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Title Classification of symmetric toroidal orbifolds

Permalink https://escholarship.org/uc/item/6b2575rq

Journal Journal of High Energy Physics, 2013(1)

ISSN 1126-6708

Authors Fischer, M Ratz, M Torrado, J et al.

Publication Date 2013

DOI 10.1007/JHEP01(2013)084

License https://creativecommons.org/licenses/by/4.0/ 4.0

Peer reviewed

eScholarship.org Powered by the California Digital Library University of California JHEP01(2013)084 b = 1, i.e. Springer N etc. We also 4 January 11, 2013 metric toroidal A December 12, 2012 : September 27, 2012 , : : 4 D , 3 S isms (local or non-local) 10.1007/JHEP01(2013)084 Published Accepted Received doi: Abelian point groups such as , n point groups and s in six dimensions. We find in and Patrick K.S. Vaudrevange a Published for SISSA by Jes´usTorrado [email protected] a , 1 supersymmetry in 4D for the heterotic string. Our strategy [email protected] , Michael Ratz, N ≥ a 1209.3906 Superstrings and Heterotic Strings, Superstring Vacua We provide a complete classification of six-dimensional sym -I etc. and 358 with non-Abelian point groups such as [email protected] 6 Z , 4 Z , Deutsches Elektronen — Synchrotron DESY, Notkestraße 85, 22607 Hamburg, Germany E-mail: Physik Department T30, Technische M¨unchen, Universit¨at James Franck Straße, 85748 Garching, Germany [email protected] 3 b a ArXiv ePrint: specify the Hodge numbersof and gauge comment symmetry on breaking. the possible mechan Keywords: briefly explore the properties of some orbifolds with Abelia Classification of symmetric toroidal orbifolds Open Access is based on a classification of crystallographic space group Abstract: Z total 520 inequivalent toroidal orbifolds, 162 of them with Maximilian Fischer, orbifolds which yield JHEP01(2013)084 3 3 4 4 5 6 7 8 8 2 2 6 12 15 15 17 18 20 24 25 27 34 44 11 15 22 24 31 34 11 – 1 – point group 2 Z G S P -classes of space groups -classes of space groups 5.1.1 Our5.1.2 results Previous5.1.3 classifications Fundamental groups Z É C.1 Abelian pointC.2 groups Non-Abelian point groups A.2 Introducing anA.3 additional shift Bravais types and Lie lattices 5.2 Non-Abelian toroidal orbifolds A.1 Bravais types and form spaces 5.1 Abelian toroidal orbifolds 4.1 Classification strategy 4.2 Residual SUSY 3.1 Affine classes3.2 of space groups 3.3 3.4 Some examples 2.1 The space2.2 group The lattice2.3 Λ The point2.4 group Examples:2.5 space groups with The orbifolding group B Two-dimensional orbifolds C Tables 6 Summary and discussion A Details on lattices 5 Results: classification of toroidal orbifolds 4 Classification of space groups 3 Equivalences of space groups Contents 1 Introduction 2 Construction of toroidal orbifolds JHEP01(2013)084 , 4 ]. 1 we 12 , the – (2.1) C 9 G , which ect more 2.2 ]. There are two 2 . Note that , G 1 symmetries in terms of contains some details on and non-Abelian heterotic R ance of matter as complete A up topographies as well as l features of the respective compactifications have been lassification is due to Donagi e (supersymmetric) standard au manifolds one has a clear bifolds [ we discuss the tools used to is devoted to a survey of the art from the Euclidean space ovide us with a large number lts from the literature [ ix 2 attention in the past few years. on Abelian toroidal orbifolds [ 5 been fully explored up to now. pactifications. Then, in section nough to produce a large number ]). In fact, the geometric properties /G . 8 , is in general not equal to the point n – 6 Ì 2.5 orbifolds. The main purpose of this paper , we present a way from crystallography to 2 = ) some of these classifications are mutually 3 Z × /S – 2 – 5.1.2 2 n Z Ê , the so-called space group. (ii) Alternatively, one S = ]). At the same time, symmetric orbifolds have a rather 5 Ç = 1 SUSY in 4D. Section . That is, a toroidal orbifold is defined as N 2.3 1 supersymmetry (SUSY) in four dimensions. ], who focus on we survey the already known 2D orbifolds, and in appendix and divide out some discrete symmetry group 10 N ≥ B n we briefly discuss our results. In various appendices we coll -dimensional lattice Λ, to be defined in detail in section Ì n 6 ]. As we shall see (in section ] (for a review see e.g. [ 4 12 , – 3 9 The structure of this paper is as follows: in section Despite their simplicity, symmetric toroidal orbifolds pr and divide out a discrete group n ]. Unlike in the supergravity compactifications on Calabi-Y so-called orbifolding group as defined in section is to provide a complete classification of symmetric Abelian orbifolds that lead to can start with an string theory description. In addition,of the candidate scheme models is that rich e maymodel yield [ a stringy completion of th of different settings, whichIn have, the rather past, surprisingly, differentmade not [ attempts of classifying parts of these resulting orbifolds, and to a comparison with previous resu lattices, in appendix provide tables of our results. 2 Construction of toroidal orbifolds We start our discussionequivalent with ways the of construction constructing of such toroidal objects: or (i) one can st determines a torus often have immediate consequencesmodels. for the One phenomenologica obtains an intuitive understanding of discrete construct toroidal orbifolds. Later,classify in all section possible space groups and apply it to string com Finally, in section group introduced in section straightforward geometric interpretation (cf. e.g. [ not consistent, and incomplete. Theand perhaps Wendland (DW) most [ complete c Ê we impose the condition of remnants of the Lorentz groupGUT of multiplets compact space, due of toflavor the structures. localization appear properties and gauge gro 2 1 Introduction Heterotic string model building hasThe perhaps received simplest an heterotic increasing compactifications are based detailed information on our classification program. Append JHEP01(2013)084 f a a ]. ϑ ϑ B ◦ 13 (2.4) (2.2) (2.3) λ (with is the leaves and i = is given S e S e g g B the vector ◦ Λ are called S h , i.e. ∈ λ ∈ , summing over i arbitrary. In the λ / ) e i n n ϑ, λ and define with = ), closely following [ } n S = ( span the same lattice by λ 2.1 f g ∈ )) as roup of the space group. Z ) are objects whose symmetries , i.e. every element of ion ( , . . . , f by some vector , respectively. Then the change n n, and 1 f f ϑ, λ Ê e { GL( = = ( ) is also common, since the normal subgroup and = 6 we will keep ∈ f e linear independent translations, every g . n , λ . λ, ϑ f consisting of all translations in linearly independent translations, then it M ) n B + and = ( n S g } ϑ, λ = ϑ v (i.e. n Λ can be written as – 3 – and the full lattice is spanned by the M = ( can be written as a composition of a mapping M g } and checking whether or not it is an element of e ∈ 7−→ S g f contains , . . . , e B λ 1 v ,...,n B e S 1 1 { ). Such groups appear already in crystallography: they − ∈{ e n i = } B i as e e as a discrete rotation or inversion). This suggests to write n { = ϑ Ê = S ∈ . Clearly, the choice of basis is not unique. For example, for e M v Z 1 of a space group ∈ be another space group element. Then the composition g i n S ). ∈ τ ) and + (one can think of ω, τ ). S be a space group. The subgroup Λ of Z be a discrete subgroup of the group of motions in = ( , . . . , n ∈ S Every element Since a space group is required to contain h n, S ω ϑ, ω λ g In the mathematical literature the reverse notation 1 does not need to be a lattice vector. Elements = 1 of basis is given by a unimodular matrix as matrices whose columns are the basis vectors in lattice of the space group. Note that for a general element λ given lattice Λ take two bases 2.1 The space group Let computing the matrix that leaves (at least) one point invariant and a translation Even though we are mostly interested in the case the metric of the space invariant. If On the other hand, one can decide whether two bases 2.2 The lattice Λ Let GL( is called a space group (of degree by ( comprise discrete translations. are the symmetry groups of crystal structures, which in turn for element is usually written to the left, and the lattice Λ is a normal subg space group element as following, we will properly define the concepts behind equat and it acts on a vector roto-translations. Let i integer coefficients), i.e. an element lattice contains a basis JHEP01(2013)084 . ) } } S of S n Z A.3 G (2.9) (2.7) (2.5) (2.8) (2.6) ⊂ ∈ point i 2 P , . . . , e , n Z 1 2 e e { 2 n = + e 1 we will see in an e ) (i.e. unimodular) 1 forms a finite group n Z pace group and thus ϑ { ormal subgroup of the n, mensions with of all GL( lattice Λ (see appendix . p can also contain inversions P ts point group with its lattice. 2 , ections. is a subgroup of it, i.e. , is generated as i /P . P in the lattice basis = translation), the space group S 0) n e = 1 1 Ì ϑ ϑ, − i ( , ), the set , to itself. Hence, similarly to the change M. ) e 2 Λ , i.e. it acts on the lattice basis vectors as S ϑ ϑ, λ π is not equal to the orbifolding group , e ⋉ 1 Λ) = ½ − . Furthermore, under a change of basis as in for ( point group P P e ⋉ , . The elements of a point group are sometimes ) M – 4 – 2 (rotation) 1 S P = ϑ B Z ( The first of our examples is the well known two- ◦ )) representation of the twist is denoted without an , e i / = 1 ) basis, we append the twists by an index indicating n e S ½ n ) is denoted as − e ( f Z n − h Ê B ϑ n, O( = (a). The space group = = = 1 e ∈ i . In detail, the lattice is given as Λ = is a rotation by ϑ S ) (or SO( } ϑ Ç maps the lattice of n ϑ translation ϑ e ). , ϑ . ◦ S n ½ and } P { 2 j of O( e , e = . 1 ∈ P ji e with elements of the form ( P ) ϑ { 2.4 e S ϑ = e = ( ) because of the possible presence of roto-translations, as ) the twist transforms according to i 2.1 ϑ e 2.4 The point group In general, however, the point group is not a subgroup of the s Given these definitions, and because the lattice is always a n , p. 15]), the so-called point group of 13 of lattice bases, point group elements can be represented by called twists or rotations. However, in general a point grou and reflections, i.e. group in order to illustrate the discussionA of simple the previous example: s the “pillow”. dimensional “pillow”, see figure and the point group using the basis 2.4 Examples: space groups with example in section the lattice basis, while the O( matrices. When written in the GL( space group (i.e. rotation In this section, we give two examples of space groups in two di 2.3 The point group For a space group index. For example, the twist ([ equation ( and can be realized as the semi-direct product of the oblique such that has a semi-direct product structure iff the point group In that case the space group is notMore necessarily precisely, a in semi-direct general product the of i point group and one can write the orbifold as equation ( JHEP01(2013)084 , , 2 } 2 Ê . , e 1 (2.10) (2.12) (2.11) ! e { 1 − 1 0 0

also contains a = g e ϑ le. In case (a) the blue generates a finite group . ct product of a lattice and g 1 -lattice translational parts. re are no roto-translations tions. fore, we will need to define e ), 2 1 and ½ rectangular lattice Λ) + ar that in general space groups = xed points. 2 . In other words, the orbifolding e 2 2 (b) P  . ϑ v 1 ]). Loosely speaking, the orbifolding e ! − 2 1 10 Λ, but not on the Euclidean space 1 1 (i.e. / e ϑ, − 2 1 2 (b). Here, the space group is generated by  1 0 v ) matrix Ê 1 Z 0 that have a non-trivial twist part, identifying − Z = , Let us take a look at a more advanced example: = S – 5 – =

g 1 e 2 = 2 1 and the torus lattice Λ generates the space group Ì e . + ϑ G as 2 2 Z G e ϑ v with 2 v 0). In other words, since the generator g , , + g 7−→ ½ 1 i is equal to the point group e v 1 , g v 6= ( ) G 2 ) = 1 , e v only on the torus ½ Λ, it is not a point group element but a roto-translation. , e ( 2 , ½ / (a) ∈ ) Z 1 1 . e , the point group is , e = ( 1 2 ½ }i ½ ( 2 Λ h . Two-dimensional examples: (a) “pillow” and (b) Klein bott is generated by those elements of = g 2 G, G = ϑ h{ Obviously, this space group cannot be written as a semi-dire S = acts on a vector and an additional element arrows indicate a wrap-around and the red symbols indicate fi Therefore, it can be represented by a GL(2 isomorphic to Since translation group may contain spaceCombining group elements the with orbifolding non-trivial, group non S because group elements which differthe by orbifolding a group lattice translation. Hence, if the two orthogonal lattice vectors (which thus span a primitive Another example: the Kleinthe bottle. space group of a Klein bottle, see figure Notice that even though the point group is g Figure 1 a point group, as is always the case2.5 when we have The roto-transla orbifoldingDue group to the possiblecannot presence be of described roto-translations, byan it lattices is additional and cle object, point the groups only. orbifolding group There (see [ JHEP01(2013)084 ) if 2 (3.1) , that S (2.13) A ∼ 1 and hence equivalent S S ical moduli; /G . ]. n affine classes is advant- 13 breaking). Ì features are common to and hence the nature of contained in ical properties. Using the -classes S = (a)). P ysical properties of a given 1 and a linear mapping il in [ tries and space groups which É t lent choices for the orbifolding /G is is standard knowledge among n detail, there are three kinds of -class can contain several affine Λ) Z definition enables us to distinguish ⊂ / n . Ê 2 with certain physical and corresponding S S = ( = }i f -classes Λ – 6 – (b)), but in general, they come with singularities 1 belong to the same affine class (i.e. such that Z 1 S n G, n 1 − h{ Ê -classes and each f / ) determines the flavor group and the nature of gauge ⊂ consists of a translation n Z → n Ê 3.1 ) determines the lattice Λ of n ) determines the point group Ê Ê = 3.2 of degree 3.3 : . Sketch of the classification of space groups. 2 ) on f S /S n A, t Ê and = ( 1 = Figure 2 S . In other words, for every point group there can be several in f 2 affine classes Ç -class (see section -class (see section Z É -class can contain several É the geometrical moduli; the number of supersymmetries in 4D and the number of geometr symmetry breaking (i.e. local vs. non-local gauge symmetry In the following, we will discuss in detail why the concept of Hence, we can define the orbifold as 1. the 3. the affine class (see section 2. the ageous to classify physically inequivalentcrystallographers space and groups. can Th for instance be found in more deta there is an affine mapping mathematical properties. These classes are: Orbifolds can be manifolds (see e.g. figure An affine mapping classes, see figure lattices and for everygroup lattice (i.e. there with can or without be roto-translations). several inequiva 3.1 Affine classesTwo of space space groups groups 3 Equivalences of spaceIn groups the contextmodel of directly string depend orbifold onwhole compactifications, sets the some of choice space ph of groupslatter, and its one can can space be define group. related equivalenceequivalence classes to classes of some These suitable space mathemat to groups. sort I space groups which can not be endowed with smooth maps (see e.g. figure is, it allows for rescalingsbetween and space rotations. groups Therefore, that this actually describe different symme Each JHEP01(2013)084 ½ − (3.3) (3.2) (3.4) = (3.5b) ϑ ) (3.5a) i e i ose any linear n ( moduli space is . 1 − by choosing ! A S ) example with ) α + 0) with α 2 s. This can be refined . 1 angle or distance. Then, Z 1 tan( ϑx ansformations amount to A, sin( / ties of their point groups. 1 ! 2 2 r n. Hence, we will only be ber of dimensions. In this trivial affine transformation r 2. Then ) . ) Ì = ( ) corresponds to a change of ifications this corresponds to , − assification of space groups. α α ) = P f ! 1 i 1 0 r e = 1 0 1 i sin( cos( i are two different orbifolds. This =

n 2

2 r and there is only one affine class of r 3 + for = Z e =

S / PA Z 1 2 2 ∼ with basis vectors 1 − = e e ϑAx Ì ∈ ( − e A S orbifold, where the angle between the basis S , there exists only a finite number of affine i 1 2 A n n e 3 − f Z and and – 7 – / 2 2 and e g x Z Ì and / ) with ) = , the elements of the point group can be written in ! i = 2 e 1 0 ! i i Ì 2.3 ) Ax ) ! e e (

