Calabi-Yau varieties with small Hodge numbers. The point LW < The point cloud parametrizes 23 Calabi-Yaus with ) , table 1], and their mirrors. We stress again that we are parametrizing Calabi-Yaus X We have seen an instance of one of the main themes of this paper, that special vacua We can try to give a topological rephrasing of the philosophy of [ ( 31 1 , 2 are associated withbeen characteristic rather topological limited; features. howeverthat we topological Clearly think interesting it the structuresphysical elucidates analysis interesting at the settings. the so principle level far that of has one persistent might homology expect correspond to Hodge numbers has certaincontinue distinctive to topological exist features. if we Indeed restrictas our shown these in point specific figure cloud 1-cycles toeven Calabi-Yaus with if rather short-lived.symmetry. The fact that they come in a pair is a consequence of mirror are larger than the rest, as they appear at larger values of Calabi-Yau varieties with small Hodgetranslate numbers into are the “special”. statement In that our the language distribution this of would Calabi-Yaus with small values of the by their Hodge numbers, whichthe is same a Hodge very crude numbers.Hodge approximation since numbers In distinct figure varieties have consider the figure on theto left. the others, We signaling see that the in formation degree of one two 1-cycles two bars at stand a out larger with length respect scale. Such cycles from the point ofin view the of KS persistent and homology. CICYin lists, To [ and begin add with the we other collect small all Hodge number the Calabi-Yaus manifolds pointed out cloud is constructed as inh figure complex Right. homology computation employssimplices. the Vietoris-Rips complex Figure 2 JHEP03(2016)045 , ) . i  12 12 p = 2 p = 2 (3.1) (3.2) ) are 2 . This + N N ) = Z 2 11 ) vanish, ); X ( , p ,  1 ) 3.2 , and L 1 11 , h 1 p X ( − ], which results in ). We will assume representation and ) i 12 LW i ) as a function of ( p 34 ] by classifying all the i gauge symmetry and (Φ ,  (Φ 27 L H 6 33 ] by the classification of , W W = 2( E ) and result in 2997 differ- X 32 ( χ 3.2 . We have then computed the , LW = ( j X ¯ t x i t ij = 9. Certain orbifolds of . p ] c N ij i X ) Φ i = (Φ 0 J ,..., t 1 0 ∂W J h – 15 – t [Φ R C = R is the number of fermion in the ) = tr t , then we can identify the Hodge numbers , obtained by taking the quotient of the space spanned by t, 27 ( R X n P = 11 p . We will assume that the other non-trivial coefficients of ( representation. The fermionic spectrum is determined by the chiral ring . The distribution of abelian orbifold vacua, on the left, appears to be 11 3 p 27 interacting via a quasi-homogeneous superpotential ), namely ) = . If the Landau-Ginzburg model has a geometrical interpretation as strings ) has isolated and non-degenerate critical points only. Ground states are related i 3.2 X 27 ( (Φ 1 n , 2 = 9 model. The massless spectra can be computed from ( ,...,N W h The barcodes resulting from the homology computation for We consider the collection of 10839 models constructed in [ = c from ( = 1 12 shown in figure the most complex from ais topological most viewpoint, apparent also in withIn degree respect degree one, to zero where figures the several figure 1-cycles shows appear the with existence no of obvious many regularity. long-lived connected components, a list of models can138 be extra spectra. obtained by We have consideringorbifolds studied the discrete and persistent discrete torsion homology torsion as forhave the models, in labeled distributions taking [ of a as abelian Landau-Ginzburg input spectrumas the by explained set above, the and of vectors used inequivalent theseassociated spectra. to homology define We our groups point using cloud the lazy witness complex non-degenerate quasi-homogeneous polynomials which can playof the a role of a superpotential ent spectra (or pairs of Hodgeabelian numbers). symmetries of This the list above was potentials extendedThis which in can results [ be in used 3798 to inequivalent construct spectra, abelian which orbifolds. we will use for our analysis. An additional p propagating in a Calabi-Yau and if necessary discarding the respective models from the lists of consistent vacua. Landau-Ginzburg models describe conformal fieldYau theories moduli at spaces. certain pointswhere Such of they models the engineer Calabi- can an alsomatter in be effective the four used in dimensionalR heterotic theory string with compactifications, expressed in termsonly of Landau-Ginzburg generators models of with the central superconformal charge algebra. We will also consider the chiral superfields by the Jacobian ideal of the superpotential The degeneracies of chiral primaries are encoded into the Poincar´epolynomial Now we turn tosuperconformal Landau-Ginzuburg vacua. field Such theories. modelsi describe They two are dimensional governedthat by a numberto of elements chiral of the superfields chiral Φ ring 3.2 Landau-Ginzburg vacua JHEP03(2016)045 . ) X ( 1 , Right 1 h consists of 3792 points and X are constructed out of vectors X ]. In general one can use D3 branes, 17 7 branes, wrapping holomorphic cycles – D 15 ) complex structure deformations and , and – 16 – ). To pass to the homology we use the lazy witness X parametrizes 128 models, 100 of which constitute the X ( 12 1 X , p 2 h + 11 , p Abelian orbifolds. The point cloud ) , with 11 = 1 model where moduli are stabilized by fluxes and quantum p X Left. − N . 