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R. M. Kaufmann, D. Li & B. Wehefritz-Kaufmann and the 5d topological WZW term Γ(g ̃r) is defined as usual, −i Γ(g ̃ ) = tr(g ̃−1dg ̃ )5 r 240π2 S3×D2 r r which only takes values that are a multiple of π, i.e. Γ(g ̃r) ∈ Z · π. Actually, after integrating out the extra variables θ and ρ, we recover the second Chern number again, 1 24π S3 We stress that the 5d WZW term is only well-defined up to a multiple of 2π because of the dependence of embeddings. In general, the embedding to SU(3) is not unique. Another possible way is to start with the mapping G:S3 ×S1 →SU(2); (x,θ)→r−1(θ)g(x)r(θ)g−1(x) where r(θ) is the diagonal matrix r(θ) = diag(e−iθ/2,eiθ/2). Since π1(SU(2)) = 0 and G(x,0) = G(x,2π) = 1, the map G : S3 ×S1 → SU(2) factors through S4 in this case, analogously to Sec. 3.4. As a result, G falls into two classes according to π4(SU(2)) = Z2. Now extending G as before, we obtain a map G ̃ from D5 into SU(3), G ̃(x, θ, ρ) = g ̃ (x, θ, ρ)g ̃−1(x, 0, ρ). rr It was proved in [34] that the corresponding WZW term is the same as before Γ ( G ̃ ) = Γ ( g ̃ r ) . If we further choose a different 5-disc D′5 that bounds the same S4, and then we consider the difference ΓD5 (G ̃) − ΓD′5 (G ̃) = 2 tr(G ̃−1dG ̃)5. Γ(g ̃r) = tr(g−1dg)3 = πc2. (4.29) −i 240π S5 The right-hand side is a multiple of 2π because of π5(SU(3)) = Z. This explains why we have to modulo 2π to eliminate the embedding ambiguity. Hence the 5d WZW term is well-defined, i.e. independent of the embedding, modulo 2π, and by the relation (4.29) the 3d WZW term is then Z2-valued, ΓS3 (g) ≡ 0, π mod 2π. (4.30) In other words, the mod 2 second Chern number computes the topological Z2 invariant after embedding SU (2) × U (1) ⊂ SU (2) × SU (2) into SU (3). Summarizing the results: the phase ambiguity enlarges S3 to S3 ×S1, which can be replaced by S4 when a special boundary condition is satisfied [34]. With the fixed target space SU(2), more room is made in the parameter space and the topological invariant originally assumed to lie in π3(SU(2)) ≃ Z is now in π4(SU(2)) ≃ Z2. In this language, a strong Z2 topological insulator is characterized by a transition matrix (3.16) with odd winding number. 1630003-42 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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