Let us illustrate this at the . α . Therefore, α i 1
0 e ½ r S ϑ, n n g = as basis vectors. Define a space group − 1

sin( ∈ cos( + 2 − = ( 1 2 2 = ) e e e f , p. 10]. Hence, classifying all affine classes of space groups r i r g = ϑx e e ϑ i 1 13 0 r 1 = and e

) = ϑ, n 1 x e ( = . As discussed there, the lattice is oblique, i.e. one can cho
= ( e g 2.4 A g f 1 − and f and their length ratio are fixed, such that the corresponding 2 i Ê e -classes of space groups space groups with Z ∈ 2 It turns out that, for a given dimension This should be compared with the Z x / that leaves the point group invariant (i.e. 2 different. Hence, it is clear that This can be seen explicitly using the affine transformation further by grouping affineFollowing classes the according argument in to section common proper As discussed above, we can sort space groups into affine classe vectors This space group is in the same affine class as demonstrates the advantages of using affine classes for3.2 the cl enables a complete classification of orbifolds for a fixed num paper, we focus on the six-dimensional case. Example in two dimensions. independent vectors classes of space groups [ Ì given in section are just the onesfor we a already given know, representative looked spaceA upon group from of a an different affine class a non- the geometrical data. Ina the change context of of valuesmoving superstring of in compact the the geometrical moduli moduli.interested in space That one of representative is, for the affine every respective tr affine compactificatio class. Take an arbitrary element for JHEP01(2013)084 i ). Z (3.7) (3.6) n, GL( = 1 SUSY ⊂ N i ce basis (using -classes do not P equivalent. ). Then, the two É Z 2, the point group , contains a lattice Λ , i.e. n, i i -class share a common P S = 1 É GL( i same number of moduli. generators, it is sufficient -classes, . trical moduli. However, as Z ⊂ esponding point groups are )) such that (cf. the parallel B ic point groups. They also i ame ) such that Z efinitions with some easy ex- P ther words, space groups from É . For n, -classes classify the inequivalent 2 Z n, S o space groups belong to the same , i.e. -class are not necessarily equivalent i GL( -class they also belong to the same . , is denoted by Z P Z GL( ∈ and -class (or in other words to the same 2 i 2 ∈ S 1 P P É U -class. S 2, the space group V , É = = -class, they have the same form space and, . In contrast to (i.e. 2 Z = 1 V U U i – 8 – -class. Hence, 1 1 P P Z ). Therefore, a point group is a finite subgroup of 1 1 Z − − . For U V 2 n, -class (or in other words to the same arithmetic crystal S Z )) orbifolds, taken from appendix and ). This allows us to identify settings that yield 2 3.1 1 4 . S Z Z , take two space groups 3.2 ). That is, if the point groups are related by a change of latti 2.5 -class can belong to different affine classes and can hence be in Z -classes of space groups É Take two space groups If two space groups belong to the same -class, hence the inclusion sketch in figure -class, the commutation relations and the orders of the corr ), the space groups belong to the same 3.3 Obviously, if two space groups belong to the same holonomy group (cf. section amples of two-dimensional 3.4 Some examples Before going to six dimensions, let us illustrate the above d distinguish between inequivalent lattices. However, if tw the unimodular group on class) if there exists an unimodular matrix physically, they possess the same amount and nature of geome space groups belong to the same discussion around equation ( U we have stressed before, spacebecause groups of from the the same possiblethe presence same of roto-translations. In o see equation ( É É the lattice basis as elements of GL( As before in section and its point group in the lattice basis is denoted by in 4D. In particular, into order consider to only determine one the representative from number of every SUSY in the lattice basis associated to the space group Then, the two spacegeometric groups crystal belong class) if to there the exists same a matrix lattices. the same. Therefore,possess they form are spaces isomorphic ofWhat is as the important for crystallograph same physics dimension, is that i.e. all they space groups have in the the s JHEP01(2013)084 ) ), 2 / 3.7 1 (3.8) (3.9) ′ 1 − (3.12) (3.11) (3.10) , f 2 , and the / } ≡ 1 2 ′ 1 e , f , 1 = ( f , ≡ { ! 2 . However, it is 1 , , f 1 ! f = − f Next, consider the from equation ( ! ! ), ≡ s equal to the point 2 1 0 0 0 1 1 0 1 1 ′ 2 2 1 / –II–1–1, as defined in V 1 f e f 2 − − and

, –II–1–1 and the second 2 and second spanned by Z } 1 0 0 1 2 0 2 / 1 2 , } = = just defines the associated Z f

2 , e f ϑ e 1 , , U , e e = ( 1 = = -classes. { e 2–1 = 1–1 1 Z f e { = ϑ f ϑ ϑ ϑ e -class. Hence, as we actually knew U = e Z e ϑ 1 − 1) and U with , ) transformation with with with 2 ′ 2 Z e e , ). = (0 i i – 9 – i ) i ) with Consider the affine class ) ) 2 2.4 2 2 2 2 e , f , e , f , e 1 1 ! 1 1 ( ( 0), ( ( , , , , , ) ) -class, but different ) ) 1 1 1 0 1 1 1 1 . If we try to find the transformation -class. É , f , e

B , f , e Z 1 1 = (1 1 1 ( ( , a reflection at the horizontal axis. Now, let this reflection ( ( = , , 1 ϑ , , e 0) 0) 0) 0) , , U . The two corresponding space groups read ϑ, ϑ, 3 2–1 1–1 ( ( h h ϑ ϑ ( ( h h = = f e = = ). Therefore, they belong to the same S S because they are given in their corresponding lattice bases f 3.6 2–1 1–1 ϑ . As there are no roto-translations, the orbifolding group i S S , see figure } B 6= 2 –II–2–1, see appendix . Two different bases for the p-rectangular lattice: 2 e , f ϑ Z 1 f { = with lattices spanned by space groups, respectively. The first space group belongs to the affine class change of basis precisely as in equation ( from the start, they act on the same lattice and theSpace matrix groups in the same group and is generated by one to appendix easy to see that they are related by the GL(2 action of the point group generator (primed vectors). Space groups in the same Figure 3 where cf. equation ( f act on a lattice, first spanned by the basis vectors JHEP01(2013)084 ). M Z and , τ (3.15) (3.13) -class, but inequivalent is a reflection É ϑ ), respectively. τ 1 − e 1 has non-integer entries. . Performing this basis ) can be spanned by the 2 , 1 . / 4 of the original lattice is red, 1 − τ, e ! É re V τ 2 mple: one can amend one of = on as a “change of basis”, see 2 in detail in the following. / he lattice, we notice that this his gives rise to a new lattice, 1 / ) instead of one from GL(2 − y ∈ or 1 e rectangular lattice, while the − 1 É τ e = V ) and ( , 2 2 e to a centered rectangular lattice, 2 re gray. The action of / / 1 1 x There is an alternative way of seeing x, y to its negative and interchanges , e 1

2 ) (3.14) e e GL(2 2 e = ∈ –II–2–1 belong to the same + 2 2 1 e M M Z with e ( exists, either 2 1 to itself, – 10 – 1 1 ) with values − e = with ! V y τ 3.13 τ − –II–1–1 affine class and add the non-lattice translation B 2 -class. y x x we see that –II–1–1 and Z Z f 2 ,

= Z for which 2–1 = ϑ y M . e τ = V B − and V 1 e , x e are matrices whose columns are ( 1–1 ϑ τ and 1 B -classes. In other words, these space groups are defined with − τ ), but now generated by a matrix Z V 2.4 and . Change of a lattice by an additional translation: the basis e B . τ We can interpret the inclusion of this additional translati In our case, let us take the − 1 Therefore, the space groups to its space group. If we incorporate this translation into t element changes the original primitivewith rectangular a lattic fundamental cell of half area. The new lattice (see figu basis vectors The transformation looks like lattices. Indeed, the firstsecond one space has group a possesses centered a rectangular primitiv lattice,The as effect we of including will additional see the translations. relationship between the twothe space space groups groups of by theand an last consequently additional exa to translation. a In different general, t to different where But for all values of is precisely the matrix in equation ( Figure 4 the basis of the new one blue. The additional lattice points a equation ( at the horizontal axis. Therefore, it maps that fulfills e JHEP01(2013)084 6 ]. , 2 4 , 3 (3.17) (3.16) , –II–2–1. 2 ] in order = 2 Z –I–1–1 and wist of the 2 12 -classes (and bifolds. is Z , N É τ ! B for 1 = 4 in 4D. On the ] and [ -classes are related ations can be found − N Z ]. In detail, for the 10 N [ 1 0 0 Z = 1 SUSY. ). In these works, the 14

p to the point group [ ited by the 4D effective × N he second point group is a V oth cases: it is a reflection N 5.1.2 oup. For example, a trivial Z the two ally. Then, in a second step uivalent space groups for the re, adding such translations is = x, y . d hence with ∀ -class. Hence, if one considers all V ersymmetry if the point group is a h are the orthogonal sum of three É ! ) 1 − É , 1 0 0 − –II–1–1 belong to different 2

GL(2 Z Finally, consider the affine classes – 11 – / ∈ ⇔ ! ! 1 ), y x − ). That is the reason for the name geometrical crystal and the new space group with lattice -classes in the same 0 0 4 . If we try to find a transformation between both space 3.7 1 0 0 Z –I–1–1 and B

-classes. M 2

of residual SUSY in 4D is given by the number of covariantly Z É = N = V ] is not fully exhaustive, see section V = 1 SUSY in 4D for SU(3) holonomy. 12 ! N 1 -classes) additional translations do not give rise to new or -classes). That is, the point groups are inequivalent: the t − Z Z 1 0 . 0 − ) transformation -classes. A general method for including additional transl A.2