2 12 Z p = 2( of 200. The computation of filtered homology requires 123057 simplices. ) over χ L ,  L . The homology computation runs over 57090 simplices. , = ( L X x ( . Barcodes for Landau-Ginzburg models. The point clouds LW We take a Calabi-Yau From this perspective the two type of vacua appear topologically very different, despite to obtain a quasi-realistic corrections. We willissues be of mostly phenomenological viability interested of in the the model distributions or of ofK¨ahlermoduli. the For flux simplicity simplifying vacua, we assumptions. will and fix ignore the K¨ahlerclass, so that Calabi-Yau metrics are 4.1 Flux compactifications Before discussing the uses offlux persistent compactifications. homology, we We briefly will review considera some II Calabi-Yau basic B/F-theory compactification, elements vacua as of obtained reviewedextended by in in an [ the orientifold of directions transverse to compact cycles andand a axion-dilaton collection moduli, of stabilized byinterested D-branes. the (yet) flux in induced We fully superpotential. willanalyze stabilized We only simple and are examples not consider phenomenologically really to viable complexwhich show models, kind structure how but of to results we one use wish can techniques to expect. of topological data analysis and 4 Persistence in fluxNow vacua we proceed torealized apply by our an techniques to orientifold flux of vacua a of Calabi-Yau the compactification, type with II B fluxes string. turned These on are along signal that the vacua tendof to cluster discrete in torsion certain models regions. is On extremely the simple, other with hand noarising the features distribution from in similar degree constructions. one. between In two physically this similar sense situations. the topological analysis can discriminate of the form complex the landmark set Discrete torsion models. Thelandmark point set cloud Figure 3 JHEP03(2016)045 a } ), as a β Z a h , β (4.3) (4.4) (4.5) (4.6) (4.1) (4.2) a where α , where 2 − G ). Since { and given by SL(2; T a Z / 2 f defines the α × T c a X, z X ( ) and × −→ M × H 3 = R ) c H X 2 z . . T M X, −→ M ∈ ( ) b a ( is a function of the 0 , δ ) 3 , 3 X 1 s = τ ( H X )) = 3 ( , H )) /g ,H ¯ τ b 0 0 z ¯ 3 , τ M β 3 ⊗ z . F − H , . Equivalently such a space − ) + i ) , ∧ s ) plays an important role in H τ ) ⊕ a a g τ ) 0 a X a i ( β ( ) z is the upper half plane where β α C i ( , z ( 0 , ) a − X 3 z +1 − z ( X +1 H G = 1 ( 1 2 , Z , H , b 2 2 1 z τ = h G h log ( log ( + z H + + 2. a − a Ω − 1 h f ⊕ , ,  2 a ) = ( ) a can be represented as Ω b  + + h Π z B δ ¯ c X Ω · a a Z ( − α α 1 ∧ Σ , a a 2 · determines the Hodge decomposition z = Ω f h . – 17 – † ∈ M c a ( ( H y , 0 0 X β z + 1 be a symplectic basis of d a i Π Z ⊕ α α 1 b z τ 2 2 , i ) ) ) 2 − B ∈ M for + Z π π =   X . z ( z z x ). Similar arguments hold for the axion-dilaton modulus, (2 (2 0 X , Ω log log 3 z − − X ( , . . . , h a = d H − − , 0 = = , A a b τ 3 z and the string coupling constant Z 3 3 = δ ω = 1 0 H F ) = ) = H = C ) is one dimensional and the fibration C . 0) form Ω ∈ ). The period vector Π( b , a, b x ) + 1. We assemble these fluxes in the 4-form on X z, τ ) as z z α ( ( d 2 ( X, X 0 takes values. The axion-dilaton ( , a T G K ( ∧ 3 3 z ( A 1 a , τ 3 0 with , Z 2 H ∂ H 1 τ H } a H ) can be expressed as the derivatives of a single function, the prepotential + z ) = ( ∈ ,B y z is the complex structure moduli space and a ( a , . . . , h d τ a G are local projective coordinates on the complex structure moduli space, and the c A ω ∧ G { a 3 = 1 is elliptically fibered over M is z F 2 a Now we turn on fluxes in the NS and RR field strengths In our basis the (3 Let T ), as = M z 3 ( with G where and Σ is the symplectic matrix of rankfluxes are 2 quantized, they can be expressed in terms of the integer valued vectors a natural hermitian metric, theon Weil-Petersson metric and the associated K¨ahlerpotential where functions G the determination of the K¨ahlerpotential and flux superpotential. The Hodge bundle has Hodge bundle. Take Ω so the actual Hodge bundlewe write of physical interest here is in terms of its periods A choice of the complex structure On a Calabi-Yau the its Poincar´edual integral cohomology basis, so that ing physical configurationswhere is non-compact andthe has axion-dilaton the form Ramond-Ramond scalar can be interpreted in F-theory as the moduli space of Calabi-Yau metrics on parametrized by complex structure deformations. The relevant moduli space parametriz- JHEP03(2016)045 , 13 (4.7) (4.9) (4.11) (4.12) (4.13) (4.10) on the ] which L 35 ≤ , . The critical the orientifold flux M g ) (4.8) N z )) = 0 Π( · z, τ ( ) , with K . h a , consistency of the com- τ z h X/g ∂ X · − ) ) Σ z f , · ) ( 0 3 brane charge z, τ f )) ( α z, τ D 2 , = ( K ) ) + Π( z, τ a 3 π K z ( z τ L. H ∂ G ∂ Π( ) ) = 0 ∧ ) = W a = (2 z i z 3 3 z ∂ τ z, τ F D D Π( ( z, τ ( ( Ω ω ( · · X G z N ∧ ) ) G ∧ δ Z – 18 – W ) h h + z 2 W 3 ) i ¯ τ τ Ω 0 3 branes transverse to X α ) + flux ∧ − − ( D ) + τ H 1 4 3 N 24. In particular this sets a bound f f ) − ) = / ( ( z, τ G π ) z, τ ( 0 0 z 3 2 ( ( G α α Z (2 T F G µ ( 2 2 ) define supersymmetric vacua. ( × ) ) d W χ = W a π π X X z τ which is an elliptic fibration over Z Z = of the Hodge bundle. 4.10 ∂ ∂ flux Z = τ L ) in a certain region of the moduli space, as the fluxes take values = (2 = (2 N ω ) = ) = ) = the number of ∧ 4.10 z, τ z 3 ) and ( z, τ z, τ ( ( ( D G G G 4.9 N runs over axion-dilaton and complex moduli and we are again neglecting W W W i a τ z D D is the connection associated with the Weil-Petersson metric on ) are D ]. We will have nothing to say about this approach, but we mention a few points for z, τ Compactifications of this sort have been widely studied since they provide workable Turning on fluxes gives a contribution to the total In local coordinates the F-term equations which follow from the superpotential The presence of non-trivial fluxes generates a non-trivial superpotential [ ( 36 dual to the Hodge bundle, given by the functional , G which is just a sumvacuum of (here delta function contributions, eachK¨ahlermoduli). one This centered distribution at is a in supersymmetric general pretty intractable and often it is simpler to satisfying the tadpole constraint. models where ideas about14 the statistical distributionscompleteness. of A string simple vacua distribution can of string be vacua tested is [ the density In our cases,This the involves orientifold a can four-fold involution. be In seen this astotal case arising amount of from flux an