Discrete remnants of orbifolding

We revisit the residual symmetries that survive the orbifold projections, and find additional transformations that have been overlooked in the past. Some of these transformations are outer automorphisms of the downstairs continuous symmetry group. Examples for these transformations include the left-right parity of the Pati-Salam model and its left-right symmetric subgroup.


Introduction
Gauge symmetry breaking via orbifolding [1,2,3] is a popular alternative to spontaneous breakdown of gauge symmetry in four dimensions. This has many reasons, including the observation that the infamous doublet-triplet splitting problem has a simple solution [4,5,6,7,8,9,10]. The low-energy continuous gauge symmetry in these models is well studied [2,9]. The main purpose of this Letter is to point out that there are additional discrete symmetries that have not been identified, nor discussed, in this context thus far.
This Letter is organized as follows. In Section 2 we review some basic facts on orbifolding. In Section 3 we revisit the conditions for residual symmetries and shall show that in the past some symmetries were missed. We illustrate this important fact by a few examples in Section 4, i.e. we give one example where a higher-dimensional SO (10) GUT is broken by an orbifold to Pati-Salam including left-right parity (a.k.a. D-parity). In addition, we present two examples which could be of relevance for flavor model-building from orbifold GUTs. Finally, Section 5 contains our summary. Some details are deferred to the appendices.

Orbifold GUT breaking
Let us collect some basic facts on orbifolding. For the sake of definiteness we consider sixdimensional settings in which two dimensions get compactified, but our findings do not depend on the number of dimensions. Consider a six-dimensional Yang-Mills theory with upstairs gauge group G, where we denote the generators of the Lie algebra in the Cartan-Weyl basis H I and E w collectively by T (CW) a . In a first step, this theory is compactified on a two-torus Ì 2 defined by the lattice vectors e 1 and e 2 , see Appendix A for more details. We can choose the torus-lattice such that it exhibits a N rotational symmetry ϑ with ϑ N = ½, where for N = 3, 4, 6 (i.e. the allowed orders N = 2 of the wall-paper groups in two dimensions) we set ϑ e 1 = e 2 , while in the case N = 2 the basis vectors e 1 and e 2 have to be linear independent. In order to orbifold the two-torus Ì 2 to a Ì 2 / N orbifold we mod out this N symmetry, i.e. we identify points y on Ì 2 which are related by a (360/N ) • rotation, Note that under this geometrical action our six-dimensional fields transform as where the χ fields transform as the internal components of a 6D vector V M (x, y) of sixdimensional Lorentz symmetry. Moreover, the N orbifold can be extended from its pure geometric action Equation (1) to include a discrete N transformation from the gauge symmetry G, i.e.
where P ∈ G acts as a discrete gauge transformation 1 , see Equation (49) with U = P = constant.
Since we restrict ourselves to Abelian orbifolds, we can choose the Cartan generators H I of G such that P can be expanded as where the vector V is "quantized" such that P N = ½.
Orbifold conditions. Next, in addition to the torus boundary conditions (48), we impose orbifold boundary conditions Using where w denotes the root vector of E w , we obtain

Residual gauge symmetries
We consider the possibility of unbroken discrete symmetries from G. In this case, a symmetry transformation from G remains unbroken if it commutes with the orbifold boundary condition (5), i.e.
for a global, unbroken transformation U ∈ G, see Equation (49). Consequently, we obtain the condition Due to Schur's lemma, it follows that P −1 U −1 P U ∝ ½. Furthermore, P is of order N (i.e. P N = ½) yielding our main condition for unbroken symmetries after orbifolding where ω = e 2πi N and we use the definition of the (grouptheoretical) commutator of two group elements (as opposed to Lie algebra elements), [A, B] = A −1 B −1 A B [11]. Since P, U ∈ G also [P, U ] must be from G. Moreover, [P, U ] ∝ ½. Thus, [P, U ] must be from the center of G, i.e.
This constrains the allowed values of k. For example, the center of SU(M ) is M , while ω is of order N . That is, these additional residual symmetries require the order of the orbifold twist and the dimension of the group center to be not coprime.