É É , 1 − V The method of using additional translations has been used in In the context of orbifolds, one can relate the holonomy grou –II–1–1 defined in appendix 2 other hand, one gets Therefore, the space groups also to different first point group isreflection a at reflection the at horizontal the axis. origin and the twist4 of t Classification of spaceIn groups this section wecompactification describe of the our heterotic strategy string to to four classify dimensions all ineq heterotic string the number Orbifold compactifications preserve four-dimensional sup by a GL(2 groups generators, see equation ( classes for to classify six-dimensional space groups withauthors point start groups with factorizedtwo-dimensional lattices, sublattices, i.e. on lattices whichadditional whic the translations are twists introduced. act Asequivalent diagon to we have switching shown between he Space groups in different The geometrical action ofat the the twist, horizontal however, axis is (see the figure same in b we obtain constant spinors and, therefore, depends on the holonomy gr holonomy group yields four covariantly constant spinors an Z change, the twist has to be transformed accordingly. Hence, 4.1 Classification strategy As is well known,theory the is amount related of to residual the supersymmetry holonomy exhib group of the compact space [ (the classification of [ in appendix possible lattices ( JHEP01(2013)084 as ven e sing (4.2) (4.1) QtoZ -class en, for Z -classes, software É lists all affine . This allows us to carat Extensions amount of unbroken SUSY x dimensions classified into -classes. roup, and therefore the possible Z -classes, the command É n general, a reducible represent- 2 of it. ... . lists all offers representatives for all in some (unspecified) lattice basis P ¯ 3 ⊕ . However, as the determinant is invariant is a subgroup of SU(3) is to consider the ) form given by 1 P originate from the six-dimensional repres- -class. b ⊕ Z − e P , É 3 P ⊕ . This decomposition can be computed using B e P a carat )) one can check whether or not the determinant ∈ – 12 – ϑ e catalog → -class. Therefore, we start our classification with e e ϑ of Q ϑ → B É 6 6 = ,... . b SO(6). B in the GL(6 ϑ , ) = det( a ⊂ SO(6). P ϑ -class and, finally, the command -classes (i.e. lattices) and finally construct for each Z ⊂ É P -class inside that . Hence, it can be decomposed P Z -classes (i.e. point groups) that are subgroups of SU(3). Th P as discussed in the following. É P -class is a subgroup of SU(3). . In principle, one can transform them to matrices from O(6) u ]. Furthermore, one can access this catalog easily using the e -class. Hence, the main open question is to decide whether a gi is a subgroup of SO(6) rather than O(6). is a subgroup of SU(3). É ϑ Z 15 into representations of SU(3), -class and find a representative P P 6 É of SO(6). One way to check that ]. In detail, the command -classes [ -classes of a given 6 16 Z ) matrices [ É -class we identify all Z É , Next, we recall that the matrices In more detail, our strategy reads: There exists a catalog of every possible affine class in up to si A discussion about the possible orders of the elements of the point g 1. Choose a 3. Verify that 4. Find every possible 2. Check that 5. Find every possible affine class inside each one of those 2 - and into irreducible representations On the other hand, theation six-dimensional of representation the is, point i group entation under this transformation (det( the character table of breaking of the equals +1 for all generators of determine whether or not carat lists all classes of a given representative of a 4.2 Residual SUSY We start by verifying that all affine classes (i.e. roto-translations). the (unspecified) lattice basis, i.e. point groups, can be found in appendix Z discrete subgroup of SU(3). Theis holonomy the group and same hence for the all members of a given GL(6 i.e. it gives the generators of the point group each the identification of all JHEP01(2013)084 of in 3 a (4.3) (4.4) (4.5) (4.6) (4.7) in units i v = 1 SUSY , we have U(3). If this . So for each N β P . ⊂ } is the same for all P equals the number and P . c ρ nal angles , , rators can be diagon- α ∈ ) and the orthogonality χ g β β f matrix. In order to de- ( ξ originating from the ... c = 6= rix representation of range the representations χ ) P × acter α α is given by the trace of the g ) + s defined over all elements of ( c i g for all corresponding orbifold. ρ is the order of ( ρ b 1 | of the irreducible representation χ te them as so-called twist vectors esentations χ − 1 for 0 for SU(3) and at least i P P | n ] and the entry is the corresponding ∈ . ( X j g ⊂ ) + g | f g f g )) P ( ) of the six-dimensional representation 1 , { P g | g a ( ¯ ( a ) = χ ρ ξ g ( ⊕ = χ ] = β contains one row for each irreducible repres- a i g ) = χ ). Then we know at least [ of SU(3)). So, the first check is to see whether g n ) P ( – 13 – g ¯ 3 6 ( → 4.7 ∈ χ ) = Tr( α g is a subgroup of (S)U(3) it is necessary to find only 6 h χ ( ρ with P P ) in the representation ∈ χ g X g ( i | ρ ρ 1 , P ) we use i χ | g ) for ) into a three-dimensional representation plus its complex n h 4.2 ( = c 4.6 ρ =1 i ) of M χ i g ( β ) is of the form ( ρ , → , and determine the multiplicities α 4.6 ) = h ξ ξ g ] we compute the character ( g ρ ). In fact, the number of irreducible representations j in equation ( χ , the character g ), three-dimensional vectors containing the three rotatio ( its complex conjugate (from P 6 i 3 and one column for each conjugacy class [ ρ ¯ a ∈ , v i χ 2 ρ denotes some (in general reducible) representation of g , v a 1 v is a subgroup of SU(3) this decomposition has to be of the kind Let us make a short remark. If a point group is Abelian its gene For of the decomposition ( in the decomposition, P = ( i i , now denoted by elements of a conjugacy class, i.e. order to check that the determinant is +1. Then ρ survives the compactification of the heterotic string on the alized simultaneously. In this case,v it is convenient to wri 6 As the trace is invariant under cyclic permutations, the char Now, the character table of a finite group entation character conjugacy class [ where the overline indicates complex conjugation and of conjugacy classes. Hence, the character table is a square matrix representation the decomposition ( compose the is possible, then thereρ are in generalconjugate. many But combinations in order toone to ar combination. see that However, one needs to know the explicit mat If where SU(3) and of the rows of thethe character conjugacy table classes). (where In the detail, scalar for product two i irreducible repr JHEP01(2013)084 of ξ (4.9) (4.8) ways (4.10) 3) = 0 − . by the GAPID is generated by          (1 + 2 7 1 P SU(3) is particular ] for these computa- ⊂ ucible representations nsional representation carat and follow the steps in 1 0 0 0 1 1 0 0 18 P . ] contains three elements 3 − − e S 3) with ϑ 0) 1 0 0 0 0 − , entified by GAP as GAPID d by Repsn to a unitary one by = − , 2 crete group 2 consecutively enumerates the 1 s except for one case (point group − P , 0 1 0 0 0 0 0 00 0 0 0 1 0 0 0 0 1 0 0 0 0 , − (1 M 1 7          = (6 ] = e 1 . ω  − ) ) a ξ ξ ( ][ e ( e 1 1 e ω -class obtained from ω ϑ − É tr ][ are a one- and a two-dimensional (non-trivial) ) and compute the explicit matrix representa- , 1 11 1 2 0 ½ ) 3 [ – 14 – nd ). If the decomposition cannot be arranged ac- ξ is not a subgroup of SU(3). Otherwise, we create 3. In this case, the check and e ( ρ 4.7 , ϑ 2 P 3 1 2 4.6 , Then we can easily compute the determinant of the ρ ρ ρ such that they add to 0. For example, the generator tr irrep and ,          i 3 6 v = 1 2 1 ] and the GAP package Repsn [ ½ i ρ − 17 As an example we consider tr 1 0  reads (in the ordering given by GAP) generate the six-dimensional (reducible) representation − ) read 3 ) = . Then we perform the decomposition of the six-dimensional S ξ ( e 4.8 N 1 0 0 0 1 1 0 0 SU(3). The 2262 ω ξ ) we know that χ − − ⊂ 1 0 0 0 0 3 4.7 and S − , respectively. Note that the conjugacy class [ ) = 0 mod 1 so that the determinant is +1. More precisely, it is al denotes the order of the group and 3 ξ 0 0 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ( e S 3 in the (reducible) representation ϑ v SU(3). N point group.          . P + 3 ⊂ 3 ) matrices, both of determinant +1, = 2 S 7 S v Z point group corresponds to the twist vector Z ) denotes the singlet and , ξ ] contains two. Furthermore, the characters of the six-dime + 7 e ( e 1 1 ϑ Z ω ρ 2)). In this case we had to transform the representation obtaine ], where v in the three complex planes , . The character table of We use the software GAP [ 3 π In our case, Repsn automatically created unitary representation 1] being 3 S . In what follows, we figure out how this decomposes into irred generated by equation ( , 3 N,M discrete groups of order representation according to equation ( while [ representation of Example: order to check that where [ of of the S cording to equation ( generators of tions. In detail, first we use GAP to uniquely identify the dis tion using the GAP package Repsn. such that The group generated by these[6 (non-commuting) matrices is id ξ of 2 hand. possible to choose the signs of the all combinations that fit with equation ( easy: two GL(6 PSL(3 JHEP01(2013)084 (4.12) (4.11) . More . Using 1 SUSY 2 3 ρ = 1 SUSY, ⊕ = 1 it turns = 1 SUSY in N ≥ N 2 has been con- . ρ N N 3 decomposes into n orbifolds, rank S    ) can be found in ξ n the following we
we know that there 1) = 4 supersymmetry. zed in table i = 1 SUSY (and not , -classes we construct is always divisible by 3 π (2 -classes have not been 2 N É N 0 É 1) ed on , h , 1) -classes with (2 , tation of this, resulting in SU(3). Furthermore, since carat h h Abelian point group and ian (1 É esentation is nt groups with eep (at least) h
− ese 60 ⊂ i , the nature of gauge symmetry 3 π 3 1) 2 , G S ) we find that (1 − h 4.9 . , the amount of unbroken supersym- , we see that 3 1 3 ρ ρ 2 10 exp 0 0 0 0 exp ⊕ for a summary of the results. The 60 cases    2 are devoted to the Abelian and non-Abelian 1 -classes, where the Abelian cases were already ρ = 2 – 15 – É 5.2 ) 3 ( → ω and ξ as 5.1 and 3 -class (i.e. point group). Using = 1, see table S orbifold compactification. É N 3    S . Furthermore, we have plotted the 138 pairs of Hodge numbers - and affine classes. For the 17 point groups with 11 Z 1 0 0 0 0 1 0 1 0 − = 2 and one case (i.e. the trivial point group) with    N does not contain the trivial singlet = - and affine classes (i.e. lattices and roto-translations). I 3 Z ]. Among other things, such settings feature, unlike Abelia ) in the appendix, visualizing the fact that -classes in six dimensions. Out of those, we find 60 ρ 3 ( 19 7 É ϑ ⊕ 2 ρ Recently, an explicit example of a non-Abelian orbifold bas = 1. Many of them were unknown before. The results are summari → more) is preserved by an 3 structed [ four dimensions unbroken. As discussed in section out that there areN in total 138 inequivalent space groups wit As both generators have determinant +1, we see that the GAP package Repsn we create the explicit matrix represen the appendix in table where 52 lead to precisely details including the generators of the orbifolding group in figure known in the literature. By contrast, most of the 38 non-Abel reduction of the gauge symmetry already at the string5 level. Results: classification ofWe perform toroidal a orbifolds systematic classification of space groups that k metry depends only on the split into 22 Abelian and 38 non-Abelian all possible discuss them in detail:case, respectively. sections 5.1 Abelian toroidal5.1.1 orbifolds Our results Restricting ourselves to Abelian point groups, we find 17 poi breaking (i.e. local or non-local) and the Hodge numbers ( are 7103 Comparing this to the character table in equation ( The only combination that fits into a three-dimensional repr irreducible representations of four cases with used in orbifold compactifications before. Starting from th Next, we classify all JHEP01(2013)084 -class, ]. É . orbifolds. 26 – 2 Z 21 × 2 have a geometric Z hibit settings with = 1 SUSY. Out of 7103 ], a tool to study the ometric’. On the other N 0 0 0 0 0 3 0 0 3 3 0 3 e model file per 30 eated input files for the 32 35 35 38 [ st e existing three generation ssible, due to the possibility geometric interpretation. To 1 SUSY where 52 have exactly compactification spaces allow non-Abelian f generations without changing N ≥ ] appears appropriate. The models ]) are related to es/ClassificationOrbifolds/ orbifolds ‘geometric’ although they ] discrete Wilson lines and/or (gen- 10 27 1 4 9 0 0 8 8 0 0 0 0 1 4 orbifolder 14 17 22 2 26 ] and/or discrete torsion [ Z Abelian 20 × , 2 2 Z ] as these models make use of non-trivial back- 0). Note that this does not say that Standard , – 16 – 29 = 4 = 4 = 4 = 2 = 1 = 2 = 1 = 2 = 1 = 2 = 1 = 4 , N N N N N N N N N N N N ] and also in [ 28 , 20 7 ) = (20 1) , (2 , h 1) , there are 60 point groups with orbifolds [ 2 3 1 (1 2 h total: Z × carat 2 # of generators # of SUSY Z equal to three, that realistic free fermionic models cannot 1) , which we have made available at , (2 ] conclude from the fact that their classification does not ex h . Summary of the classification of all point groups with at lea 10 − The results are also available as input for the At this point, a comment on a statement in DW [ 1) , = 1. (1 DW [ N 6, except for the case ( have non-trivial background fieldsfor as a their clear six-dimensional geometric interpretation. eralized) discrete torsion allows usthe to Hodge control numbers. the numberhand, o Hence, we one use might the call terminology such to models call ‘non-ge these There is a geometry file for each of the 138 affine classes, and on obtained in the free fermionic construction (such as [ h ground fields, i.e. discretebe Wilson more lines, precise, which do as not pointed have out a in [ low energy phenomenologyorbifolder of heterotic orbifolds. We have cr interpretation. On the onemodels based hand, on this is in agreement with th http://einrichtungen.physik.tu-muenchen.de/T30e/cod Table 1 Models with three generations ofof quarks introducing and so-called leptons discrete are Wilson impo lines [ cases obtained from JHEP01(2013)084 1 3 2 4 1 3 2 2 1 4 4 4 1 carat s were 35 41 15 15 138 an point classes # of affine . However, as ]. Again, this in GAP, “ = 1 SUSY pre- P 31 1 3 2 4 1 3 2 2 1 4 5 5 1 2 2 ion of geometries. 12 10 60 N # of -classes Z ] give a classification 9 . IdGroup f rank six as lattices Λ = 1 SUSY. Columns # 3, 4 lasses) with Abelian point N le index” gives the index in the corresponding affine classes 944 871 ], see also [ 1965 4667 1997 2950 5600 5567 3346 3307 4625 2377 1745 1964 2629 1859 1759 index carat e point group 11 ces and, in addition, neglects d Love [ -dimensional carat carat symbol min.290 min.201 min.296 min.403 min.665 min.475 min.467 min.562 min.553 min.185 min.258 min.424 min.429 min.278 . 