available.solve Therefore F-theory the to equations compactification. find ( supersymmetric critical points, one has to If we denote by pactification requires the tadpole cancellation condition where point equations ( acting on sections Ω W stabilizes complex structure moduli.L Such a superpotential is a section of the line bundle JHEP03(2016)045 = 2 + (4.15) (4.14) 3 ). One b ) and are z, τ Z ( , corresponds G X, L ( W 3 2 H D ). A slightly different i,j z . The F-term equation ( ) only in that each delta ρ L ) and therefore to a large , ≤ ) z, τ 4.13 2 2 ( f f G 1 = sgn det vanishes and therefore h W , 1 F + i , ) − 2 ) take values in 1 1) 2 h 2 f ). The integral of such a distribution h − z , h 1 ( + ( 1 f ρ ) takes values. Physically such an integral h i ). Therefore the flux superpotential is τ M ⊗ L , ∗ ) or of the dual Hodge bundle 2 = ( z, τ T ( ( h i h G M ) is almost a topological quantity (it would be which differs from ( – 19 – top − c I 1 DW 4.14 h M ) and Z − 2 , f 1 I ∼ f ) = ( ], suggests that techniques based on persistent homology, τ ( 36 ] = ( G 14 W f of the moduli space corresponds to the number of supersymmetric U ) can be taken to be Π = (1 4.1 ). ]. Having no complex structure moduli, of ( M } 14 A, B { = 2. This implies that the periods of the holomorphic three form Ω over the symplectic The point which is important for us is that a high degree of topological complexity of 1 , 2 h basis where the fluxes vectors constrained by the tadpole cancellation condition 4.2 Rigid Calabi-Yau We begin with the simpleststudied case, in a [ rigid Calabi-Yau with2 no complex structure moduli, as have a high degree ofcan topological measure complexity. the Persistent topological homology is complexityobtain precisely of interesting a a information tool distribution by which of applyingof points. techniques string We of vacua. therefore statistical expect In topologyin to the to simple remaining ensembles models. of this section we will see examples of such applications for compact the moduli space of physicalto configurations a large number ofnumber critical points of for supersymmetric the superpotential vacua. In other words we expect a distribution of vacua to the top Chern class of the bundlegives where an estimate ofby the counting number vacua of with specific supersymmetric properties,We vacua. say will a One certain not can value consider refine ofnote the such such cosmological that a counting constant. the formula problems number in of this vacua paper. from ( However it is interesting to term equations, and leave the analysisof of such statistical a distributions to index the density future. is An [ example Morse index for dealingthe with vacua total in number supergravity. ofcan Indeed vacua be while is created in often or rigid destroyed givendiscussed supersymmetry in by more pairs, a precisely in topological the depth situationwhich formula, of in can in Morse [ supergravity be theory. vacua This thoughtstatistical analogy, of distributions of as vacua. a In version this of paper Morse we theory, will could focus be on usefully actual solution applied of to the the F- over a certain region vacua satisfying certain properties, which enterapproach in is the to definition define of anfunction index density is d weightedmotivation by for the doing so sign is that of such an the index Jacobian appears to ( be the proper generalization of the deal with an approximate continuum distribution JHEP03(2016)045 , X  F ]: the (4.16) 14 and study X ). To solve this implies that the Z  , ). The corresponding ,  L , X ( LW which lie in a fundamental domain . τ 2 1 f h ) to construct a point cloud modulo the action of SL(2 τ + i + i 2 2 H f h – 20 – Im consisting of 140 points. The computation runs over τ, = L τ . From the distribution of the barcodes we learn the = (Re to generate random vectors of integral fluxes obeying the 4 x ) orbit. time to cover them up. Z  , ) with a landmark set . This restriction is to avoid overcounting of physically equivalent vacua mathematica ,  H L , X ( ) in Z ) = 0 has the simple solution . Barcodes for the distribution of flux vacua in a rigid Calabi-Yau. The point cloud LW , τ ( G Then we use the values W τ some are smaller (short-livedof bars) several and degree one othercorresponding bars are cycles which larger have are roughly (long-lived born the bars).cycles, and same since die size. The it at The takes presence the long-lived more bars same correspond value to of bigger following facts. Theof structure of relatively the long-lived barcode andThese with short-lived corresponds Betti regularly to number the distributeddistribution one “voids” 1-cycles imply contains in in the the holes, the presence distributionThe at point of presence the vacua, cloud. of already center thesenon-trivial noted of holes in 1-cycles which [ implies will there form that is in at a the certain filtered big values homology. of degeneracy the These of filtration cycles vacua. parameter have different sizes, for SL(2 which lie in the same SL(2 its persistent homologybarcodes via are the shown lazy in witness figure complex The moduli space isequation the we upper use half plane tadpole constraint and only retain those values of complex 203362 simplices. D Figure 4 consists of 1064958 inequivalent vacua. We compute the persistent homology using the lazy witness JHEP03(2016)045 4)) = times , ) orbit. 