Unbroken continuous gauge symmetries
There are two related ways to identify the unbroken gauge symmetries after orbifolding. First, as is well known, the unbroken gauge interactions are mediated by the zero-modes of the gauge bosons. These are the modes with trivial boundary conditions Equation (7). Thus, the gauge bosons V µ I (x, y) and V µ w (x, y), which are associated to the Cartan generators H I and to those roots w for which V ·w ∈ , have trivial boundary conditions and hence massless modes in four dimensions.
Second, we can use our main condition (10) to identify the unbroken continuous symmetries [9]. The unbroken continuous symmetries are continuously connected to the identity U = ½.
Hence, we have to set k = 0 in Equation (10) and In this way, Equation (10) yields the condition for a generator of the unbroken gauge symmetry Since the boundary condition P is expanded in terms of the Cartan generators H I , Equation (4), we can use Equation (6) to confirm that the Cartan generators H I and the generators E w with V · w ∈ satisfy Equation (12), i.e. they remain unbroken after orbifolding.

Unbroken discrete gauge symmetries
In addition to the unbroken continuous gauge symmetries, our main condition (10) can have additional solutions which then lead to further discrete remnants from the higher-dimensional gauge symmetry G. Importantly, these discrete symmetries can originate from our main condition (10) either for k = 0 (see the example in Section 4.1) or for k = 0 (see the examples in Section 4.2).

Examples and applications
In this section, we illustrate our general findings in a few examples.

Gauge origin of D-parity and left-right parity
The Pati-Salam model [12]can have, in addition to the continuous gauge group a 2 symmetry D that exchanges the SU(2) factors and acts on SU(4) representations as complex conjugation. This symmetry is part of the SO(10) supergroup containing G PS , and can be preserved in 4D models of grand unification if one breaks SO(10) by giving a VEV to a 54-plet [13,14]. At the level of the left-right symmetric subgroup of G PS , this 2 is the well-known left-right parity [15]. That is, the symmetries of these settings are The purpose of the following discussion is to show that this 2 factor is a residual symmetry of the corresponding orbifold GUT, which to our knowledge has not been pointed out before. To this end, consider a theory with SO(10) symmetry in higher dimensions compactified on a 2 orbifold such as Ë 1 / 2 or Ì 2 / 2 . We choose the GUT breaking boundary condition As is well known, the continuous low-energy gauge symmetry is G PS [10]. However, there is an additional 2 symmetry. In more detail, our main condition (10) yields and we search for the unbroken elements U (k) ∈ SO(10). For k = 0 condition (16) reads The most general SO(10) matrix satisfying this condition reads Consequently, we find the conditions Hence, while Let us remark that setting k = 1 in our main condition (16) does not yield further unbroken symmetries.
Here, we write D in this suggestive way because this will make it very obvious how this 2 acts. We could have represented it by any diagonal matrix with entries ±1 subject to the condition that the number of −1 on either sides of the semicolon is odd. How does this 2 act on representations? Consider first the SO(4) subblock. There, the transformation D can be understood by analogy to parity acting on spinors ( 1 /2, 0) ⊕ (0, 1 /2) of SU(2) × SU(2) in 4D Euclidean space-time: parity interchanges the SU (2) representations. Translated to Pati-Salam, D acts on (r L , r R ) of SU(2) L × SU(2) R as Altogether a representation (r 4 , r L , r R ) of SU(4) × SU(2) L × SU(2) R transforms under D as So this 2 exchanges (4, 2, 1) and (4, 1, 2), i.e. the left-and right-handed fermions of the standard model. That is, this simple orbifold GUT gives rise to the well-known left-right parity [15], where it originates from SO(10) and is hence clearly a discrete gauge symmetry. Ironically, the representation of its generator (23) supports the naming in [15], where this transformation has been called parity. Even though it is not the ordinary space-time transformation that gets broken spontaneously there, as the left-right symmetric model is chiral and even in its unbroken phase does not preserve parity, this transformation does act on the SO(6) ≃ SU(4) and SO(4) ≃ SU(2) L × SU(2) R representations in an analogous way as space-time parity does. Altogether we have found that the breaking pattern of the SO(10) orbifold GUT is where the 2 corresponds to the left-right parity and is in particular a nontrivial outer automorphism of G PS . It is amusing to see that the same mechanism that breaks the gauge symmetry and provides us with a solution to the doublet-triplet splitting problem automatically leads to this symmetry. This parity has a simple geometric interpretation in terms of root lattices, which already can be obtained from a lower-dimensional example. Consider the breaking of SO(5) to SO(4) with a twist P 5 = diag(1, −1, −1, −1, −1) ∈ SO(5). This breaking removes a simple root from the root lattice (see Figure 1), and the simple roots of su(2) L ⊕ su(2) R span a sublattice of the original so(5) lattice. However, the Weyl reflection w.r.t. the plane orthogonal to the "broken" root α is a symmetry of the su(2) L ⊕ su(2) R sublattice, and exchanges (the generators of) the su(2) algebras.
The analogous statement holds in the full Pati-Salam example, but depicting the transformation D as a Weyl reflection is more difficult since the rank of so(10) is 5. As we shall see, the residual transformations in the examples in Section 4.2 can also be related to elements of the Weyl group.
Discussing the phenomenological implications of this symmetry is beyond the scope of this work, we only note the revived interest in this transformation in [16] and references therein.