3 group.3664 group.2702 group.3567 and, finally, “ 2] 2] 5] 5] 5] 2] 1] 1] 2] 2] 1] 1] 1] 2] 2] 2] 14] , , , , , , . , , , , , , , , , , , – 17 – [3 [4 [6 [6 [7 [8 [8 [4 [8 [9 [12 [12 [12 [12 [18 [16 [36 GAPID carat catalog 2) 1) 1) 1) 1) 1) 1) 1) Q − − − − − − − − , , , , , , , , 1 0 0 0 0 0 0 0 , they overcount the geometries and, in addition, miss a few , , , , , , , , 5) 6) 2) 2) 3) 3) 3) 4) 2) (1 (1 (1 (1 (1 (1 (1 (1 − − A.3 6 3 6 4 6 2 4 6 1 1 1 1 1 1 1 1 − − − − − − − . However, the inequivalent lattices and roto-translation , , , , , , , , , 6 4 5 , , , , , , , , 1 1 2 2 2 3 1 command , , Z , , , , , , , twist 1) 1) 1) 1) 1) 1) 1) 1) (1 (1 (1 (1 (1 (1 (1 (1 (1 vector(s) − − − − − − − − 4 1 6 1 6 1 7 1 8 1 8 1 1 1 3 1 -classes: “GAPID” is obtained using the command 12 12 , , , , , , , , and 1 1 1 1 1 1 1 1 É = 1 4 carat , , , , , , , , Z orbifolds there have been two approaches for the classificat -classes obtained from (0 (0 (0 (0 (0 (0 (0 (0 N = 2. These space groups are based on the well-known four Abeli , 2 1 2 1 3 1 3 1 4 1 6 1 2 1 2 1 2 É 3 Z Z N × , 2 -class. –I 4 6 2 4 3 6 2 –II 6 Z orbifolds using root lattices of semi-simple Lie algebras o 6 Z Z Z Z Z Z . Summary of all space groups with Abelian point group and É Z –I Z –I –I –II 7 3 4 Z –II –II 8 6 × × × × × × N 8 12 6 Z Z Z 12 -class × In addition, we find 23 inequivalent space groups (i.e. affine c For Z Z × 4 6 2 2 3 3 Z Z Z Z Z 2 label of É 2 Z Z Z Z Z Z Z # of Abelian Z serving Abelian toroidal orbifolds. For example, Bailin an for cases. A detailed comparison to our results canIn be found the in first tab one,classification the is classification somewhat is incomplete: based on it Lie misses lattices four [ latti group and 5.1.2 Previous classifications There are several attempts in the literature to classify six and the (generalized) Coxeter element as the generator of th unknown before. They are summarized in table 4 and 5 identify the groups Table 2 symbol” using the also discussed in appendix list of all 7103 in that that contains a model with standard embedding for each of the JHEP01(2013)084 , 5 ]), a (5.1) in the carat 32 ] 11 34 , oints, hence 2 . In addition, P ]. See table ve some points 10 for a comparison. ry but the Hodge 5 5 9 4 5 23 ] (based on [ he identifies 8 out of classes 10 symbol min.170, 4 # of affine 0). is given as [ Z , ussed here are trivial, for S × 0)). Finally, in the case of models 3–1 and 3–2 of DW. 4 , p carat ). The Hodge numbers agree Z 3 3 3 1 by DW [ 10 11 # of 1], -classes , and Z 3 with (38 Z , 4 ], there is an (incomplete) classification × i Z 5 3 10 F × h Z 1968 4668 1970 index 0) and we have (84 carat 4 , 1 SUSY for Abelian point groups S/ Z ] to our table > -class as already calculated in [ – 18 – 12 -class. In detail, the affine classes 1–6, 2–4, 3–6, = N É É 1 2 π 4 Z Z fundamental group. ] finds (80 ]. For both × carat symbol min.202 min.174 min.291 2 × 2 12 12 = 4 SUSY (i.e. GAPID [1 group.1611 2 Z Z fundamental group Z N 2 = 2 fundamental group Z 2 1] 2] 1] 1] N ) , , , , fundamental group ⋉ 2 4 and 6 [ is given, which, as we find, is complete, see table [4 [6 [2 [3 2 2 . The only non-trivial cases are the following (see table Z , 2 GAPID Z Z S - and one affine class. ) Z Z } -class with = 3 × ½ 2 É i ≡ { Z N 4 6 2 3 F = h Z Z Z Z -class means a ( i means a means a means the fundamental group equals the space group (no fixed p for label of # of Abelian É ] correctly identifies that there is only one possible geomet F is the group generated by those space group elements that lea A C D h 0 means a trivial fundamentalS group N 12 i [ Z – – – – – . Summary of all space groups with F 6 h × 4–4, 6–5 and 8–3 posses a where 21 space groups from the 6 space groups from the Z The fundamental groups of most of the Abelian orbifolds disc Furthermore, based on the strategy of DW [ N • • × Z 6 15 affine classes (compare section 2.3 of [ in those cases appendix): of the possibility of roto-translations. In a second approach classification for In addition, we were able to resolve an ambiguity between the with our findings except for case IV.7 (i.e. fixed. Z Table 3 there is the trivial index 2709) with one where 5.1.3 Fundamental groups The fundamental group of a toroidal orbifold with space grou numbers disagree with ours, i.e. [ JHEP01(2013)084 those 2 they are Z A × 2 2 ] and table D.1 Z 9 SU(2) × on-decomposable freely ed freely acting. A non- 2 SU(2) and SU(3) r hand, for ologically appealing feature × × ing. In total we find 31 affine amental groups. These cases , the non-decomposable freely e written as a combination of posable freely acting elements 2 , while in the cases died elsewhere. 3 [2] SU(3) SU(2) ing group. Z D SO(5) × × × SU(3) 2 3 × SU(2) SU(3)
× and Z k six, see e.g. table 3 of [ [2] × [2] × C space groups and the traditional notation of [3] [3] and and SU(3) N 4 SU(3) 2 Z Z SU(2) and SO(9) × SU(2) × 2 – 19 – -class. In detail, the affine classes 1–4, 2–4, 3–3 2 2 × corresponding root lattice(s) Z É SU(2) 3 and SO(8) × Z SU(2) SU(3) 4 SU(2) SU(2) F SU(4) SU(3) and SO(7) SO(5) and SO(8) × SU(2) × × SU(3) and × × 3 × × × × × × [2] 2 2 [2] 3 SU(3) Z SU(3) and SO(8) 2 ) × × 2 2 6 4 fundamental group. — — SO(8) SO(10) F E SO(4) SU(4) (G — G — SO(8) SU(4) SU(6) SU(7) SO(9) SU(3) SO(5) SO(5) 3 Z 2 3 1 2 1 2 1 3 1 2 1 2 3 4 1 1 1 1 2 -class Z –I –II –II –I –II –I -class . Matching between our classification of 8 12 12 7 8 6 3 4 6 Z Z Z Z Z Z Z Z Z É and 4–3 posses a 4 space groups from the ]. Cases previously not known are indicated with a dash. In the context of heterotic compactifications, the phenomen Elements of the space group that leave no fixed points are call • 33 of non-local GUT symmetryacting breaking space group is elements due withclasses to a based non-trivial the gauge on presence embedd Abelian ofare point of n groups special with interest, and non-trivial their fund phenomenology will be stu both pure lattice translations and elements of the orbifold trivial fundamental group signals thein presence the of space non-decom group,non-freely i.e. acting freely elements. acting In elements the that cases cannot b of [ acting elements belong to the orbifolding group. On the othe elements are pure lattice translations in the cases lattices as root lattices of semi-simple Lie algebras of ran Table 4 JHEP01(2013)084 ] 1 0 0 0 0 0 0 0 S C C C C C A A D D 10 π ] and Γ with 36 / 5 –II 2 Ë SU(2) Γ with non- × × / 2 3 3 5 ] C SU(2) 11 — — — — — — — — — — — — — point groups have [ × SU(3) 4 SU(3) SU(4) 2 r example, the point A × ¨rt etF¨orste al. = 1 SUSY and 27 inequi- and SU(3) eory on AdS 4 SU(4) N D y. For example, in the context ] , belian point groups and central ave not been studied systemat- 3 . = 1 SUSY and three cases with 10 S = 2. Most of them were unknown 7 3–2 ) have been discussed in [ 2 N ≡ N 1–2 1–4 1–5 2–6 2–7 2–8 3–3 3–4 4–1 2–3 2–5 2–4 3–5 3–6 2–1 2–2 n Donagi - and affine classes, respectively. 3–1 et al. [ Z between our classification and the ones in [ ]. In addition, conformal field theories of 2 and table Z 5–3 5–4 5–5 6–1 6–2 6–3 7–1 7–2 8–1 9–1 9–2 9–3 37 6 ) or ∆(6 10–1 10–2 11–1 12–1 12–2 Here 2 × . Surprisingly, the order of non-Abelian point 2 n – 20 – C Z 1 0 0 0 0 0 0 0 S S S C C C A A A A D ]. π 38 –I 3 4 2 6 ] SU(2) SU(2) in appendix SU(2) - and affine classes. It turns out that there are in total 331 — — — — — — — — — — — — — 11 — × × [ Z × 12 2 SU(2) ¨rt etF¨orste al. ]. Furthermore, non-compact examples of the form SU(4) SU(3) 35 SU(3) SU(3) of order up to 31 [ ] ⊂ 10 SU(3) focusing on Γ = ∆(3 ⊂ 1–1 1–3 0–3 0–4 1–6 1–8 1–7 1–9 2–9 0–1 0–2 2–11 2–12 2–13 2–14 1–10 1–11 2–10 = 2 have been classified in [ Donagi . Comparison of the affine classes of c et al. [ ]. In our case, the two numbers enumerate the 11 Next, we classify all Our classification results in 35 point groups with = 2 SUSY, see table 4–2 5–1 5–2 3–1 3–2 3–3 3–4 4–1 2–1 2–2 2–3 2–4 2–5 2–6 1–1 1–2 1–3 1–4 Here N two-dimensional toroidal orbifolds with Abelian and non-A and [ some related work has been carried out for IIB superstring th charge groups has a muchgroup wider ∆(216) has range order compared 216. to the Abelian case. Fo Abelian Γ non-Abelian Γ before. The results are summarized in table Table 5 been constructed [ valent space groups with non-Abelian point group and inequivalent space groups with non-Abelian point group and ically up to now andof the free literature fermionic is limited constructions to compact examples models onl based on 5.2 Non-Abelian toroidal orbifolds Six-dimensional orbifolds with non-Abelian point groups h JHEP01(2013)084 . 6 D (5.3) (5.4) (5.5) (5.2) it de- and the 6
D 0 . , 3 1 8 P ) 11 48 15 18 14 55 16 3 ( classes ω , 3 4 # of affine 20 27 ,
classes    continued ... 0 1 6 , # of affine i ) is a non-Abelian finite 3 π 6 ( 0 2 3 1 6 9 9 2 6 4 5 6 ϑ − D # of -classes , . 1 = 5 twisted sectors, all of Z 1 6 5 1 1 7   i − 3 1 # of 0 π -classes sentation is generated by , i 2 3 1 Z , π i 2 1 00 e 0 0 0 e orbifold. e π 1 SUSY for non-Abelian 0) , 2 6 , 2935 6266 2262 4682 4893 2258 6222 5650 5645 4235 e 0    index > carat 0 , , , 0 0 /D N -classes) and in total eight affine classes, = 6   5750 3374 5669 index Z Ì carat ) 3 = = (i = ( orbifold has 6 ω 4 2 6 6 – 21 – = 2 subsectors. carat symbol /D min.300 min.207 min.664 min.511 min.430 min.506 min.613 , e , e , e 6 group.1637 group.4474 group.4469 and N Ì 0) 0) 1) carat symbol min.487 min.565 , , , is a non-trivial, one-dimensional representation of 1 0 0 group.4493 ′    , , , 4] 6] 8] 3] 1] 1] 3] 1] 3] 13] 1 , , , , , , , , , , = 2 [6 [8 [12 [16 [16 [18 [21 [24 [12 SU(3). In terms of irreducible representations of 3] 1] N [16 = (0 = (1 = (0 4] Let us consider the GAPID 1 0 0 , , , 0 0 1 0 1 0 3 1 5 ⊂ − e e e [8 [24 [12 6 , where GAPID    ′ D 1 2 7 = Z ⊕ T 2 8 3 ) ⋊ = 1 SUSY in 4D. 2 3 S Z Z 3 ) ( . Summary of all space groups with 16 3)–II 8 6 4 4 2 3 Orbifold. ϑ N , → × ⋊ ⋊ S Q Z A D D -class -class 6 Dic 3 8 3 3 QD label of É × label of É D Z Z Z 4 SL(2 # of non-Abelian . For example, consider the space group generated by 7 Frobenius Table 6 Z ( has six conjugacy classes, the 6 There are two inequivalent lattices (i.e. two D group of order 12. The (reducible) three-dimensional repre them have fixed planes and hence are lattice As Hence, we find Example: and one can see that see table composes as JHEP01(2013)084 , or M 3 Z S × N Z 3 2 4 1 3 2 1 7 2 6 2 2 1 3 4 8 4 3 2 1 = 1 SUSY. or 12 19 10 30 10 331 classes N N s, such as Z # of affine d . 1 SUSY. 3 4 1 2 1 1 2 1 4 1 2 2 2 6 3 5 1 3 2 4 1 1 3 1 1 108 # of -classes N ≥ Z 1 supersymmetry in four dimen- = 1 SUSY. 2924 2802 6512 2806 2810 2934 2846 2851 6743 3414 3408 4326 6735 4895 2864 6337 4353 2875 4356 2774 5713 5712 2897 6988 4533 index N ≥ carat N = 1 SUSY, and N – 22 – carat symbol min.536 min.616 min.528 min.659 min.620 min.661 min.651 group.7614 group.7498 group.5290 group.7500 group.7504 group.7622 group.7540 group.7545 group.5943 group.5937 group.3770 group.5125 group.6834 group.4532 group.4531 group.7587 group.5746 group.7007 8] 8] 3] 6] 3] 3] 5] 42] 88] 95] 15] 22] 29] 33] 25] 30] 42] 64] 67] 10] 11] 12] 11] 11] 12] , , , , , , , , , , , , , , , , , , , , , , , , , [54 [24 [27 [36 [48 [24 [24 [48 [48 [72 [72 [72 [96 [96 [24 [24 [24 [32 [36 [36 GAPID [168 [216 [216 [108 [108 = 1 -classes (point groups) that lead to É N 8] , ) 2 2 4 2 4 Z Z 3) Z Z Z , 4 8 4 4 3 3 ⋊ ⋊ ) ) 2) 3) ⋊ ⋊ ⋊ , S S S A ) ) Q φ φ D , 3)–I 2 4 3 4 , × × × 3) 3) × × × SL(2 S -classes decompose in Z Z Z -class , , ( 3 6 4 3 3 3 ∆(54) ∆(96) ∆(27) ∆(48) É × Σ(72 Σ(36 ∆(216) ∆(108) × × label of É Z Z Z Z Z GL(2 Z GAPID [24 × PSL(3 SL(2 ∆(216), out of which 35 lead to exactly 38 with a non-Abelian point group with two or three generator out of which 17 lead to exactly 22 with an Abelian point group with one or two generators, i.e 3 6 4 3 × Z SL(2 SL(2 Z Z • • : Summary of all space groups with non-Abelian point group an ( Z ( 3 # of non-Abelian Z 2. These 1. In total we find 60 sions. Our main results are as follows: 6 Summary and discussion We have classified all symmetric orbifolds that give Table 7 JHEP01(2013)084 = 1 = 1 N N -classes c stand- Z ilding are = 1 SUSY, = 1 SUSY, actly actly N N ly or SO(32)) of the ]. Explicit MSSM 8 45 E , × ompactifications of the 44 8 he data for all 138 space classification of all roto- omenology. For instance, eometry. However, as we en argued that ‘non-local’ -translations, a combination ss, the most suitable objects i.e. E tices. non-local GUT breaking have odels discussed so far, the larger s so far only been performed for 1 SUSY. N ≥ – 23 – = 1 SUSY required to construct the corresponding -classes (or point groups) can come with inequivalent ]. Among other things, this allows one to obtain a N É -classes. In the traditional orbifold literature, 30 -classes, or, in other words, orbifold geometries that lead Z ]. As we have seen, there are 31 affine classes of space Z 29 , 28 -classes that can lead to models yielding the supersymmetri ], has certain phenomenological advantages [ -classes potentially relevant for supersymmetric model bu ]. É 43 É 39 – 40 -classes decompose in 1 SUSY. Z SUSY. SUSY. 71 with an Abelian point group, out of which 60 lead to exactly and 358 with a non-Abelian point group, out of which 331 lead to ex and 115 with a non-Abelian point group, out of which 108 lead to ex 162 with an Abelian point group, out of which 138 lead to exact N ≥ • • • • –II orbifold [ to 6 As we have explained in detail, An important aspect of our classification is that we provide t Our classification also has conceivable importance for phen Our results on Z 3. We find that there are 186 5. We find 520 affine classes that lead to 4. These 6. These affine classes decompose in heterotic string gets broken by orbifolding.symmetry In gets most of broken the locally m atGUT some symmetry fixed breaking, point. asheterotic Yet string it utilized [ has in be the context of smooth c candidate models, based on thebeen DW constructed classification, featuring recently [ lattices, classified by the so-called groups with Abelian point group and statistical survey of the propertiesthe of the models, which ha ard model. as follows. models with the C++ orbifolder [ That is, there are 52 Furthermore, space groups can be extended by so-called roto of a twisttranslations and in a terms (non-lattice) ofto affine translation. classify classes, inequivalent which space We are, groups. provide as we a discu full one of the questions is how the ten-dimensional gauge group ( are given by Liehave lattices pointed out, and not a all given lattices choice can fixes be described an by orbifold Lie g lat JHEP01(2013)084 - ⊂ T ]. G 46 ], asymmetric ] and discrete ng to work out 48 20 , ]. In addition, our R” (267104). Both . J.T. would like to 47 . This research was tation on Moorea for 27 ds, which have a rather t of geometries that can ions. M.F. would like to R. would like to thank the carat act with the free fermionic y from the self-dual point. our classification. Further, r a dynamical stabilization luster of excellence “Origin etical Underground Physics ects. On the other hand, it his is the concept of Bravais ork was partially supported idal compactifications [ We thank the Simons Center article Physics at the Energy ructed [ , that lead to an orbifold with oup of some lattice Λ as uce a Wilson line that breaks Wilson lines [ ring theory on orientifolds (see s T-folds [ strategy is independent of this 3 tion of this work. pens if one sends one or more Z ic methods can be applied in a ond the scope of this study. Let × 3 Z , we call them Bravais equivalent. This and ] for a review). G , is a subgroup of it. Now, if two lattices 4 52 G Z × ⊂ – 24 – 2 Z P , 2 Z × 2 Z -classes É ], play a crucial role in model building. It will be interesti 26 – 23 ] for some interesting models and [ , 51 21 ). Obviously, the point group – Z 49 In this study, we have focused on symmetric toroidal orbifol n, thank Sebastian Konopka forby very the useful Deutsche discussions. Forschungsgemeinschaft (DFG)and This through Structure the w of c the Universe”Frontier and of the New Graduiertenkolleg “P Phenomena”.done in P.V. the is context supportedM.F. of and by J.T. the SFB would ERC grant likeUC Advanced to 676 Irvine, thank Grant where DESY part project for offor “FLAVOU its this Geometry hospitality. work and was Physics M. in done, Stonyand for Brook, Related hospitality. the Areas Center (CETUP* for 2012) Theor their in hospitality South and Dakota for and partial the Gump support s during the comple A Details on lattices A.1 Bravais types andOne can form classify spaces lattices byequivalent the lattices. symmetry groups In they more obey. detail, T denote the symmetry gr thank Wilhelm Plesken for helpful advice regarding the use of Acknowledgments We would like to thank Pascal Vaudrevange for useful discuss torsion [ and/or non-toroidal orbifolds, whoseus classification also is mention, bey weAs implicitly we assumed are using that crystallographicassumption. the methods radii Still, our it are classification might be awa interesting to study what hap results may also be appliede.g. to compactifications [ of type II st the conditions on suchit background is, fields of in course, the geometries clear of that there are other orbifolds, such a moduli to the self-dualformulation, values. where In this also case interesting one models may have make been cont const a non-trivial fundamental group,the thus GUT allowing symmetry. us In togive other introd rise words, to we have non-localof identified GUT some a breaking. of large the se This moduli might in also the allow early fo universe, similar as in toro give rise to the same finite unimodular group GL( groups, based on the clear geometric interpretation,straightforward such way. that We have crystallograph focusedis on known the that geometrical background asp fields, i.e. the so-called discrete JHEP01(2013)084 . ]. n 13 . By (A.4) (A.5) (A.1) (A.3) (A.2) as P 2.2 } ∈ ) of some n ϑ G ( F ]. But in order , . . . , e 1 e 13 ) is an element of . { e , in a second step, . } )[ , c.f. section = roup defined by G P ! M e s defined by the lattice ( e a given lattice belongs 1 ] (though one should be ) s tangular lattice (cf. ap- ype of lattice, i.e. every e ∈ − 54 to be in this special basis, ∈ F g 1 0 0 nvariant, i.e. for tices for every dimension . Gr( ) , ge of lattice basis, represented

