Preprint typeset in JHEP style. - PAPER VERSION hep-ph/0601015 Classification of 1D and 2D Orbifolds Lars Nilse School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, UK Email: [email protected] Abstract: We present a complete classification of all 1D and 2D orbifold compactifications. There exist 2 one-dimensional and 17 two-dimensional orbifolds. The classification includes orbifolds such 1 2 2 as S /Z2 or T /Zn, as well as less familiar ones like T /Dn or the M¨obius strip. We derive the explicit form of the basis functions and prove their orthonormality and completeness. Our study is based on the classification of space groups, which is well-known from crystallography. We define these groups in a novel, purely algebraic way. That enables us to determine all possible parities 2 that can be defined on the orbifolds. We discuss field theories on T /Zn with brane kinetic terms, and describe the derivation of their mass eigenstate bases. arXiv:hep-ph/0601015v2 1 Mar 2006 Keywords: orbifold, dimension 5, dimension 6. Contents 1. Introduction 2 2. 1D Orbifolds 4 2.1 Circle S1 6 1 2.2 Interval S /Z2 7 3. 2D Orbifolds 9 3.1 Torus T 2 16 3.2 Klein Bottle R2/pg 18 3.3 Real Projective Plane RP 2 20 2 3.4 T /Z2 22 2 ′ 3.5 T /Z2 23 2 ′′ 3.6 T /Z2 24 2 3.7 T /D2 26 2 ′ 3.8 T /D2 27 2 3.9 T /F2 28 2 3.10 T /Z3 30 2 3.11 T /D3 31 2 3.12 T /F3 32 2 3.13 T /Z4 33 1 2 3.14 T /D4 35 3.15 R2/p4g 36 2 3.16 T /Z6 38 2 3.17 T /D6 39 4. Conclusions 41 1 A. Sum rules on S /Z2 43 2 B. Scherk-Schwarz mechanism on T /Z2 46 2 C. Brane kinetic terms on T /Z3 49 1. Introduction In the late 1990s it was realized that extra spatial dimensions are not necessarily linked to the Planck scale and could potentially be relevant for T eV scale physics [1]. Since then, extra di- mensional models have enjoyed great popularity among physicists. In many of these models, the number of dimensions is reduced by compactifying them on so-called orbifolds. Orbifolds are quo- tient spaces of a manifold modulo a discrete space group. In crystallography, the classification of these groups is well-established. There exist two one-dimensional (1D) and 17 two-dimensional (2D) space groups [2]. Based on the classification of these space groups1, we present in this paper a complete classification of 1D and 2D orbifolds. We derive the explicit form of the corresponding basis functions and prove their orthonormality and completeness. Higher-dimensional fields de- fined on the orbifolds can possess various parities including Scherk-Schwarz (SS) phases. Using a novel algebraic definition of the space groups, we derive the set of all possible parities for each of 2 the orbifolds. For example, we show that a complex field compactified on T /Z6 cannot possess any SS parities. – In this paper, we present a complete catalogue of all possible 1D and 2D orbifold 1The classification has previously been used in a study of conformal field theories [3]. 2 compactifications. We hope it will prove to be a helpful tool for model builders. The main results are summarized in Tables 1 and 2. The concept of extra dimensions led to new perspectives in nearly all areas of current Beyond Standard Model research [4, 5]. Take for example neutrino physics. The observed light neutrino masses are conventionally explained by the seesaw mechanism [6]. It requires a heavy mass scale that might not be available in some of the extra-dimensional scenarios. One possible alterna- tive explanation assumes sterile bulk neutrinos coupling to brane-localized Standard Model (SM) fermions [7, 8]. The light neutrino masses arise now from the volume-suppressed Yukawa couplings. Any neutrino mass model is intimately linked to other research areas, such as leptogenesis or neu- trinoless double beta decay [9], and therefore very likely to be experimentally testable. Another good example is electroweak physics [10, 11]. In the SM, the unitarity of high energy scattering pro- cesses is ensured by Higgs exchange. In extra dimensional models, no Higgs is necessary. Unitarity is the consequence of the exchange of an infinite tower of Kaluza-Klein (KK) modes [12, 13, 14]. In a similar way, extra dimensions play an important role in the study of dark matter [15] and dark energy [16, 17]. – The majority of the models cited in this paragraph are 5D. They rely on a single extra dimension. Some more recent