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R. M. Kaufmann, D. Li & B. Wehefritz-Kaufmann Cl0,n(R), which has a direct generalization into Clifford bundles. In general, for a pair of spaces (X,Y), given two vector bundles E,F on X and a bundle map σ : E → F such that σ is an isomorphism E|Y ≃ F|Y when restricted to Y, the Atiyah–Bott–Shapiro construction [5] defines an element [(E, F ; σ)] in the relative K-group K(X, Y ) := K ̃ (X/Y ). In [51], Kitaev applied the Atiyah–Bott–Shapiro construction to the Clifford extension problem on Clifford bundles and proposed the K-theory classification scheme for topological insulators. 5.4. Twisted equivariant K-theory An even more general framework has been set up in [31] for studying topological phases of quantum systems with additional symmetries by twisted equivariant K- theory. For a symmetry group G, a homomorphism φ : G → {±1} is defined to keep track of the unitary or anti-unitary symmetry, and a group extension τ, 1→T→Gτ →G→1 is introduced to take care of the phase ambiguity in quantum mechanics. As a topo- logical invariant, the group of reduced topological phases RT P (G, φ, τ ) is defined to be the abelian group of generalized quantum symmetries (G,φ,τ). In this subsec- tion, we will see how to apply twisted equivariant K-theory to classify topological phases closely following [31]. For time reversal invariant topological insulators, the reduced topological phase group can be computed by the shifted KR-group of the Brillouin torus, RTP(G,φ,τ) ≃ KR−4(Td). (5.10) Similarly, if the parity is reversed (i.e. with particle-hole symmetry P and P2 = −1), it can be computed by the equivariant K-group, RTP(G,φ,τ) ≃ KZ2(Td). (5.11) Finally, if T2 = −1 and P2 = −1, then the twisted equivariant K-groups are applied, RTP(G,φ,τ) ≃ KR−4(Td) ≃ KO−4(Td). (5.12) Z2 Z2 For the torus Td, the KR-groups can be computed by the stable splitting of the torus into wedge products of spheres and the twisted Thom isomorphism. Using the reduced KR-theory defined as usual KR(X) ≃ KR(X) ⊕ Z, we have −4 2 −4 3 ×4 KR (T )≃Z2, KR (T )≃(Z2) . Let us look closely at two natural maps appeared in the computation of −43 32 KR (T ). One is the projection map, i.e. three projections pij : T → T , which 1630003-50 Rev. Math. Phys. 2016.28. Downloaded from www.worldscientific.com by PURDUE UNIVERSITY on 08/11/17. For personal use only.PDF Image | Notes on topological insulators
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