Some examples Sample Clauses

Some examples. Before going to six dimensions, let us illustrate the above definitions with some easy examples of two-dimensional Z2 orbifolds, taken from Appendix 5.B. Space groups in the same Z-class Consider the affine class Z2-II–1–1, as defined in Appendix 5.B. As there are no roto-translations, the orbifolding group is equal to the point group and is generated by ϑ, a reflection at the horizontal axis. Now, let this reflection act on a lattice, first spanned by the basis vectors e = {e1, e2} and second spanned by f = {f 1, f 2}, see Figure 5.3. The two corresponding space groups read 0 −1 ⟩ f Se = ⟨(ϑ, 0), (1, e1), (1, e2)⟩ with ϑe = 1 0 , (5.21) Sf = ⟨(ϑ, 0), (1, f 1) (1, f 2 0 −1 , (5.22) where ϑe = ϑf because they are given in their corresponding lattice bases. How- ever, it is easy to see that they are related by the GL(2, Z) transformation 0 1 U = 1 1 with U−1 ϑe U = ϑf , (5.23) cf. Equation (5.19). Therefore, they belong to the same Z-class. Hence, as we actually knew from the start, they act on the same lattice and the matrix U just defines the associated change of basis precisely as in Equation (5.4). Space groups in the same Q-class, but different Z-classes Next, consider the space groups, 0 −1 S1–1 = ⟨(ϑ1–1, 0), (1, e1), (1, e2)⟩ with ϑ1–1,e = 1 0 , (5.24) S2–1 = ⟨(ϑ2–1, 0), (1, f 1), (1, f )⟩ with ϑ2–1,f = 0 1 , (5.25) 1 0 2 with lattices spanned by e1 = (1, 0), e2 = (0, 1) and f 1 = (1/2, 1/2), f 2 = (1/2, 1/2), respectively. The first space group belongs to the affine class Z2- II–1–1 and the second one to Z2-II–2–1, see Appendix 5.B. If we try to find the transformation V from Equation (5.20) that fulfills V −1 ϑ1–1,e V = ϑ2–1,f we see that y −y with x, y ∈ Q . (5.26) But for all values of x and y for which V −1 exists, either V or V −1 has non-integer entries. Therefore, the space groups Z2-II–1–1 and Z2-II–2–1 belong to the same Q-class, but to different Z-classes. In other words, these space groups are defined with inequivalent lattices. Indeed, the first space group possesses a primitive rectangular lattice, while the second one has a centered rectangular lattice, as we will see in detail in the following. There is an alternative way of seeing the relationship between the two space groups of the last example: one can amend one of the space groups by an additional translation. In general, this gives rise to a new lattice, and consequently to a different Z-class. In our case, let us take the Z2-II–1–1 affine class and add the non-lattice translation...