4 m C (1 , 8 X ( 1 , 1 h denotes the 4) are m , 4 w time as the 1-cycles. This ], where the moduli space  (1 40 is a consequence of the data 8 –  X = 0 (4.17) 38 4 x , figure 6]. We have discussed this ) is invariant under the acton of the 3 x 36 2 4.17 x 1 x 0 it is possible that interesting physical features parametrizes complex structure deformations ψ x ) action is a symmetry which relates physically 8 ψ C – 21 – M , 4) (where the notation − , 2 0 4 x . Since the data set is limited, points at higher values (1 τ 4 P + 4 8 i x 4 =1 i X . The variable ). The relevant Hodge numbers of Z w × 4), defined as the hypersurface , 2 8 4 Z , which contains precisely a single representative for each SL(2 (1 8 F 4)) = 149. The defining equation ( , X 4 (1 8 X . This identification follows from the fact that a number of these connected com- ( 1 4 , will be more rare when the sample of vacua is generated at random, and therefore 2 h τ parametrizing complex structure deformations is one dimensional. We will consider All of these conclusions could easily be drawn just by looking directly at the plot of On the other hand the formation of long-lived bars in degree zero, which implies the The presence of a big degeneracy of vacua at the center of these holes shows up as a c in the weighted projectiverepetition space of the weight 1 and discrete group Γ = Now we turn to aM more sophisticated example, studiedthe in Calabi-Yau [ dataset in detail as anof example the of barcodes. how these Nowis features we not would turn available. show to up higher from dimensional the point perspective clouds4.3 for which a A simple Calabi-Yau plot hypersurface example of Im will appear as longnumber lived one). connected components (uncorrelated with any structure at Betti flux vacua on the upper half plane, see for example [ in our example. Most importantly we have now learnedexistence how of to more recognize connected such componentsset features. being at limited. larger values This of isto accumulate a at sign small that values the of flux Im vacua are not uniformly distributed but tend equivalent configurations. To avoid overcounting onedamental domain has to restrict theHowever attention nothing to guarantees a that fun- topologicalfundamental interesting domain, features which will are remain prominent sotifying in in these a another. features is One to way cross in correlate which the we barcodes’ can avoid plots misiden- at different degrees, at least perspective is typically not common inbars topological in data the analysis, barcode where in thecareful very degree in short-lived zero applying would such havemapping been techniques class interpreted to group as on string noise. the theoryare moduli We mapped vacua space must to since be short-lived very due homology classesimportant to in and the the worth fundamental stressing. action domain. of The This SL(2 the mechanism is connected component in the degreein zero barcode. figure Such bars areponents the disappear very from short-lived the classes barcode’sis plot roughly an at example the same ofbut also what at we the can correlations learn between not the just barcodes by in looking different at degrees. the Note barcodes that by such themselves a JHEP03(2016)045 ) ]. 40 , 4.20 ] is a ) and (4.21) (4.20) (4.19) (4.18) ˜ Z 39 Π. The and the 37 , ψ 4) , ) and ( 4 . (1 ) 8 4.19 C , can be identified X ( 3 e X H SL(2 , as well as worldsheet , section 5]. We define ). Note that contrary τ ψ ∈ 39 , i ! Im A τ, , but only its projection onto ) −→ − c d ]. a b i X

ψ Re 38 [ τ h , x, ˜ − X . and ··· i Im 4) are not invariant under Γ. If we f and we shall use the result of [ 0 , taking values in ) have the form , ( + x 4 x, x ! 0 f 0 4 3 c (1 3 =1 c = 972 i 3 8 4.10 X 3 F H h f X provides a new way to look at the full set of ) −→ − and 2 flux + + = (Re X 0 ! 0 0 N h x x – 22 – τ h b b i ) and ( 2 2 + c d a b − f π h 3 ) symmetry acting on the axion-dilaton and on the 2 2 4.9

+ D + Z f ( , ) is now 0 0 N 1 a a d −→ 1 1 f = 1. It will be useful to introduce an auxiliary variable h 4.12 − ! 3 3 = = con F H τ x ψ

Γ. The complex structure moduli space of ]. They will be neglected in the following. To summarize the log 38 X/ , b d , where the three special points correspond to a Landau-Ginzburg point, ) which as a first approximation is not a function of the remaining moduli: , which according to the Greene-Plesser orbifold construction [ 2 + + where each point represents a vacuum, a solution given by ( e . X , z ∞} τ X 1 , c τ a τ 1 , z , which measures the distance from the conifold locus. In this region the period 2 0 G ψ −→ , \{ 1 1 − τ G P To be more concrete, we will also consider a particular region of the moduli space, We are interested in an orientifold of this model which breaks supersymmetry down to In general complex structure moduli of = 1. The orientifold in question acts as = 1 ˜ physical values of thehave moduli however without to any deal projection. withflux the vectors. Before SL(2 Its analyzing action the is barcodes, we in terms of a seta of point known cloud constants, which theand reader can parametrized find by in vectors [ to of the the previous form case, wespecial cannot planes. plot The the persistent whole homology point of cloud nearby the conifold point x vector can be explicitly computedExplicitly as the a solutions function of of equations ( Complex structure moduli whichperpotential transform only at non-trivially higher underthe order Γ as fixed can Γ-invariant point monomials appear andonly of in relevant will Γ moduli the be for [ stabilized su- theaxion-dilaton at model zero, at hand are the complex structure modulus parity reversal. Wecompatible turn with on the flux Γ action, vectors tadpole along cancellation the condition 3-cycles ( associated with the period vector sponding Picard-Fuchs equations simplify greatly.Π One = is ( left with athose reduced can period only vector appearThis as period higher vector order coincides monomials with which the are period invariant under vector of theN Γ action. crepant resolution of with a conifold point and the large complex structure pointrestrict respectively. our attention to the periods which are invariant under the Γ action, the corre- of the mirror JHEP03(2016)045 . Then we algorithm, 4 parametrizes X consisting of 200 mathematica ) for a randomly L 4.19 min-max also shows three bars ) using 5 4 , h 3 ). In the persistent homology τ , h 2 4). The point cloud Im , , h 4 1 τ, (1 h 8 Re . The only non trivial barcode is for ) with a landmark set X = ( x, ,  5 L h , Im X ( x, LW – 23 – ) and 4 ). Isn’t this a contradiction with the clustering of until we reach the fundamental domain. = (Re  , f 3 1 τ x , f 2 , f algorithm. The computation runs over 1099256 simplices. 