papers [8, 11, 17, 18] explore the possibilities of 6D models. As we will see, 2D orbifold compactifications are more varied and possess a richer structure. They can also be employed in string compactifications [19]. The paper is organized as follows. In Sections 2 and 3, we discuss in detail each of the 1D and 2D orbifolds. We derive the basis functions and study the possible parities of complex fields on 1 these orbifolds. In Appendix A, we revisit our study of high energy unitarity on S /Z2 [13, 14]. We present a simplified proof of important sum rules that were central in our discussion. The proof relies only on the orthonormality and completeness of the basis and can therefore easily be adapted for any of the orbifolds discussed in this paper. In Appendix B, we discuss Scherk- Schwarz phases on 2D orbifolds that were previously ignored in Section 3. In Appendix C, we 2 describe how to derive the mass eigenstate bases for quantum field theories compactified on T /Zn with brane kinetic terms (BKT) at the orbifold fixed points. Some important compactifications such as S1/(Z Z′ ) [20], T 2/(Z Z′ Z′′) [21] or the Chiral Square [22, 23] do not appear in 2 × 2 2 × 2 × 2 Tables 1 and 2. They are being discussed in Section 2.2, Appendix B and Section 3.13 respectively. 3 2. 1D Orbifolds One-dimensional orbifolds are very simple. There are only two of them, the circle and the interval. Higher-dimensional fields compactified on them can be expanded in terms of exponentials and sines/cosines respectively. Despite this simplicity we present here a more detailed discussion in order to prepare for the arguments of Section 3. Orbifolds are quotient spaces of a manifold modulo a discrete group2. We are not free to choose any arbitrary discrete group, but are restricted to so-called space groups. An n-dimensional (nD) space group is defined as a cocompact discrete group of isometries of Rn [27]. These groups are also known as crystallographic groups or Bieberbach groups, and in 2D as wallpaper groups. Their classification3 is known for dimensions n 6. ≤ In the one-dimensional case, we consider quotient spaces R/Γ where Γ stands for one of the two 1D space groups Z and D∞, see Table 1. The most intuitive way of defining the groups is by identifying the isometries of R that act as generators of the group4, see Fig. 1(a) and 1(b). Alternatively, the groups can be characterized in a purely algebraic way. Z R/Z S1 circle 1 D∞ R/D∞ S /Z2 interval Table 1: The first column lists the 1D space groups Γ. The following three columns state the corresponding quotient spaces R/Γ = S1/Γ′ and their geometry. Z = t (2.1) h i Z = r r2 = 1 (2.2) 2 h | i D = t, r r2 = 1, (tr)2 = 1 (2.3) ∞ h | i Z, Z ⊇ 2 The set of generators and the relations among them define uniquely the structure of the groups. Note that D∞ is a natural extension of the definition of the dihedral group Dn (3.19) that we will encounter in Section 3. The advantage of this representation-independent definition will become apparent when we discuss the possible parities on these orbifolds. It is important to note that 2A formal definition can be found in William Thurston’s lecture notes [24]. The terminology orbifold was first used in one of his lecture courses in 1976. The concept itself occurs first in 1956 as V-manifold in [25, 26]. 3For n =1, 2, 3, 4, 5, 6 there are 2, 17, 230, 4895, 222 018, 28927922 space groups respectively [28, 29, 30]. 4 It is not difficult to convince onself that there are no further 1D space groups apart from Z and D∞. Given a fundamental ’pattern’ , there exist only two possible 1D ’wallpapers’, and . → →→→→ →←→← 4 the choice of generator in the definitions is not unique. For example, the space group D∞ can be defined equally well in terms of two π-rotations5, see Fig. 1(c). D∞ Z2 Z2 ≃ ∗ (2.4) = r r2 = 1 r′ r′2 = 1 with r′ tr h | i∗h | i ≡ Let Γ′ Γ be the largest subgroup of Γ that does not include translations. By writing the 1D ⊆ orbifolds as R/Γ= S1/Γ′ (2.5) 1 1 we make contact with the standard notation. The circle S and the interval S /Z2 are the only one-dimensional orbifolds, since the classification of 1D space groups is complete. The notation in the above paragraphs might appear unfamiliar, but the relations in them are well known. Equations (2.3) and (2.4) correspond to the results (3.42) and (3.43) in [4]. Note that the circle in Fig.