e T ) sible basis for that lattice [ e ij thonormal) lattice vectors with que). ) M ]. The form space e = f. e.g. [ for all B 15 ϑ T . ]. e F = Gr( B 53 ! = ), i.e. Gr( b ϑ 0 Z ) 0 a with e n, F g ) = (

j 2 ) is exactly the number of (untwisted) moduli T Gr( g Z , e T P – 25 – i ( | ϑ e ) = F ∼ = of GL( n such that its Gram matrix Gr( P × ( } } n sym P = ( ϑ n F 7−→ Ê , ϑ ij 2 ) ) ϑ ∈ e e , . . . , e = F 1 Gr( ) denote the standard scalar product. By definition, the Gram Gr( { e j , the Gram matrix changes as ) is defined as the vector space of all symmetric matrices left { ½ , e { Z M i = e n, ) = = e G ( P GL( F ⊂ , i.e. G G ), consisting of two arbitrarily long, orthogonal vectors. ] starts with an orthonormal lattice in six dimensions. Then A.3 10 Let us consider an example in two dimensions. Take the point g Note that physically the Gram matrix is the metric of the toru The interesting task would now be to decide which Bravais typ finite group the orbifold offers. It leaves invariant the form space That form spacependix corresponds to the Bravais type called p-rec to see that onewhich lattice is belongs canonically to chosen a to be given the form so-called space, shortest it pos needs A.2 Introducing anDW additional [ shift additional shifts, which are linear combinations of the (or Fortunately, algorithms for precisely thatcareful: task do the exist, shortest c basis of a lattice isΛ in and general the not dimension uni of the form space invariant by contrast, elements of the point group leave the Gram matrix i equivalence generates a finite number of Bravais types of lat On the other hand, we define the Gram matrix of the lattice basi the form space of a finite subgroup Hence, a formlattice space Λ is has in a direct basis correspondence to a Bravais t by a unimodular matrix matrix is a symmetric, positive definite matrix. Under a chan where the parentheses ( to. This can be done using the notion of form spaces [ They have been classified for dimensions up to six [ JHEP01(2013)084 ). Z (A.8) (A.6) n, (A.7a) (A.7b) xample: ) and equa- . 2.4    , 1 cients of the linear itself. Here we show −    asis, but using trans- -class from the same ], which consists of an d Z at appears in the linear 10 1 0 0 an perform a basis reduc- coefficients of some of the is 0 10 0 0 1 0 0 0 1 0 0 0 1 − is not a basis of the lattice Λ,       ave seen in the second example f . = ub-)lattice Λ spanned by the basis . e ) =    e ) and an additional shift 2 2 2 6 ) vector for the additional shift: the / / / e e 1 1 1 th Gr( + 4 e 1 0 0 1 0 0 ]), which is a transformation from GL( accordingly: see equation ( . Notice that +    54 } 2 M e : start with the identity matrix and substitute – 26 – , τ = ( 4 M 1 2 , e (originally 6 2 model named (1–1) in DW [ M e = , ω 2 rd { Z τ =    , × f 1 2 −    Z 1 0 ). Hence, we are selecting a different − É 1 0 0 0 1 0 0 0 1 0 0 1 00 0 n,       . Let us list the necessary steps and illustrate them with an e 3 = = is spanned by f e e . ). ϑ B } , those additional shifts can be incorporated to the lattice 6 2.5 , e 3.4 4 , e 2 tion (e.g. using the LLL algorithm, cf. [ lattice vectors. Exchangecombination) one by of the the new old additional shift. lattice vectors (th tion ( the column corresponding to the exchanged vector by the coeffi combination. but one of a new, different lattice Σ. new basis e The perhaps most elegant procedure is to perform a change of b The basis matrix, Gram matrix and point group generators rea Let us follow the steps described above: As an example, take the { 4. (Optional) In order to see the geometry more clearly, one c 1. We choose to exchange the 3 1. The additional shift is a linear combination with rational 2. Write the transformation matrix 3. Transform your space group using 2. In accordance with our choice, the transformation matrix -class, cf. section = in detail how to transform the space group accordingly. orthogonal lattice (p-cubic) with orthonormal basis We will restrict the discussion toe the three-dimensional (s É rational coefficients, are included in thein space section group. As we h formations from GL( JHEP01(2013)084 , , f ). r 8 ertain (A.9a) (A.11) riately (A.9b) (A.10a) (A.10b) . 5 , ], i.e. to the Lie .    11 point group in the , 1 1    to a reduced one 2 .    − − 1 Z 2 2 4 − / / /    1 1 3 1 1 3 × − − 2 . 2 oxeter element. However, entify the point group, i.e. / Z 1 0 1 0 1 1 ). Some more examples are 1 3 1 1 3 5    al lattice, which contains as rent reasons: -equivalent (semi-simple) Lie − − − − − 1 00 0 10 1 0 2 example, the lattices spanned 1 0 0 1 c . We see that introducing the / 1 − 1 0 0 8    attices as root lattices of (semi- − 1 4       b    of et F¨orste al. [ (both one-parametric, see table 4 − = = A a f r ) = ) = − f r c a . For the new lattice Σ in the new basis Gr( − Gr( ) with table b r M − – 27 – a − , are the same (see figure b a c a b c , ω    2 2 2 2    ,    / / / ) is 1 1 1 1 , ω    − − − 2 2 2    / / / 2 2 2 ] corresponds to model ) = 1 1 1 A.10 1 1 1 1 0 / / / 1 1 1 P 10 − − − − ( − − F 1 0 0 1 0 0 0 0 1 00 1 2 2 2 where the SU(4) part is an f-cubic lattice, see table / / / 1 1 1 0 1 1 0 0 0       3       into the p-cubic lattice is equivalent to work with the approp . A root lattice is the lattice spanned by the simple roots of a c = = 8 f f = = τ SU(2) ϑ B r r × ϑ B the quantities we are interested in look like and transform the point group elements accordingly, A remark is in order. The form space left invariant by the Last, we compare the Gram matrix Gr( 3. We perform the transformation using 4. Next, we perform a LLL reduction, which is a change of basis transformed point group in an i-cubic lattice. A.3 Bravais typesIt and is Lie common lattices insimple) the Lie string-orbifold algebras. literature to Ona describe the discrete one l subgroup hand, this ofwe makes SU(3), find it using this easy practice Weyl to reflections to id be and problematicRedundancies. the for C at least three diffe provided in table additional shift Therefore, model (1–1) in [ lattice SU(4) (semi-simple) Lie algebra. Even ifalgebras the are simple different, roots the of latticesby two non they the root span systems might not. of SU(3) For and G (reduced) basis of equation ( This form space ispossible the realizations one the i-cubic of and a the three-parametric, f-cubic lattices i-orthogon JHEP01(2013)084 ns think ). The form space ce point in the center A.1 stem, and the green lines undamental cells (shaded). ted set of values. So, for lows for, and what is the ctors). The realization of is three and the most basic hat the situation could be f the system (e.g. the edge root lattice and calculating attice has no description as zes for different sublattices, endix t there is no root lattice that lattices of Lie algebras allow ded and made explicit in the of Lie root lattices. udes semi-simple Lie algebras or i-cubic to crystallographers t lattices does not exhaust the gle between the basis vectors is s overcounting. But the problem f lattices which are not generated – 28 – Every Bravais type allows for a set of continuous deformatio ). 8 When considering the redundancy of root lattices, one might root system. Simple roots are also indicated, as well as the f ). The bcc lattice is a cubic lattice with an additional latti 8 2 . The hexagonal lattice: the blue lines form the SU(3) root sy Continuous parameters. of the fundamental cell.length of Its the cube). only freecan One possible parameter generate way is this to the convince Bravaiswhich oneself size lattice Bravais tha lattice o is it takingroot generates. every lattice (see rank We table find three that the i-cubic l which conserve its symmetries.form Those space deformations that are defines enco tells that us particular how Bravais many typeeffect deformation (cf. of parameters app them one (tothat Bravais change freedom type lengths in al of thefor or context just angles of one between root parameter, basis the lattices(direct size ve is products of very the of limited: system; simplebut and if never ones), one the one incl angles canexample, between a choose vectors, two-dimensional oblique different which lattice, si arearbitrary, in could fixed which never the to be an a unambiguously expressed limi in terms Missing lattices. that there are moreresolved by root introducing some lattices clever convention than toexists avoid types in thi the of other latticeswhole direction family and too: of Bravais t types, the i.e. setby there any are of Bravais root types all system. o possible Theexample roo lowest is dimension the body in centered which cubic this(see lattice, table occurs also known as bcc Figure 5 form the G JHEP01(2013)084 a a clas- a 8 SU(2) b a ⊕ b a SU(2) continued ... ,a 2 indicated as a ⊕ roduct there is derstood ortho- SU(2) SU(2) SU(2) a ,G c b SU(2) a ⊕ ⊙ ⊙ ngth stands for the ⊕ a a a SU(2) a ossibilities are written ar products. SU(3) een our classification of ogonal product of semi- imensions, together with SU(2) SU(2) SU(2) Lie algebra notation SO(5), SU(2) nd ambiguous, and is lacking types and form spaces, which , we present in table have been used: rder to overcome the discussed means that the shortest simple SO(7), SU(2) ,a 2 – 29 – 1 dimension 2 dimensions 3 dimensions . lattice name a Obliquep-Cubic mp cP RulerSquareHexagonal r p-Rectangular tp hp op c-Rectangular oc    2    0 a  has length squared b b
0 0 a a a a/ 2 0 0 a a ± a c a a a b a  a       means free-angle product. The scalar product of the roots is Gram matrix means orthogonal product. Unspecified products should be un means a product with the leftmost factor. ֓ root of G squared length of the shortest simple root, e.g. G gonal. subindex. Noticeactually that no in equivalent thesimple Lie cases Lie lattice in algebras description: is whichin not we italics. a a have semi-simple non-orth used Lie algebra. this These p p Equal subindices mean equal length of the roots or equal scal A subindex in an algebra whose simple roots are of different le Nevertheless, in order to justify some of the matchings betw • ⊕ • ⊙ • • ← • In conclusion, the language of root lattices is incomplete a space groups and the ones already existing in the literature sification of all oftheir equivalent the root Bravais lattices, typesambiguities if of in there the lattices are root in any. lattice 1, There, language, some 2 in conventions o and 3 d geometrical insight with respectis, to therefore, the the language one of used Bravais in this paper. JHEP01(2013)084 b ֓ ֓ ֓ ← ← ← b d f b d c b SU(2) ⊙ ⊙ ⊙ b a c ⊕ centered (in SU(2) SU(2) SU(2) SU(2) c ⊕ ⊕ SU(2) SU(2) SU(2) ⊕ ⊕ b a ] b d e b ,a ⊙ ⊙ ⊙ 2 or SO(5)] b a a a SU(2) SU(2) (none) SU(2) c c or G primitive, ⊕ ⊙ ⊙ a SU(4), Sp(6) SU(2) SU(2) SU(2) p a a a b d c possible root lattice expres- SU(2) (no simple expr.) (no simple expr.) (no simple expr.) rhombohedral. ⊙ ⊙ ⊙ Lie algebra notation ⊕ r a a a a [SU(3) SU(2) SU(2) SU(2) SU(2) SU(2) SU(2) ice names: [SU(2) body-centered, and i – 30 – lattice name i-Orthorhombic oI p-Monoclinic mP c-Monoclinic mC Triclinic aP i-Tetragonal tI p-Orthorhombic oP c-Orthorhombic oC f-Orthorhombic oF i-Cubicp-Hexagonal cI hP r-Hexagonal hR p-Tetragonal tP f-Cubic cF face-centered,   f c +   b c   b   b − − c   3 3 b b   + 2 2 + 2 − − + b 0 a a a/ a/               a c b a a/ b b b c c 0 0 0 − − a 0 0 0 d b c c 2 0 + a a b b b 0 0 0 0 3 b e a a a a d a b + a/ 2 + 2 − − a a/ a ± a + a/ a a c a a c a a c d a b b a d f b a b b a − c a               + a a/ Gram matrix + 2 + a a   : List of Bravais types in 1, 2 and 3 dimensions, together with   b a     +   a   2D) or base-centered (in 3D), Table 8 sions. The following prefixes and suffixes are used for the latt JHEP01(2013)084 n ir (B.1) h order . They are ts extended to ously be the least 2, one can find point dimensions are given orm space contain as ]. d in two and three di- n ≥ 13 he basis vectors to have n six dimensions there exist For example, if we set the nts of dimensions that add wallpaper groups of a four-dimensional order e lattice (mp) (i.e. we take paper, we reproduce here the op) lattice. If we set now the ices form the embedding chain ce in [ wn as is the dihedral group of order 2 n D . This can be realized by building a n , > n ) ≤ m ) ( . There, – 31 – φ 9 m ( φ . Nevertheless, in dimensions } . For further information about this topic, the standard 6 18 such that , , one can classify the 17 two-dimensional space groups by the 14 m 3 , of (irreducible) point group elements in . Graph of Bravais types embeddings in 2D and 3D. 12 m , -function. For dimension two, this leaves only elements wit 10 φ , 9 , 8 Figure 6 , as possible point group elements. In six dimensions, this ge ]. 7 } , 55 6 6 , , 4 5 mp. , , is the Euler 3 4 . In that case, the order of the point group element would obvi → , ֒ φ , n 2 3 , op , The possible orders A graph containing all of the existing embeddings of that kin As discussed in section In general, Bravais types with two or more parameters in the f 1 2 { → , ֒ -classes. Those can be found in table 1 well-known, and their classification can be found for instan reference is [ by the equation É list of all possible two-dimensional space groups, also kno B Two-dimensional orbifolds In order to illustrate some of the concepts addressed in this mensions can be seen in figure specific cases other typesoff with diagonal a parameter lower number tothe of zero basis parameters. vectors in to the bediagonal two-dimensional orthogonal), elements obliqu we of get theequal a length), p-rectangular form we ( space get to atp square be lattice equal (tp). (i.e. These we three latt take t up to point group element as the direct sum of two point group eleme where group elements with order common multiple of thepoint orders groups of with the elements of factors.10 order element For 30, and example, which a are in two-dimensional a order direct 3 sum element. in { JHEP01(2013)084 (B.2) ation and image continued ... 1) by simple matrix- x, matrix of some element π 1 1 3 4 1 2 1 2 1 1 classes the specific information of every # of affine 10 , ! name & description Torus Manifold Pillow Orbifold, 4with cone-angle singularities Pipe Manifold, 2 boundaries Klein bottle Manifold, non-orientable i 1 1 2 2 1 2 1 2 1 1 1 λ # of -classes e 0 Z ϑ

!. In table – 32 – 3 n D = -classes in two dimensions. ∼ = e 2 4 6 g É 3 . D D S D - and affine class to which they belong, its Bravais type ∼ = ∼ = ∼ = ∼ = -class Z       2 4 3 6 0 0 1 2    label of É -, 0 1 / Z Z Z Z 1 1 0 0 1 Table 9 É − 1 1 × ⋉ ⋉ ⋉ –II –I − − 2 2 4 2 3 2 6 2 2 1 0 0 0 0 . This matrix acts on an augmented vector ( Z Z Z Z Z Z Z Z id Z 0 0 0 0 0 0 1 0 − 1 0 e ), its orbifolding group generators in augmented matrix not    generators       8 is given by S ∈ ) i e i is the symmetric group of order , λ e –aff. class n ϑ Z S –II–1–1 –II–1–2 –I–1–1 – 2 2 2 = ( Z Z id–1–1 Oblique Z É p-Rectangular p-Rectangular Oblique lattice e and of lattice (cf. table a name, description andg image of its topology. The augmented affine class is shown: the using the lattice basis vector multiplication. JHEP01(2013)084 image continued ... π 4 / , 1 , 1 , 1 π 2 / π π 3 , 1 sin- π 2 / π / 2 π π and 2 of 2 / 3 π / π Triangle pillow Orbifold, 2with cone-angle singularities Triangle Manifold, one boundary, 1 angle of Jester’s hat Orbifold, 1with cone-angle singularity Triangle pillow Orbifold, 3with cone-angle singularities Triangle Manifold, 3 boundary, all angles name & description strip M¨obius Manifold, non-orientable, 1 boundary Rectangle Manifold, 1 boundary Cut pillow Orbifold, 2with singularities cone-angle Cross–cap pillow Orbifold, 2with cone-angle singularities Jester’s hat Orbifold, 1with singularity cone-angle boundary gularity with cone-angle boundary boundary       2 2 2          0 1 1 / / / – 33 – 0 0 1 0 0 1 1 1 1       0 0 1 0 0 1 0 0 1 1 1 1 1 1 1 − − − 1 − − − − 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0                   , , , , , ,    ,                   0 0 1 1 0 0 1 0 0 1 0 0 1 0 0 2 2          1 / / 0 0 1 1 1 0 0 1 0 0 1    1 1 1 1 1 0 1 0 − − − − 1 1 1 − 1 1 − − − − − 1 0 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 − 0 − − − 0 1                            generators    –1–2 –1–1 –1–1 –1–1 –1–2 –1–3 –2–1 3 4 4 2 2 2 2 Z Z Z Z Z Z Z –aff. class Z ⋉ ⋉ ⋉ × × × × –1–1 –1–1 –II–2–1 – 3 2 2 2 4 2 2 2 2 2 Z Z Z Z Z Z Z Z Z Z É Hexagonal Square c-Rectangular lattice Square Hexagonal Square p-Rectangular p-Rectangular p-Rectangular c-Rectangular JHEP01(2013)084 ) 1) , (B.3) (2 1) 0) 7) 3) , , , , , h 1) , (25 (36 (31 (27 (1 h ( continued . . . image 3 / , the pipe and the π and 2 , 1 , ) 3 3 / 3 / Z the resulting orbifolds. π / π 2 π 2 . , sectors -classes are separated by of ( 2 ntal groups are trivial with /