1 to its fundamental domain (and transform the flux vectors −→ − f τ τ 1. = ( ). To fix the conifold monodromy we only consider points such f min-max ) and x < 4.20 τ ), and because of the validity range of the approximated periods, we in the fundamental domain and the value of the transformed fluxes to τ π, π from ( − [ x ∈ Round(Re ) and the barcodes are shown in figure x . Barcodes for the distribution of flux vacua on − ,  as the fact that generic bars are very short-lived. Figure L τ 5 , We have computed the homology of the point cloud using the lazy witness complex X In practice, to reduce the computation time we put a cutoff to the number of iterations, and simply ( 4 −→ the features of the approximationlazy scheme witness we are scheme using wewhich for have selects our chosen points computation. the spread Indeed landmark apart in selector as the much with as the possible.discard Therefore the points point which if are it has outside not reached the fundamental domain by the time the cutoff is reached. of flux vacua to clusterfigure around the conifold singularity.which Indeed are this clustering rather shows long-lived,overall up with connected in respect component at tovacua? large the The others answer (a is no, fourth and bar the corresponds reason to is that the we must also be careful in understanding that arg only keep points with LW Betti number zero, higherthis persistent case homology the dataset groups does showing not scant present any or appreciable no “voids”. structure. This is In due to the tendency Then we map eachaccordingly). value of To doτ so we useuse the a value standard of algorithm,compute log by iterating the transformations computation we use thepoints lazy selected witness using complex the To deal with this redundancy we proceedgenerated as set follows. First of we fluxes compute ( Figure 5 14585 vacua with vectors of the form JHEP03(2016)045 heterotic ]. 8 41 E is constructed X × 8 (or more generically ˜ V and V and a number of NS5 branes . Since the selector is random, L 8 4). The point cloud , 4 is then used to construct a standard (1 8 V X = 1 vacua. The data required for the consists of 200 points randomly selected. The L N – 24 – . The bundle heterotic X 8 in E C . × ; in particular we note the different range of the proximity 8 5 6 . The heterotic string has very appealing features in the con- X are two holomorphic vector bundles X ), but this time the landmark set ,  L , respect to figure X . Also in this case the persistent homology computation is done using the lazy witness (  5 . Barcodes for the distribution of flux vacua on LW Let us quickly review E This example shows which kind of information we can get from the persistent homology To clarify this point further, let us repeat the persistent homology computation, this struction of phenomenological models of physics beyond the standardcompactification model on [ coherent sheaves) with structurewrapping a group holomorphic a curve subgroup of E 5 A first lookFinally at we heterotic conclude with models ainteresting rather heterotic preliminary glance vacua. at astring We collection over will of a consider phenomenologically Calabi-Yau compactifications of the E this is what weparameter see in figure of a point cloud ofwith vacua, but care. also how different approximation schemes have to be handled since this is relatively empty the bars are longertime than using the a average. randomfrom selector the to clustering choose hypothesis the weclustering expect landmark that region, set almost and all therefore the landmark the points barcodes’ will plot be will in the show almost no structure. Indeed complex computation requires 852691 simplices. the clustering region will bewe privileged see with these extra respect bars. to points These inside, bars and correspond this to is points outside the the reason clustering region, and Figure 6 as in figure JHEP03(2016)045 ˆ Γ, X X/ . The . The to the -stable ˆ . Then V = X µ we have V V G ˆ X V for on . 8 deg rank . The choice ˆ V H ˆ X rank with the SU(3) group in E ) = < V H H ( µ gut J has to be non-simply X rank hidden sector. By turning 8 < to the standard model group gauge symmetry of the visible G 8 with 0 and then working equivariantly with , the commutant of X G -polystable bundle, a direct sum of to be trivial and the number of NS5 branes µ corresponds to a solution of the Donaldson- J ⊂ V ˜ ], a large class of realistic models were con- V V . More general non-standard embeddings are 21 – 25 – – Γ corresponds to a 1-cycle. Wilson line operators 19 ∈ gut γ 6 corresponds to the hidden sector. The low energy ef- has to be a ˜ V V , for example SU(5) or SO(10), the commutant of the G . A similar setup holds for the E group will be SU(5), up to abelian factors. The latter are typ- 8 gut group in E giving an E obeys a stability condition. To every holomorphic bundle or sheaf V X V = 1 supergravity coupled to a number of gauge and matter multiplets. of gut . Typically one considers to have a rank five special unitary structure group, so that the first Chern . Physically the Wilson line operators break the X theory is then based on a group V N H . Solving this condition determines a viable structure group -stable if for any sub-bundle V . The choice of this bundle breaks the E ) = Γ. Each element µ equivariant with respect to the Γ-action descends to a bundle X ˆ X X ]. ( and is ). A poly-stable bundle is a direct sum of stable bundles, all of them with the V gut and 1 V 42 on π and V C ) vanishes, the ( , X V V V ( . A poly-stable holomorphic bundle ˜ V 1 < µ µ , c If we choose In an impressive series of works [ The compactifications that we will consider rely on the choice of a poly-stable vector The simplest choice is the standard embedding, which identifies ) can be defined along these cycles in terms of a flat rank one bundle on V we associate its slope as the ratio between its degree and its rank J γ ( database [ class ically Green-Schwarz anomalous and