Non-Abelian residual symmetries
In what follows, we present two examples in which the higher-dimensional gauge group gets broken by the orbifold to a semi-direct product of an Abelian gauge symmetry with a discrete N factor. Such symmetries naturally contain non-Abelian discrete groups that can be used as flavor symmetries.
The associated gauge embedding P is chosen as Then, the unbroken symmetry is given by those U (k) ∈ SU(2) that satisfy Since P, U (k) ∈ SU(2), the right-hand side of Equation (29)  To find all solutions of Equation (29) we parameterize a general element U (k) ∈ SU(2) using p, q ∈ as Then, Equation (29) reads which is equivalent to Now, since |p| 2 − |q| 2 ∈ Ê we see explicitly that Equation (31) has no solutions for k ∈ {1, 3}.
w : In summary, the six-dimensional SU(2) gauge symmetry is broken by this 4 orbifold according to Let us remark that this unbroken symmetry after orbifolding contains, for example, the binary dihedral groups Q N with N = even as subgroups [17], including the quaternion group for N = 4.

Ì 2 / 3 Orbifold GUT
Next, we choose a six-dimensional gauge symmetry G = SU(3) and |e 1 | = |e 2 | with e 1 · e 2 = − |e 1 | 2 /2. This lattice has a 3 rotational symmetry ϑ that we divide out in order to construct a Ì 2 / 3 orbifold. The associated gauge embedding P is chosen as where ω = exp 2πi /3. Then, the unbroken symmetry is given by those U (k) ∈ SU(3) that satisfy Since P, U (k) ∈ SU(3), the right-hand side of Equation (40) and two discrete transformations where U (2) = (U (1) ) 2 . Since (U (1) ) 3 = ½, U (1) generates an unbroken 3 . Consequently, the six-dimensional SU(3) gauge symmetry is broken by the 3 orbifold according to (cf. [18,19]) Again, the 3 can be understood as a remnant of the Weyl group: if we denote the Weyl reflection w.r.t. the root α by w α , conjugating with U (1) has the same action on the generators as the Weyl transformation w α (1) w α (2) , where α (I) , I = 1, 2, denote the simple roots of SU(3). The U(1) factors emerge from the standard gauge symmetry breaking by orbifold boundary conditions to the commuting subgroup, see for example [9,Equation (6)]. However, to our knowledge, there is no systematic way in the previous literature how to derive the (non-commuting) 3 factor. We also note that if one breaks the U(1) factors down to 3 symmetries, this leaves us with ( 3 × 3 ) ⋊ 3 , which is known as ∆(27) and has been proposed as a flavor symmetry.

Summary
We discussed how gauged discrete symmetries emerge from orbifolds. Although we used the field-theoretic constructions the discussion is purely group-theoretical and applies to stringderived orbifolds as well. We identify residual discrete symmetries. These include the so-called left-right parity of the Pati-Salam model or its left-right symmetric subgroup, which, to the best of our knowledge, have been overlooked in the literature so far. These symmetries are inner automorphisms of the upstairs symmetry group but outer automorphisms of the orbifolded setup.
Notably, we find that these symmetries do not have to commute with the orbifold twist. Rather, the transformations U have to fulfill the weaker condition where P is the orbifold twist and Z(G) the center of the group G. In accordance with the usual expectations, all these symmetries are gauged, i.e. local.

B D-parity in Pati-Salam from orbifolding
In this appendix, we give an explicit example how one can compute the action of a residual symmetry transformation on the unbroken gauge symmetry. To do so, we consider D-parity from the Pati-Salam example Section 4.1 and work out the consequences of this 2 on SO(4). The so(4) algebra is generated by six antisymmetric matrices that fulfill An explicit representation can be chosen as These generators can be "disentangled" by making a basis change W ± i := 1 2 (M i ± N i ), for i = 1, 2, 3, such that we arrive at the relations