T É π and the Klein bottle’s one is b 1 2 − and Z a U = π 6 / ) from 2 name & description Jester’s hat Orbifold, 1with cone-angle singularity Triangle pillow Orbifold, 3with singularities cone-angles Triangle Manifold, 1with angles boundary, boundary π and 2 2 1) T , b a , T T 6) (2 Z , 2) 0) , , , h ∈ 1) , – 34 – (1 + (10 + (6 + (4 h 1 1 1 1       m, n T T T T 0 0 1 0 0 1 | 0) 0) 0) 0) G 1 1 1 , , , , m − − − b n 1 1 0 0 0 0 0 0 1 a + (16 + (16 + (16 + (27        , , U U U U    1) 1) 1) 0) 0) 0) 0) 0) 0 0 1 =       , , , , θ, θ, θ, θ, 0 1 0 0 1 0 1 ( (5 ( (5 ( (5 generators of contributions to ( ( (9 S − 0 0 1 0 0 0 0 0 1 0 1 0 1 0 1 0 1       generators    1 1 1 1 local local local local class, affine breaking , the cross-cap pillow (a projective plane) – Z 1 2 3 1 Z (Λ) class : List of all possible two-dimensional orbifolds. –1–1 –2–1 6 3 Z Z ) – 3 4 –aff. class P Z Z É Z ⋉ ⋉ ( class –1–1 Sometimes it is of interest to know the fundamental groups of – 6 2 2 Z Z Z É Hexagonal lattice Hexagonal Hexagonal C Tables C.1 Abelian point groups Among the two-dimensional space groups,the most of following the exceptions: fundame the torus has a fundamental group its own space group, with group structure ¨bu strip M¨obius double lines. Table 10 JHEP01(2013)084 ) 1) , (2 1) 0) 7) 3) 5) 1) 7) 3) 5) 1) 7) 5) 1) 0) 3) 19) 11) , , , , , , , , , , , , , , , , h , , 1) , (25 (24 (31 (27 (29 (25 (31 (51 (29 (25 (31 (29 (25 (24 (27 (19 (35 (1 h ( continued . . . 4 4 4 4 5 5 5 4 4 4 4 4 T T T T T T T sectors T T T T T 3) 3) 0) 0) 0) 0) 0) 1 , , , , , , , , 2) 4) 2) 0) 0) T 1 , , , , , T 0) + (6 + (6 + (3 + (3 + (4 + (4 + (4 10 and , + (3 + (6 + (4 + (9 + (9 3 3 3 3 4 4 4 3 3 T 3 4 3 3 3 3 T T T T T T T T T U 0) T T T T T T , 4) 0) 4) 0) 3) 1) 0) 5) 1) , , , , , , , , , 0) 0) 0) 0) 1) 0) + (16 , , , , , , 0 , + (1 1 1 ) from + (8 + (4 + (8 + (4 + (6 + (4 + (3 + (6 + (2 T , 9 9 8 + (8 + (7 + (8 + (8 + (2 + (1 1 2 2 2 2 2 2 2 2 2 1) T T T , 0) T 2 2 2 2 2 2 T T T T T T T T T , 1) 0) 2) (2 T T T T T T 8) , , , 0) 0) 0) 0) 0) 3) 3) 0) 0) , , , , , , , , , , 0) 0) 1) 0) 0) 0) , h , , , , , , 1) , + (16 + (2 + (1 + (3 – 35 – + (8 (1 + (6 + (6 + (3 + (3 1 0) 7 7 6 , + (1 + (7 + (10 + (10 + (10 + (3 + (2 + (3 + (3 + (15 + (15 h 1 0 1 1 1 1 T T T , 1 0 1 1 1 1 1 1 1 1 1 1 ω, T T T T T 0) 0) 2) T T T T T T T T T T T T ,( , , , 0) 0) 0) 0) 0) G
, , , , , 0) 8) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) ) 0) , , , , , , , , , , , , 6 e + (4 + (3 + (3 ω, 0) 5 6 6 6 6 + ,( + (4 + (16 + (8 + (8 + (3 + (3 + (7 + (4 + (4 + (4 + (8 + (3 + (3 + (12 + (12 + (12 + (12 T T T T T
2 ω, 2 e U U U U U U U U U U U U U U U U U 0) 0) 3) 1) 1) e ( , , , , , 1 1 2 2 1) 3) 3) 1) 0) 0) 0) 0) 0) 0) 1) 0) 0) 1) 1) 1) 1) 0),( 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) , , , , , , , , , , , , , , , , , θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, (3 +(4 ( (3 (3 ( (3 +(2 ( (3 +(4 ( (3 +(2 ( ( (3 ( (3 ( (3 ( (3 ( (3 +(3 (5 ( (5 ( (3 ( (3 ( (3 ( (3 generators of contributions to ( ( 1 2 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 local local local local local local local local local local local local local local local local local class, affine breaking – 1 2 1 2 1 1 1 2 3 1 2 1 2 3 4 1 Z (Λ) class 2 Z ) –I – –II –I –I 7 –II –II 8 6 × P 8 6 12 Z É 12 ( class Z Z 2 Z Z Z Z Z JHEP01(2013)084 ) 1) , (2 9) 3) 7) 7) 3) 9) 7) 3) 7) 7) 3) 3) 7) 3) 11) 11) 15) 11) 15) 11) 11) , , , , , , h , , , , , , , , , , , , , , , , 1) (9 (7 (3 (7 (7 (3 (3 (7 (3 , (21 (27 (19 (31 (27 (11 (11 (15 (11 (15 (11 (11 (1 h ( continued . . . sectors T 1 1 , , 1 1 1 1 1 1 , , , , 1 1 1 1 T T and T T T T 0) 0) , , U 2) 0) 4) 0) , , , , + (4 + (8 + (6 + (8 + (4 + (8 0 0 , , 0 0 0 0 1 1 1 0 1 1 ) from 1 1 , , , , , , , , , , 1 1 1 1 1 1 1 1 1 1 T T 1) , T T T T T T T T T T 0) 0) (2 , , 0) 0) 4) 4) 4) 4) 2) 0) 4) 4) , , , , , , , , , , , h



5 5 4 5 1) , + (4 + (8 e e e e – 36 – + (4 + (8 + (4 + (4 + (4 + (4 + (2 + (8 + (4 + (4 2 2 2 2 1 1 1 1 (1 1 1 0) 0) , , h 1 1 1 1 1 0 1 0 0 1 1 1 1 1 0 0 0 , , , , , , , , , , , , , , , ω, ω, ω, ω, 0 0 0 0 0 1 1 1 1 0 0 1 0 0 1 ω, ω, T T


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, , 2) 2) 4) 0) 4) 4) 2) 0) 4) 4) 4) 8) 8) 8) 4) ) ) ) ) ) ) 2 2 2 1 1 1 0) 0) 0) 4 5 5 , , , , , , , , , , , , , , , 3 6 6 6 6 6 e e e e e e e e e 2 2 2 1 1 1 ω, ω, ω, ω, ω, ω, 0) 0) 0) 0) 0) + + + + + + ,( , ,( , ,( , + (6 + (2 + (10 + (2 + (8 + (4 + (4 + (4 + (4 + (8 + (4 + (4 + (12 + (8 + (8 + (4 + (8





ω, ω, ω, 2 4 2 3 3 4 ω, ω, ω, ω, ω, 6 6 4 4 3 3 e e e e e e U U U U U U U U U U U U U U U U U U U U U ( e ( e ( e e ( ( e ( e 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 3) 3) 3) 3) 3) 3) 3) 3) 3) 3) 3) 3) 3) 3) 3) 3) 3) 3) 3) 3) 3) 0),( 0), 0),( 0), 0),( 0),( 0),( 0), , , , , , , , , , , , , , , , , , , , , , θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, (3 ( (3 ( (3 ( (3 ( (3 ( (3 (3 (3 (3 (3 (3 ( (3 (3 (3 (3 (3 ( (3 (3 (3 ( (3 generators of contributions to ( (3 5 1 2 1 2 1 2 3 4 6 1 2 3 4 4 1 2 3 4 5 local local local local local local local class, affine breaking non-local non-local non-local non-local non-local non-local non-local non-local non-local non-local non-local non-local non-local non-local – 6 4 5 3 2 Z (Λ) class ) – P É ( class JHEP01(2013)084 ) 1) , (2 6) 3) 1) 9) 1) 9) 5) 5) 3) 5) 3) 9) 5) 7) 7) 9) 13) 11) , , , , , , , , , , , , h , , , , , , , 1) (9 (5 (7 (7 (9 , (12 (15 (61 (21 (37 (21 (17 (17 (15 (17 (15 (25 (11 (1 h ( continued . . . 0 , 2 1 , 0 1 1 1 T , , , 1 1 1 T 0) T T T
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1 1 1 1 T T + and ) T T T T T T T T T T T T T 2 4 0) 0) e e , , U 4) 0) 2) 2) 1) 2) 2) 0) 0) 0) 0) 2) 0) , , , , , , , , , , , , , + + 1 2 + (8 + (4 e e + (4 + (4 + (2 + (2 + (3 + (2 + (2 + (4 + (4 + (4 + (4 + (2 + (4 ( 0 2 , , 2 1 + 0 1 2 0 0 0 2 2 0 0 0 0 0 1 0 1 , ) from
1 0 , , , , , , , , , , , , , , , ,

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+ , , h + ) 1 1 ω, 5 5 5 1) e T e , e e + (8 ( , – 37 – + (8 + (8 + (8 + (2 + (2 + (3 + (10 + (6 + (6 + (4 + (4 + (4 + (4 + (2 + (2 + (4 + (2 2 1 (1 0) 2 0) 0) 0)
+ , + + , h 2 2 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 ) 0 , , , , , , , , , , , , , , , , , , 4 4 4 4 ω, 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 ω, ω, ω, T e

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ω, ω, 1 1 1 1 2 1 5 ω, ω, ω, ω, ω, ω, ω, ω, 0) e 4 6 e e e e e e e U U U U U U U U U U U U U U U U U U U ( , 4) ( ( ( ( e ( ( ( e 2 1 , 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 1) 1) 1) 3) 1) 1) 1) 1) 3) 3) 3) 3) 3) 3) 3) 3) 3) 3) 3) 0),( 0),( 0),( 0),( 0),( 0),( 0), 0),( 0),( 0), , , , , , , , , , , , , , , , , , , , θ, θ, θ, θ, θ, ω, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, (3 (3 ( (3 (3 ( (3 +(16 (3 +(4 (3 (3 (3 ( (3 (3 ( (3 ( (3 (3 ( (3 ( (3 ( (3 ( (3 ( (3 generators of contributions to ( 5 6 1 1 2 3 4 3 1 2 1 1 2 1 2 1 1 2 3 local local local local local local local local local local local class, affine breaking non-local non-local non-local non-local non-local non-local non-local non-local – 2 1 7 8 9 10 11 12 Z (Λ) class 4 Z ) – × P É ( class 2 Z JHEP01(2013)084 ) 1) , (2 3) 5) 6) 2) 1) 1) 5) 3) 4) 3) 3) 9) 5) 7) 3) 3) 7) , , , , , , , , , , , , , , , , , , h 1) , (27 (17 (18 (14 (37 (25 (17 (15 (16 (51 (27 (21 (17 (31 (27 (39 (19 (1 h ( continued . . . 1 0 0 2 0 2 2 1 0 2 , , , , , , , , , , 1 1 1 1 1 1 1 1 1 1 T T T T T T T T T T 0) 0) 2) 2) 0) 0) 0) 0) 0) 0) , , , , , , , , , , sectors T + (8 + (6 + (2 + (6 + (6 + (4 + (4 + (8 + (4 + (4 3 1 3 3 1 1 2 3 1 1 3 3 1 , , , , , , , , , , , , , 0 1 0 0 1 1 1 0 1 1 0 0 1


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, e e e e e e T T T T T T T T T T T T T T T T 6 6 4 , , , , , G e e e




+ + + + + + + + + 1) 2) 1) 0) 1) 0) 2) 1) 1) 0) 0) 2) 2) 4) 2) 2) ) ) ) ) ) + (6 + (8 + (8 2 2 2 1 1 1 5 2 2 , , , , , , , , , , , , , , , , 3 2 2 4 2 2 4 4 6 5 3 + (4 e e e 1 1 1 e e e e e e e e e e e , , , 1 1 ω, ω, ω, 1 1 1 0) 0) 0) , , + + + 1 1 + + + + + + + + + + + T T T , , , + (1 + (4 + (3 + (3 + (3 + (2 + (6 + (2 + (1 + (6 + (4 + (8 + (2 + (6 + (2 + (8 1 1 1 T T


1 1 1 3 1 1 3 3 5 4 1 ω, ω, ω, 0) 0) 0) e e e 4 6 6 e e e e e e e e e e e U U U U U U U U U U U U U U U U ( ( ( , , , 0) 0) ( ( ( e ( ( e ( ( e ( ( ( ( 2 2 2 1 1 1 , , 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 0),( 0),( 0),( , , , , , , , , , , , , , , , , θ, ω, θ, θ, ω, θ, ω, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, (3 (3 (3 ( (3 (3 ( (3 +(12 (3 ( (3 +(12 (3 +(8 (3 (3 (3 (3 (3 (3 +(8 (3 generators of contributions to ( +(16 4 5 1 6 1 2 3 1 2 3 4 5 2 3 4 5 6 local local local local local local local local local local local local local class, affine breaking non-local non-local non-local – 5 4 3 Z (Λ) class ) – P É ( class JHEP01(2013)084 ) 1) , (2 1) 2) 2) 1) 3) 3) 1) 2) 2) 4) 6) 4) 1) 9) 1) 7) 2) , , , , , , , , , , , , , , , , , , h 1) , (25 (32 (20 (19 (27 (15 (13 (32 (20 (22 (36 (22 (37 (21 (25 (19 (14 (1 h ( continued . . . 0 0 2 0 2 2 0 0 2 0 2 0 0 2 , , , , , , , , , , , , , , 1 1 1 1 1 1 1 1 1 1 1 1 1 1 T T T T T T T T T T T T T T
) 0) 0) 2) 0) 0) 0) 1) 0) 0) 0) 0) 0) 2) 0) 5 , , , , , , , , , , , , , , e sectors + T + (2 + (3 + (2 + (4 + (2 + (2 + (4 + (4 + (2 + (4 + (2 + (6 + (2 + (4 4 e 3 3 1 3 1 1 3 2 3 1 3 1 3 3 1 , , , , , , , , , , , , , , , 0 0 1 0 1 1 0 1 0 1 0 1 0 0 1
+ and ) T T T T T T T T T T T T T T T 2 6 e e U 1) 0) 0) 0) 0) 0) 0) 2) 0) 0) 1) 0) 0) 1) 0) , , , , , , , , , , , , , , , + +
) 1 5 5 e e e + (1 + (1 + (8 + (2 + (8 + (8 + (1 + (2 + (2 + (8 + (3 + (8 + (2 + (1 + (8 (
2 1 + 2 2 0 2 0 0 2 1 2 0 2 0 2 2 0 1 1 ) from
) , , , , , , , , , , , , , , , , , + ) 5 4 0 0 1 0 1 1 0 1 0 1 0 1 0 0 1 1 1 1) 5 3 , e e ω, T T T T T T T T T T T T T T T T T e e ( (2 ( 2 1 + 1) 0) 2) 1) 0) 0) 0) 0) 1) 0) 3) 0) 0) 2) 0) 0) 0) , 2 1 + , , , , , , , , , , , , , , , , ,
2 , h 2 2 2 2 2 2 )

4 , , , , , , e ω, ) 2 2 5 2 1) e ω, 1 1 1 1 1 1 ( , , , 3 e e ( 1 1 2 1 T T T T T T , , – 39 – e + (5 + (4 + (2 + (5 + (2 + (2 + (4 + (8 + (5 + (2 + (7 + (2 + (6 + (4 + (4 + (8 + (8 2 2 1 1 (1 T T

+ 0) 0) 1) 0) 0) 0) h 1 1 2 1 2 2 1 2 1 2 1 2 1 1 2 2 2 ) ) + , , , , , , , , , , , , , , , , , ω, , , , , , , 0) 2) 4 6 5 ω, ω, 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0



, , 1 e e e T T T T T T T T T T T T T T T T T 6 6 6 6 , , , e G e e e ( e


+ + + 1) 0) 2) 1) 0) 1) 0) 0) 0) 0) 0) 1) 1) 3) 0) 1) 0) ) ) ) + (3 + (4 + (4 + (4 + (4 + (6 2 2 2 2 2 1 1 1 1 1 , , , , , , , , , , , , , , , , , 3 3 2 5 5 4 + (2 + (2 1 1 1 1 1 1 e e e e e e , , , , , , 1 1 ω, ω, ω, ω, ω, 1 1 1 1 1 1 0) 0) 0) 0) 0) , , 1 1 + + + + + + T T T T T T , , , , , + (1 + (1 + (1 + (2 + (2 + (2 + (5 + (4 + (3 + (2 + (5 + (4 + (7 + (2 + (1 + (6 + (3 T T