decouple atergy high supersymmetry energies. implies that The requirement of low en- is based on said group and can be chosen to bestructed a along phenomenologically these lines, realistic containing model. of the the standard MSSM model and gauge nomany group, low exotics. the energy matter With properties content an of extensive such use models of were derived computational explicitly algebraic and geometry are available in the W of a bundle physical spectrum and thedata relevant low of energy quantities cansubgroup then commuting be with all computed the from Wilson the lines. The low energy effective field theory on connected, which is inintersections general in not weighted theconsidering projective case a spaces. for discrete Calabi-Yaus grouprespect constructed to Γ One Γ. from freely can In complete actingfor practice which easily on this remedy is accomplished this by working problem with by the Calabi-Yau bundle sector down tostructure a group on appropriate Wilson lines itplus is a possible number to of further U(1) break factors. To have non-trivial Wilson lines a bundle µ same Uhlenbeck-Yau equations and therefore preserves supersymmetry. low energy holonomy group of possible as long as V fective action is Consistency with the low energying Bianchi identity requires a cohomologicalto condition be relat- an adjustablebundle parameter so that the only non-trivial condition involves the visible model-like effective action while JHEP03(2016)045 , X X (5.1) of the V and computed X contains the same ∈ ] we can form more Y x 42 ∈ : it cannot be plotted y 7 R -cycles. This could imply n which are CICYs with level, or the number of physical X . Each point gut Y . This simple choice certainly does not ,
a ) ] which contain extensive geometrical and 5 is embedded in L at the L 21 ( X – 5 =1 2 ¯ 5 a M . The barcodes show plenty of structure, even 19 – 26 – -cycle. Of course to establish this decisively and 7 = n and V , . . . , c 5 ) 1 L ( 2 , c ) X -stability parameter. A clever choice is a sum of line bundles ( µ 1 , 2 . It is quite remarkable that such a simple choice still allows for a ] revealed a large class of viable models. , h ) 21 represents many physically distinct models. Such a problem can be in the low energy spectrum. We take these two parameters as inputs – X ( of vector-like pairs 1 19 X H , 1 to create an augmented point cloud 5 n h n X is a concrete, qualitative example of one of the main ideas of this note: that = 7 ) = 0 for all x ), despite other possibilities being allowed. We want to construct a point cloud a L ( ). We show the barcodes in figure 5.1 -persistence modules given by the homology of the filtered Vietoris-Rips complex µ X Let us try to explore this possibility more in detail and compare directly two distribu- On the other hand the topological analysis is very efficient in comparing different Figure On the other hand such a point cloud We want to have a preliminary look at this database from the perspective of persistent N (  tions of vacua by fixingrefined certain point parameters. clouds Out with ofis more the information. the database number For of example [ Higgs an doublets information readily available together with distributions of vacua. As weof have physically explained realistic our models working arethis assumption not idea, is random one that but could distributions havewhich start topological exhibit comparing features. more different In structure. distributions thesingled of Distribution light out. vacua with of and higher look complexity for are, the in a ones certain sense, In the distribution atfor hand example that we there seealgebraic exists the equations subtle which appearance correlations describes of between the quantitatively many one the has vacua, to in go the beyond form the of topological some analysis presented here. the VR in degree four, if very short-lived. distributions of physically interesting vacua show a certain degree of topological complexity. but we will leave such a detailed analysis forin the any future. simple waydistribution and of there vacua. ishave no We considered readily around propose available a that tool hundred persistent of to homology models gauge with provides the distinct such properties values a of of tool. this We the form do proper justice to thephysical database information built on in the [ since models. a point In in particularresolved by such adding a more parametrization and is more very parameters coarse in the construction of the point cloud related) done in [ homology. In particular weform will ( only consider varieties out of this dataset, and we will do so in the simplest possible way, by collecting vectors of with physically realistic models, and indeed a scan over such possibilities (and other closely bundles all with the same JHEP03(2016)045 ) ) 1 1 X Y ( ) of Y = 1 i  (  L 5 ( VR n 2 consists c VR X as follows. ] for models 3 Y 42 and contain geometrical 1 Y 3 Y = 1 is topologically much and 5 = 1, but they differ in 1 n Y H , but is also from the number of n 9 spectrum. The point cloud . However this time, instead of just -persistence modules in the barcodes
N H and gut , n 8 5 , n ) 5 L – 27 – ( 2 specifies the , we select two subsets of data a ) of the Calabi-Yau and the second Chern classes Y L X ( , . . . , c 1 ) =1 , 1 2 5 a = 3 vacua. The latter indeed shows clear regularities in h L 5 ( L 2 n ), In other words the datasets of = , c X ) ( V 1 5 . , X 1 ( 9 h 1 , 2 , h and ) . The homological computation is done using the Vietoris-Rips complex 7 8 X R ( 1 , 1 ) and collect the information about their h 3 Y . Barcodes for the distribution of heterotic vacua. The datum of a vacuum consists in ( = 3 respectively.  = 5 y n VR Although comparisons have to be made carefully since the two datasets have different These precise values have been chosen just for convenience. 5 the barcodes. This issimplices clear generated by during comparing theby figures persistent a homology factor computation, of which 50. is greater for shown in figures number of entries, one featurewith is a immediately clear: single Higgs amongmore the doublet, database complex the of than distribution [ that of of vacua with and parameters of a Calabi-Yau andwith an holomorphic fixed bundle characteristics. which We give wanttheir rise to to persistent compare different homology. their spectra distributionsand Again from we the perspective study of the filtered Vietoris-Rips complexes geometrical information as above plusform this extra information onstudying the the spectrum topological and features has of All the the points in both subsets have a single Higgs doublet Figure 7 the two Hodge numbers five line bundles, suchof that 107 points in and runs over 1786924 simplices. JHEP03(2016)045 , the only 8 which parametrize an as in figure 9 9 R R = 3. The persistent homology 5 n = 1. The computation of the persistent modules consisting of 65 points in – 28 – consisting of 34 points in 5 1 n 3 Y Y , with two additional parameters, the number of Higgs 7 ) involves 574095 simplices. 2 = 1 and of vector-like pairs Z H ); . Barcodes for the point cloud 1 . Barcodes for the point cloud n Y (  VR ( i Figure 9 difference being the valuecomputations of runs vector over pairs 10325 which simplices. is now fixed at Figure 8 heterotic compactification asdoublets in figure H JHEP03(2016)045 we have 3 = 1 would be preferred over 5 n – 29 – = 3. Again, this lends (modest) support to the idea that string 5 n = 2 supersymmetry, arising from compactifications of the type II N we have applied our formalism to certain classes of flux vacua in type 4 In section To exemplify these ideas we have studied three classes of vacua. In section If we assume that topologically rich structures correspond to physically interesting examples of the statistical approachin to the counting landscape the of flux numberhomology vacua, has of where the one vacua advantage is with ofdistribution interested allowing of certain vacua, a even properties. simple when visualization suchhow a of In distribution the the has features this high basic of dimension. framework properties such We of persistent have distributions discussed a can be extracted from their barcodes in two simple that, regardless of the interpretationbutions as could string vacua, also the be studythe seen of space Landau-Ginzburg as of distri- a two dimensional veryunderstand superconformal preliminary if field persistent attempt theories. homology at could It be the would an study be useful of very tool interesting the for to thisII topology B/F-theory direction of of compactifications. investigation. The models we have discussed are typically presented as certain Calabi-Yau manifolds, oring superconformal mathematical fixed problems points. on Boththe their of process own, these we regardless have are of learnedfeatures interest- the of how such to physical distributions. use applications. We topologicalsuch have During as also data Calabi-Yau established analysis with that to small certain reproduce Hodge distributions numbers, the of have vacua, known non-trivial topological features. Note studied vacua with string on a Calabi-Yau, orilies from of Landau-Ginzburg such models. vacua we By havepersistent appropriately constructed homologies. the labeling corresponding fam- point We clouds have and computed looked their for topological structures in the distribution of completely characterized by their barcodes.lifespan Barcodes of are the a persistent graphical homologyat representation classes which of of we the the look distribution atwhich of the characterize vacua, distribution the as is point the varied.this length cloud They framework scale by correspond patters the are to appearance emergent homological with signatures of the patterns length within scale. the dataset. In In this paper wecharacterized have by explored certain the topologicaltopological possibility features properties. that of distributions a Awork distribution of readily persistent of string homology available points vacua extracts tool is canlection topological topological to of be data invariants vacua capture at analysis. we the every associate In length persistence this scale. modules frame- in To various a degree; col- the latter in turn are vacuum selection principle based ontopologically the complex topological distributions features of ofWe vacua believe the are this distributions preferred idea of over is vacua: topologically worth further simple investigations. ones. 6 Conclusions features, one would concludecompactifications that with compactifications with vacua can be characterized by theit topological properties opens of their up distributions.complexity the In of particular possibility their persistent that homology. vacua In with other words realistic we are features led can to propose be a singled qualitative out by the JHEP03(2016)045 pairs 5 ¯ − 5 ]. One can 36 ) and study its , 18 4.14 , 14 , 13 = 1 is topologically much richer and 5 n – 30 – we have considered quasi-realistic compactifications of the heterotic 5 spectrum. The class of vacua with gut A particular interesting point in this respect is that, also due to the approximation Let us summarize the main ideas we have encountered in this note. First of all we Finally in section In this note we have obtained distributions of flux vacua by solving directly the equa- -cycles. The presence of such features suggests that distributions of phenomenologically systematically families of M- or F-theory compactifications. an example of very short-livedregarded bars in which correlation reveal with physicallyanalysis interesting barcodes is features in physically only relevant other is when at degrees.needed the to moment To understand not which clear; how extent much certainly informationit. topological more and work data One and how ideas could many are new hope insights to can learn we gain something from by analyzing much bigger datasets, or by studying schemes available, studying the persistent topology ofeasier a distribution than of vacua studying is considerably features its