1 1 1 4 4 3 ω, ω, ω, ω, ω, 0) 0) 0) 0) 0) 0) 2 6 6 6 6 e e e e e e U U U U U U U U U U U U U U U U U , , , , , , 0) 0) ( ( ( e ( e e ( e ( e , , 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 0),( 0),( 0),( 0),( 0),( , , , , , , , , , , , , , , , , , θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, (3 +(8 ( (3 +(10 ( (3 +(10 (3 (3 ( (3 +(12 (3 (3 (3 ( (3 +(12 (3 (3 +(12 (3 ( (3 +(12 (3 +(8 (3 generators of contributions to ( (3 3 1 1 2 3 1 2 4 5 1 2 3 2 1 2 3 local local local local local local local local local local local local local local local class, affine breaking non-local non-local – 8 9 7 6 10 Z (Λ) class ) – P É ( class JHEP01(2013)084 ) 1) , (2 4) 2) 0) 0) 0) 6) 0) 3) 3) 7) 5) 3) 0) 12) , , , , , , , , , , , , , , h , 1) , (40 (26 (24 (24 (84 (18 (12 (15 (51 (31 (41 (27 (36 (24 (1 h ( continued . . . 3 3 , , 1 1 1 1 T T , ,
1 0 0 0 0 1 2 4 1 4 1 1 ) , , , , , , , , , , 0) 0) 1 1 1 1 1 1 1 0 1 0 6 T T , , e T T T T T T T T T T 0) 0) , , 0) 0) 0) 0) 0) 0) 1) 1) 0) 1) , , , , , , , , , , + 2 + (8 + (4 sectors 5 2 2 , , e 1 1 T + (15 + (6 + (6 + (2 + (2 + (2 + (27 + (9 + (1 + (4 + (6 + (4 T T + 0 0 3 3 3 3 0 0 1 1 3 0 3 , , , , , , , , , , , , , 2 0) 0) 1 1 0 0 0 0 1 1 1 1 0 1 0 e , , and T T T T T T T T T T T T T ,

U 0) 0) 0) 0) 0) 0) 0) 3) 0) 0) 0) 2) 2) ) + 2 ) , , , , , , , , , , , , , 6 + (8 + (8 6 1 e e e 4 1 1 1 1 4 4 4 , , , , , , , , ( 1 2 2 1 1 1 1 1

+ 3 1 + (3 + (2 + (6 + (2 + (2 + (2 + (9 + (3 + (9 + (8 + (6 + (4 + (4 + 2 ) ) T T T T T T T T 5 6 6 2 4 2 2 2 2 2 2 2 0 2 4 2 ) from 5 , , , , , , , , , , , , , e e e ω, 0) 0) 0) 0) 0) 0) 0) 0) 0 0 0 0 0 0 0 0 0 1 0 0 0 e 1) , , , , , , , , , , T T T T T T T T T T T T T
+ (2 + 2 + 2 + 2 ) 2) 1) 0) 0) 0) 0) 0) 3) 3) 1) 1) 1) 1) 4 4 6 , , , , , , , , , , , , , 5 5 + (2 + (9 + (3 + (8 + (8 + (2 + (2 + (2 e , h e e e e 3 0 0 0 3 0 3 3 3 3 ( ( 1) , , , , , , , , , , , + + 3 3 1 1 1 2 2 1 1 1 1 1 1 1 + 2 – 40 – + (5 + (4 + (9 + (9 + (9 + (9 + (9 + (3 + (3 + (1 + (4 + (4 + (4 (1 3 5 T T T T T T T T T T 3 e e h 1 1 2 1 1 1 1 1 1 1 2 1 2 1 ω, ω, , , , , , , , , , , , , , , e 0) 0) 0) 0) 2) 0) 0) 0) 2) 0) 1 0 0 0 0 0 0 0 0 0 0 0 0 0 , ,

, , , , , , , , , , T T T T T T T T T T T T T T 4 6

+ + 2 + 2 ) ) G e e 2
0) 2) 1) 0) 0) 0) 0) 0) 3) 3) 0) 0) 1) 0) 2 6 6 2 2 2 1 1 e 6 , , , , , , , , , , , , , , e e e e + (3 + (2 + (9 + (2 + (6 + (4 + (2 + (8 + (4 + (4 e 2 1 2 0 2 1 1 1 5 2 5 2 1 ω, ω, + + + + 0) 0) 0) 0) 0) 0) 0) 0) , , , , , , , , , , , + 2 1 1 1 2 1 1 1 1 0 1 0 , , 1 5 5 1 + (9 + (5 + (2 + (9 + (3 + (3 + (4 + (2 + (2 + (2 + (2 + (1 + (4 + (1 1 T T T T T T T T T T T

ω, e e e e ω, ω, ω, ω, ω, ω, ω, ω, e 4 6 U U U U U U U U U U U U U U ( 0) 0) 0) 3) 0) 0) 0) 0) 0) 0) 0) (2 (2 (2 (2 e e 3 1 , , , , , , , , , , , 1 1 1 1 1 1 3 3 3 3 2 2 0) 0) 0) 0) 0) 0) 1) 0) 0) 0) 1) 1) 1) 1) 0),( 0),( 0),( 0),( 0),( 0),( 0),( 0),( 0), , , , , , , , , , , , , , , θ, θ, θ, ω, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, (3 ( (3 +(3 ( (3 +(2 ( (3 +(9 (3 +(3 (3 (3 +(6 ( (3 +(2 ( (3 +(2 ( (3 +(2 (3 ( (3 +(1 (3 +(6 ( (3 +(1 generators of contributions to ( ( 4 1 2 1 1 2 3 1 1 1 1 2 1 2 2 local local local local local local local local local local local local local class, affine breaking non-local – 2 4 1 1 2 3 1 2 Z (Λ) class –I 3 –II 6 6 Z ) – Z Z × P É ( × class × 3 2 2 Z Z Z JHEP01(2013)084 ) 1) , (2 1) 6) 0) 0) 1) 5) 3) 4) 6) 0) 0) 2) 0) 0) , , , , , , , , , , , , , , , h 1) , (25 (18 (12 (20 (73 (29 (51 (16 (18 (12 (36 (14 (12 (36 (1 h ( continued . . . 3 , 2 3 2 1 , , , 2 1 2 T 1 1 1 T T T , , , 0) 4 2 1 4 2 0 1 1 1 , , , , , , , 1) 0) 0) 0 1 1 0 1 2 T T T , , , T T T T T T 0) 0) 0) , , , 1) 0) 0) 0) 0) 1) + (6 , , , , , , + (2 + (6 + (1 2 sectors , 2 1 2 2 1 1 , , , , , 2 2 1 2 2 T T
+ (3 + (5 + (15 + (9 + (11 + (5 + (5 + (1 + (15 ) T T T T T 0) 4 3 1 0 0 0 3 1 2 1 0 1 , , , , , , , , , , , , 0) 0) 0) 0) 0) e 0 1 1 1 1 0 1 1 1 1 2 , , , , , and T T T T T T T T T T T +

U 1) 0) 0) 1) 0) 1) 0) 1) 0) 0) 1) 3 ) ) , , , , , , , , , , , e
6 + (15 + (6 + (1 + (9 + (3 + (1 6 ) e e 1 1 1 0 1 1 0 1 1 1 4
+ , , , , , , , , , , )
e 2 1 2 2 1 2 2 2 2 2 +

2 6 ) + (4 + (4 + (3 + (1 + (1 + (4 + (4 + (1 + (9 + (3 + (1 ) ) T T T T T T T T T T e 3 e + 2 5 +
4 6 2 0 2 2 2 2 0 1 2 2 2 ) from e e ) , , , , , , , , , , , 5 e e 0) 0) 0) 1) 0) 0) 0) 0) 0) 1) 2 0 1 0 0 0 0 1 1 0 0 1 6 e 1) , , , , , , , , , , e , + + 2 e + 2 , , T T T T T T T T T T T + +

+ 2 2 1 (2 + 5 + ) ) 3 5 1) 0) 0) 1) 0) 0) 1) 0) 3) 0) 1) + e e e 4 4 6 5 , , , , , , , , , , , e e 1 + (1 + (6 + (3 + (2 + (6 + (3 + (1 + (3 + (3 + (1 e , h 5 e e e e (2 (2 e 0 0 0 4 0 0 4 0 0 0 (2 (2 ( 1) 3 3 1 1 , , , , , , , , , , + 2 , + + 3 3 3 2 1 1 2 1 2 1 1 2 1 2 2 2 + + 2 – 41 – + 2 + (3 + (1 + (3 + (1 + (1 + (5 + (2 + (9 + (3 + (3 + (1 4 (1 3 3 T T T T T T T T T T ω, ω, e 4 3 e e h 4 1 3 1 1 1 1 1 1 3 0 1 1 1 1 ω, ω, ω, , , , , , , , , , , , , , , e e 0) 0) 0) 1) 0) 0) 0) 0) 0) 1) e 0 0 1 0 0 1 0 0 0 1 0 1 0 1 + , , , , , , , , , , , , , , , T T T T T T T T T T T T T T




+ + + 2 + 2 2 ) ) ) ) ) G e + 2 2 2 0) 1) 0) 0) 1) 0) 0) 0) 1) 1) 3) 0) 0) 0) 2 4 4 6 2 4 3 e e , , , , , , , , , , , , , , e e e e e e e 3 + (3 + (3 + (1 + (1 + (6 + (6 + (2 + (3 + (1 + (3 e + 2 + 5 4 3 2 5 4 3 2 2 1 2 + + 0) 0) 0) 0) 0) , , , , , , , , , , , + 2 0 1 1 1 0 1 1 1 1 2 1 + 2 + 2 + 2 + 2 + 2 + 1 1 1 3 + (1 + (4 + (9 + (1 + (1 + (4 + (9 + (3 + (1 + (3 + (9 + (3 + (9 + (1 e e 1 T T T T T T T T T T T e e 3 5 1 3 2 1 ω, ω, ω, ω, ω, e e e e e e e U U U U U U U U U U U U U U ( (2 (2 0) 0) 0) 0) 0) 0) 0) 0) 1) 1) 0) ( ( (2 ( ( (2 ( ( 3 3 3 1 1 1 , , , , , , , , , , , 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0),( 0),( 0),( 0),( 0),( , , , , , , , , , , , , , , θ, θ, θ, θ, ω, θ, θ, ω, θ, θ, ω, θ, θ, θ, θ, θ, ( (3 +(1 +(3 (3 +(4 (3 ( (3 +(1 ( (3 +(1 +(6 (3 +(4 (3 ( (3 +(3 (3 +(1 +(1 (3 (3 ( (3 +(3 (3 generators of contributions to ( (3 1 2 1 1 2 3 1 2 3 3 4 1 2 local local local local local local local local local local local class, affine breaking non-local non-local non-local – 2 5 1 4 3 Z (Λ) class 6 Z ) – × P É ( class 3 Z JHEP01(2013)084 ) 1) , (2 3) 1) 1) 0) 0) 0) 0) 0) 0) 0) 1) , , , , , , , , , , , , h 1) , (27 (37 (25 (54 (30 (54 (90 (54 (42 (30 (61 (1 h ( continued . . . 1 , 0 1 1 2 2 , , , , 3 2 2 2 T T T T T 0) 0 0 0 0 1 2 0 0 1 0 0 1 , , , , , , , , , , , , , 0) 0) 0) 0) 1 1 2 1 1 1 2 1 1 1 1 1 , , , , T T T T T T T T T T T T
) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 6 0 , , , , , , , , , , , , , + (1 + (8 + (8 + (9 + (10 e 1 2 sectors , 2 0 2 0 1 2 0 2 3

T , , , , , , , , + ) ) 2 2 2 2 2 2 2 2 T T + (2 + (1 + (5 + (2 + (4 + (6 + (3 + (2 + (4 + (4 + (2 + (4 5 4 6 0) T T T T T T T T e e e , 0) 3 3 2 3 3 1 2 3 3 3 3 3 , , , , , , , , , , , , , 0) 0) 0) 0) 0) 0) 0) 0) 0 0 1 0 0 1 1 0 0 0 0 0 + + + , , , , , , , , and T T T T T T T T T T T T 4 3 5 + (9 e e e U 0) 0) 0) 0) 1) 0) 0) 0) 0) 0) 0) 0) + (4 ( 3 , , , , , , , , , , , , , + (4 + (5 + (4 + (5 + (8 + (7 + (6 + (3 2 1 + + 0 1 , 1 3 1 1 3 1 0 1 3 1 1 3 3 3 T , , , , , , , , , , , e e ω, 2 1 3 2 1 3 2 2 1 3 2 T + (2 + (1 + (4 + (3 + (1 + (6 + (4 + (2 + (1 + (4 + (2 + (2 ( 0) T T T T T T T T T T T 2 1 , 0) + 2 2 1 2 2 0 1 2 2 2 2 2 , ) from
, , , , , , , , , , , , , 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0)
) 1 0 0 1 0 0 1 1 0 0 0 0 0 1) ) , , , , , , , , , , , 3 , e ω, T T T T T T T T T T T T 6 e ( ω, + (4 (2 e 2 1 0) 0) 0) 1) 1) 0) 0) 0) 0) 0) 0) 0) , ,( + (4 + 2 , , , , , , , , , , , ,

, + (4 + (2 + (2 + (6 + (2 + (2 + (7 + (8 + (2 + (2 + (4 , h + 2 ) )

1 1 , ω, ) ) 0 2 2 0 0 2 0 2 0 2 2 0 0 5 6 5 1) e 2 T , , , , , , , , , , , , , , 2 6 e e e ( 2 2 1 3 2 1 3 1 2 2 1 3 2 T , – 42 – e e + (5 + (5 + (4 + (6 + (4 + (1 + (4 + (5 + (3 + (9 + (7 + (5 2 1 (1 0) T T T T T T T T T T T T T 0)
+ + + , 0) h 1 1 2 1 1 2 2 1 1 1 1 1 ) + + , , , , , , , , , , , , , 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 6 3 5 4 ω, 0 0 0 0 0 0 0 0 0 0 0 0 ω, , , , , , , , , , , , , , 1 5 e e e e T T T T T T T T T T T T ,( , e e G ( (

+ + + + 0) 0) 0) 0) 1) 1) 1) 0) 0) 0) 0) 0) ) ) + (12 + (9 2 2 1 1 , , , , , , , , , , , , 4 2 6 2 2 4 + (4 + (8 + (2 + (4 + (3 + (8 + (2 + (5 + (8 + (2 + (3 + (8 + (7 1 1 e e e e e e , , 2 1 2 3 1 1 1 2 1 1 2 2 1 0 2 ω, ω, 1 2 0) 0) 0) 0) , , , , , , , , , , , , , , , 1 1 2 1 3 2 1 2 2 1 2 1 1 3 1 + + + + + + T T , , + (2 + (1 + (4 + (2 + (1 + (5 + (3 + (1 + (6 + (4 + (2 + (2 T T T T T T T T T T T T T T T

1 1 5 1 1 1 ω, ω, ω, ω, 0) 0) 2 6 e e e e e e U U U U U U U U U U U U , , 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) e ( ( ( e ( ( ( , , , , , , , , , , , , , , , 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0),( 0),( 0),( 0),( , , , , , , , , , , , , θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, θ, +(4 ( (3 +(8 +(5 (3 +(1 +(1 (3 +(4 ( (3 +(8 +(5 (3 (3 +(4 ( (3 +(8 +(6 (3 +(4 (3 (3 +(12 +(12 (3 +(4 +(2 (3 +(4 generators of contributions to ( ( 1 2 4 1 2 1 2 3 2 3 4 1 local local local local local local local local local local local local class, affine breaking – 4 3 2 1 Z (Λ) class 4 Z ) – × P É ( class 4 Z JHEP01(2013)084 ) . 1) , 11 (2 0) 0) 3) 0) 0) , , , , , , h . 1) , ℓ (84 (42 (21 (36 (18 ω (1 ]). The twists k h 9 θ ( ææ 1 3 2 2 0 1 , , , , , , 2 1 2 3 5 2 ææ T T T T T T 4 0 0 0 , , , , 0) 0) 0) 0) 0) 0) 0 2 1 2 , , , , , , 80 T T T T l orbifolds of table 0) 1) 0) 0) , , , , + (6 + (4 + (9 + (4 + (1 + (6 ææ sectors 0 2 1 1 2 0 , , , , , , 2 1 2 3 4 2 T + (4 + (2 + (1 + (1 T T T T T T 3 2 3 2 , , , , 0) 0) 0) 0) 0) 0) 0 1 0 1