geometrical one properties. typicallycomputationally needs In less only involved. other a wordspersistent smaller On homology to need the number sample to other of be the hand partly vacua relevant revisited the and to usual be the applied ideas whole in used this process context: in is we the have study seen of from the more persistent homologytechniques classes could in be the efficiently barcodes;and applied others algebraic in cannot. geometry conjunction Hopefully typically withto such the used call usual in the tools attention studyingdiversified of from distributions problem the statistics of posed computational vacua. by topology the community We landscape to also of the string hope extremely vacua. rich and have shown how techniques ofcharacterizing topological distributions data of analysis stringqualitative can vacua. information be from applied Persistent to homology thedepends the can set on problem of be the of questions string used one vacua. to is asking. extract The Certain usefulness features of of string such vacua information can be understood level of persistent homology.the This persistent possibility topological can featuresexample be of by made comparing different rather two classes concrete differentin of set by the vacua. of comparing vacua We which have differtherefore given in from the a our number perspective specific of should be preferred. exhibits non-trivial topological features,n in the form ofviable higher vacua dimensional, might if befrom short-lived, characterized their by persistent a homologyphysically high interesting modules. features degree are of associated In with topological topologically other interesting complexity, words structures as at it the seen leads to the possibility that easily construct a point cloudproperties by using discretizing statistical an index topology.investigation. density Indeed such as we ( expect that this should bestring. a We rich have area considered of data a of class a of certain vacua holomorphic labeled bundle. by A a striking Calabi-Yau aspect and of the such topological a class of vacua is that it can be used to determinetheir properties numbers of but the also distributions by of using flux vacua, the not invariants just oftions by persistent counting for topology. supersymmetric vacua.continuum Of approximation course to another the possibility index would density be of to vacua use as directly in a [ cases. We have given concrete examples of how the techniques of topological data analysis JHEP03(2016)045 , javaplex . Surveys on Discrete and Discrete Comput. Geom. , , in (2009) 255 On the local behavior of spaces of -cycle. If they exist, such corre- n 46 Topological persistence and simplification . (2008) 1 – 31 – . , ed. 1.0, Createspace, U.S.A. (2014). 76 Computing persistent homology Bull. Amer. Math. Soc. , Persistent homology — a survey , Amer. Math. Soc., U.S.A. (2008), pg. 257. ), which permits any use, distribution and reproduction in (2002) 511 453 28 , CC-BY 4.0 Int. J. Comput. Vis. , This article is distributed under the terms of the Creative Commons . Topology and data ]. Similar thanks are in order to Andre Lukas and Harald Skarke for main- Elementary applied topology 22 . Here the concept of topological complexity is concretely provided by persistent (2005) 249 Computational Geometry natural images Discrete Comput. Geom. 33 On a more speculative level the results of this note have led us to the idea that sets of H. Edelsbrunner and J. Harer, R. Ghrist, G. Carlsson, T. Ishkhanov, V. de Silva and A. Zomorodian, H. Edelsbrunner, D. Letscher and A. Zomorodian, A. Zomorodian and G. Carlsson, G. Carlsson, [4] [5] [6] [1] [2] [3] Attribution License ( any medium, provided the original author(s) and source are credited. References was partially supported byEXCL/MAT-GEO/0222/2012 and FCT/Portugal the and program Investigador IST-IDIF/01426/2014/CP1214/CT0001. FCT through IF2014, I under UID/MAT/04459/2013, contract alsothe thank preparation IHES of this for paper. hospitality andOpen support during Access. Acknowledgments I wish to thank theavailable in Applied [ Topology group attaining Stanford the University for databases making used in this note and keeping them available for use. This work to the possibility of formulatingtopologically complex a distributions qualitative of topological vacua vacuumple selection are ones physically principle: preferred that overhomology. topologically sim- Clearly much more work is required to turn this into a quantitative statement. analysis is needed to establishset this of concretely. equations These which force correlations the canlations vacua would assume distinguish to the a lie form family over of of a vacuaeral certain a from understanding others. of Even the if physical atfrom this significance a stage of superpotential, we such have where no correlations in gen- (exceptpersistent principle when homology it they should provides arise be a possible computational to framework derive to them visualize analytically), them. This leads us vacua with a higher degree of topologicalically complexity simpler are ones. singled out It withfeatures respect is within to tempting topolog- a to distribution suggest of vacua thatwe is have the already associated presence remarked, with the of physically presence interesting topologicallylations of features. higher interesting between dimensional the As cycles vacua can distributed imply along subtle the corre- cycle, although a more refined case by case JHEP03(2016)045 , ]. ] ]. 423 JHEP ]. , SPIRE IN SPIRE Lect. Notes ][ IN ]. SPIRE [ IN . ][ Phys. Rept. , SPIRE ]. Proceedings of the (2007) 733 hep-th/0307049 Proc. Nat. Acad. IN [ , ][ 79 Proc. Nat. Acad. Sci. , in . , ]. SPIRE (2003) 046 IN [ . , A comprehensive scan for arXiv:1106.4804 05 . [ SPIRE arXiv:0803.1194 (2004) 060 . jHoles: a tool for understanding , IN Complete intersection Calabi-Yau math-ph/0506015 01 ][ [ kreuzer/CY/ JHEP , , ∼ Rev. Mod. Phys. (2014) 5 arXiv:1307.4787 , [ ]. 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