, , , , , , and ) ) T T T T 6 5 ææ = 1 SUSY preserving Abelian toroidal e e U 60 0) 0) 0) 0) labels the twisted sector , , , , + (3 + (4 + (4 + (4 + (4 + (5 + + N 1 3 1 1 0 0 1 3 3 4 , , , , , , , , ææ k,ℓ e e 3 1 3 1 2 3 4 1 + (4 + (4 + (1 + (4 T T T T T T T T T ææ + + 2 1 2 1 ) from , , , , 0) 0) 0) 0) 0) 0) 0) 0) 2 2 0 1 0 1 1) , , , , , , , , 1 , e e , T T T T and 1 ( (2 h 2 1 + 0) 0) 0) 0) 2 ææ , , , , ææ 1 + (1 + (1 + (1 + (2 + (4 + (4 + (2 + (1 , h ææ e ω, 40 ææ 0 2 2 0 2 0 5 4 0 2 1) ( , , , , , , , , , , , ææ 3 2 1 3 2 1 1 2 4 1 2 1 ææ , – 43 – ææ + (4 + (4 + (3 + (4 (1 ææ T T T T T T T T T T
h 1 2 1 2 ) , , , , ω, 0) 1) 0) 0) 0) 0) 0) 0) 0) 0) ææ 6 0 0 0 0 , , , , , , , , , , e ææ ææ T T T T , ææ G
+ 0) 1) 0) 0) ) ææ ææ , , , , 5 4 ææ + (1 + (1 + (4 + (4 + (1 + (2 + (6 + (1 + (1 + (6 ææ ææ ææ e e 5 4 3 3 1 2 1 1 2 1 1 ææ 0) 0) , , , , , , , , , , , ææ ææ 0 1 2 3 5 2 2 1 2 2 1 + + -classes is the common one in the literature (cf. e.g. [ ææ ææ + (1 + (2 + (1 + (1 20 ææ ææ ææ T T T T T T T T T T T 2 3 ω, ω, ææ ææ É e e ææ U U U U ææ 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) ( ( ææ ææ , , , , , , , , , , , 1 1 2 2 ææ 0) 0) 0) 0) 0),( 0),( ææ ææ , , , , ææ θ, θ, ææ θ, θ, ( (3 +(1 +(2 +(4 +(4 +(1 +(5 (3 +(4 ( (3 +(6 +(3 (3 +(4 generators of contributions to ( +(6 ææ ææ ææ ææ 1 3 1 2 0 local local local local class, affine 5 0 15 10 breaking 1 , 2 – h 1 5 Z (Λ) class . Statistics of the Hodge numbers for the 138 Abelian toroida 6 : Summary of the classification of all six-dimensional Z ) correspond to the twist vectors listed in table – × P ω É ( class 6 Z Figure 7 and Table 11 orbifolds. The nomenclature for the θ

JHEP01(2013)084

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   

   

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e 0 0 0 1 0

6

i 2 −

π

4] , [12 1

   

0 1 0 0 e 0

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6

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3] , [12

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4893 4 ,

4 A

0 1 0 0 0 1 −

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0 1 − 0 e 0 0

3

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4] , [8

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5750 5 ,

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0 0 1 0 0 1

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3] , [8

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4682 5 ,

4 D

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0 1 0 e 0 0

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π

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3

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GAPID rmSU(3) from index classes

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JHEP01(2013)084

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2 2

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) e − − ) e − − 0 0 1 − 0 (e (e

12 3 12 3

1 1 i 2 i 2 i 2 i 2 π π π π

3] , [24 11 11 2 2

2 2

   

− 0 ) e + ) e + 1 0 0 (e (e

7 6743 ,

12 3 12 3

1 i 2 1 i 2 i 2 i 2

π π π π

I − 3) , SL(2 11 11 2 2

0 0 1 e 0 0

3    

i 2

π

2    

   

e 0 0 0 i − 0

3

i 2

π

1] , [24 1

   

0 1 0 0 e 0

6266 12 ,

3

i 2 −

π

8 3

1 Z Z ⋊

0 0 i − 0 0 1

   

   

   

e 0 0 0 0 1

7

i 2

π

1] , [21 1

   

e 0 1 0 0 0

, 2935 5

7

i 2

π

7 T Frobenius 2

0 0 0 1 0 e

7    

i 2

π

4    

   

0 e 0 0 e 0

3 3

i 2 i 2 −

π π

3] , [18 1 1

   

e 0 0 e 0 0

, 4235 9

3 3

i 2 − i 2 − π π

3 3 S × 1

1 Z

0 0 1 e 0 0

6      

i 2

π

1       – 45 –

      0 1 0 1 0 0 i − 0 0

13] , [16

      1 0 0 0 1 − 0 0 i − 0

5645 10 , ,

2 2 4 ) × (

Z ⋊ Z Z

0 0 1 − 0 0 1 − 0 0 1 −

   

   

   

0 e 0 0 1 0

8

i 2 −

π

8] , [16 3

   

e 0 0 1 0 0

, 5650 7

8

i 2 −

π

16 QD 1

0 0 1 0 0 1 −

   

   

    0 i − 0 1 0 0

6] , [16

    1 0 0 0 1 − 0

, 6222 10

2 8

Z ⋊ Z

0 0 i − 0 0 1 −

GAPID rmSU(3) from index classes

-class of label twists fconj. of # carat É

JHEP01(2013)084

otne . . . continued

   

   

   

0 0 1 e 0 0

3

i 2 −

π

3] , [27 1

   

0 1 0 0 e 0 11 2864 ,

3

i 2

π

∆(27) 1

0 1 0 0 0 1

   

   

    0 0 1 0 1 − 0

12] , [24

    1 0 0 1 0 0

4895 5 ,

4 S

0 1 0 0 0 1

   

   

   

e 0 i 0 0 0

6

i 2

π

11] , [24 1

   

e 0 0 0 i − 0

, 6735 15

3

i 2 − π

8 3 Q ×

1 Z

0 0 0 0 1 e

3    

i 2 −

π

1    

   

0 1 0 0 e 0

3

i 2 −

π

10] , [24 1

   

0 1 − 0 e 0 0

, 4326 15

3

i 2 − π

3 4 D ×

1 Z

0 0 1 − e 0 0

6    

i 2

π

1     – 46 –

   

e 0 0 0 1 0

6

i 2 −

π

8] , [24 1

   

0 1 0 0 e 0

3408 9 ,

3

i 2 −

π

2 2 6 ) × ( 1

Z ⋊ Z Z

0 0 1 − 0 0 1 −

   

   

   

0 1 0 e 0 0

12

i 2

π

5] , [24 1

   

1 0 0 e 0 0

, 3414 12

12

i 2 π

3 4 S ×

5 Z

0 0 1 − 0 0 1 −

   

   

2 2

   

− 0 i) − (1 − i) − (1 0 1 − 0

1 1

3] , [24

2 2

   

− 0 1+i) + (1 i) + (1 1 0 0 5669 7 ,

1 1 2) = N ( II − 3) , SL(2

0 0 1 0 0 1

GAPID rmSU(3) from index classes

-class of label twists fconj. of # carat É

JHEP01(2013)084

otne . . . continued

   

   

2 2

   

0 1 − 0 0 1+i) + (1 i) + (1

1 1

33] , [48

2 2

   

1 0 0 − 0 i) − (1 i) − (1

14 5712 ,

1 1

2 3) , SL(2

Z ⋊

0 0 1 0 0 1 −

   

   

2 2 2 2

   

) e + ) e − − 0 (e (e i) + (1 − i) + (1 − 0

8 8 8 8

1 i 2 1 i 2 i 2 i 2 1 1 π π π π

29] , [48 3 3 1 1

2 2 2 2

   

− ) e + ) e − 0 (e (e i) − (1 i) − (1 − 0

, 5713 8

8 8 8 8

1 1 1 1 i 2 i 2 i 2 i 2

π π π π

3) , GL(2 3 3 1 1

0 0 1 − 0 0 1

   

   

    0 0 1 i 0 0

3] , [48

    1 0 0 0 i − 0 , 2774 8

∆(48)

0 1 0 0 0 1

   

   

   

0 e 0 0 e 0

6 3

i 2 − i 2 −

π π

12] , [36 1 1

   

0 e 0 e 0 0

, 4356 18

6 3

i 2 i 2 − π π

6 3 S × 1

1 Z

e 0 0 1 0 0

6    

i 2

π

1     – 47 –

   

e 0 0 0 0 1

6

i 2 −

π

11] , [36 1

   

e 0 0 1 0 0

2875 12 ,

3

i 2 π

4 3 A ×

1 Z

0 0 0 1 0 e

6    

i 2 −

π

1    

   

0 e 0 0 e 0

6 3

i 2 i 2

π π

6] , [36 1 1

   

e 0 0 e 0 0

, 4353 18

3 3

i 2 − i 2 −

π π

4 3 3 ) ( × 1 1

Z ⋊ Z Z

0 0 1 e 0 0

3    

i 2 −

π

1    

    0 i − 0 0 1 0

11] , [32

    1 0 0 1 0 0

, 6337 14

2 4 4 ) × (

Z ⋊ Z Z

0 0 i − 0 0 1 −

GAPID rmSU(3) from index classes

-class of label twists fconj. of # carat É

JHEP01(2013)084

otne . . . continued

   

   

3 3 3

   

− ) e + − ) e + ) 2e + 0 0 1 (e (2e (e

3 3 3 3 3 3

1 1 1 i 2 − i 2 − i 2 − i 2 i 2 i 2 π π π π π π

15] , [108 1 1 1 1 1 1

3 3 3

   

e + ) e + − ) e + ) 1 0 0 (2e (e (2e

14 2806 ,

3 3 3 3 3 3

i 2 − 1 i 2 − 1 i 2 − 1 i 2 i 2 i 2 π π π π π π

) φ Σ(36 1 1 1 1 1 1

3 3 3

(e (2e (e 0 1 0 − e + ) 2e + − ) 2e + )

3 3 3 3 3 3    

1 1 1 i 2 i 2 i 2 i 2 − i 2 − i 2 −

π π π π π π

1 1 1 1 1 1    

2 2 2 2

   

1+i) + (1 − i) − (1 i) + (1 − i) + (1 0 − 0

1 1 1 1

67] , [96

2 2 2 2

   

− 0 i) − (1 i) + (1 i) − (1 i) − (1 − 0

, 6512 16

1 1 1 1

4 3) , SL(2

Z ⋊

0 0 i − 0 0 1

   

   

    0 0 1 i 0 0

64] , [96

    1 0 0 0 0 1 , 2802 10

∆(96)

0 1 0 0 i 0

   

   

   

0 0 0 1 e 0

6

i 2

π

42] , [72 1

   

1 0 0 e 0 0

, 2924 15

3

i 2 − π

3 4 S ×

1 Z

0 1 0 e 0 0

3    

i 2 −

π

1     – 48 –

   

e 0 e 0 0 30] , [72 0

3 3

i 2 − i 2 π π

1 1

3    

e 0 0 0 = e 0 8] , [24 GAPID ×

4533 27 ,

3 3 Z

i 2 − i 2 − π π

1 1

2 2 6 3

0 0 ) ) × (( × 0 0 1 − e

6     Z Z Z Z ⋊

i 2

π

1    

2 2

   

− 0 0 i) − (1 − i) − (1 e 0

6

1 i 2 1 π

25] , [72 1

2 2

   

− 0 1+i) + (1 i) + (1 e 0 0

, 6988 21

3

1 1 i 2 − π

3 3) , SL(2 ×

1 Z

0 0 1 e 0 0

3      

i 2 −

π

1      

      0 0 1 0 1 − 0 0 0 1

8] , [54

     

1 − 0 0 1 0 0 e 0 0 , , 2897 10

3

i 2 −

π

∆(54) 1

0 0 1 − 0 1 0 e 0 0

3

i 2

π 1

GAPID rmSU(3) from index classes

-class of label twists fconj. of # carat É

JHEP01(2013)084

wse etr o h eeoi riodcompactification. orbifold heterotic the for sectors twisted 1 − c to

eae iceegop facranodr h ubro conjugac of number The order. certain a of groups discrete merates corresponds c classes y

h iceegop(..tenme feeet)adtesecon the and elements) of number the (i.e. group discrete the ubrcneuieyenu- consecutively number d

ie h re of order the gives N number first the numbers: two of consists ] M N, [ GAPID The SUSY.

umr ftecasfiaino l o-bla on ruswith groups point non-Abelian all of classification the of Summary : 1 ≥ N al 12 Table

     

     

      0 0 1 0 1 − 0 0 0 1

95] , [216

     

e 0 0 1 0 0 1 0 0 2851 19 , ,

3

i 2 −

π

∆(216) 1

0 e 0 0 0 1 0 1 0

3

  i 2

π

1

 

3

6 6

 

i 3 i 3 + 3 + 3

√

e 1 1

12

√ √ i 2 π

5

3 3

6

 

+ 3 − − i 3

√ √

i 1 i

√

3 3

6

− i 3 − + 3

  √ √

i i 1

  √

6 6 6


 – 49 –

i 3 i 3 i 3 + 3 + 3 + 3

 

1 1 1

√ √ √

88] , [216

3 3

6

+ 3 − i 3

√ √

, 2846 16

e 1 i

 

12

√ i 2 π

) φ Σ(72 5

3 3

6

− 3 i + 3

√ √

    i e 1

12

√ i 2 π

5    

18 36 18

   

7 7 7 2i − 7 11i − 5 − 2i + 1 − 0 1 0

1 1 1

42] , [168

√ √ √

36 9 36

   

3 + 23i i + i − i + 25i − i 7 7 − 7 1 0 0 − , 2934 6

1 1 1

2) , PSL(3

√ √ √

18 36 18

4i − 1 − 5i + 11 4i + 5 − 0 0 1 − 7 7 7

    1 1 1

√ √ √    

    1 − 0 0 0 0 1

22] , [108

   

0 1 0 0 e 0 , 2810 20

3

−

∆(108) i 2 π

0 1 0 e 0 0

6

−

i 2 π

GAPID rmSU(3) from index classes

-class of label twists fconj. of # carat É JHEP01(2013)084 , ]. ] ]. e , ommons SPIRE SPIRE , IN , IN ][ ][ (2012). ]. (2007) 683 , Crystallographic , hep-th/0612044 11 [ , SPIRE IN [ , From strings to the MSSM hep-th/0606187 [ assenhaus, hep-ph/0511035 orbifold phenomenology (2007) 011 [ ]. Heterotic brane world ]. Vacuum Configurations for ]. nge, ) d reproduction in any medium, Supersymmetric standard model Supersymmetric Standard Model , 03 N er, ( (1987) 282 Non-Factorisable Z(2) times Z(2) Z SPIRE SPIRE SPIRE , (2008). ]. IN Orbifold Compactifications with Three IN IN (2007) 149 JHEP ][ ][ ][ Strings on Orbifolds. 2. Strings on Orbifolds , B 191 Adv. Theor. Math. Phys. SPIRE (2006) 121602 , IN [ B 785 ]. 96 Crystallographic Algorithms and Tables – 50 – , ]. ]. ]. SPIRE Phys. Lett. , IN [ n SPIRE SPIRE (1985) 46 hep-th/0406208 , Wiley (1978). [ arXiv:0806.3905 IN IN [ Geometrical aspects of arXiv:0809.0330 [ [ (1) Nucl. Phys. 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Comp. JHEP [ (0 [ arXiv:1205.1228 Nucl. Phys. [ mini-landscape. (II). Completing the search for MSSM vacua [ JHEP compactifications Non-Commutativity in Closed String Theory Compactifications with D-branes, Orientifolds and Fluxes Phys. Lett. Phys. Lett. Phys. Lett. [49] F. Gmeiner, R. Blumenhagen, G. Honecker, D. and L¨ust T. [39] O. Lebedev, H.P. Nilles, S. M. Ramos-S´anchez, Ratz and [53] W. Plesken and W. Hanrath, [54] A.K. Lenstra, H.W. Lenstra and L. Lovasz, [55] T. Hahn ed., [47] C. Hull, [51] F. Gmeiner and G. Honecker, [52] R. Blumenhagen, B. D. K¨ors, and L¨ust S. Stieberger, [40] V. Bouchard and R. Donagi, [41] V. Braun, Y.-H. He, B.A. Ovrut and T.[42] Pantev, V. Braun, P. Candelas, R. Davies and R. Donagi, [45] A. Anandakrishnan and S. Raby, SU(6) [46] R.H. Brandenberger and C. Vafa, [48] C. Condeescu, I. Florakis and D. L¨ust, [50] M.R. Douglas and W. Taylor, [43] L.B. Anderson, J. Gray, A. Lukas and E. Palti, [44] A. Hebecker and M. Trapletti, [38] S. Dulat and K. 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