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R. M. Kaufmann, D. Li & B. Wehefritz-Kaufmann field theory. From the 3d bulk Chern–Simons (CS) theory, one obtains a 2d topo- logical Wess–Zumino–Witten (WZW) term by the CS/WZW duality. It is shown in [31, 47, 69] that for a 3d topological insulator, the Chern–Simons invariant, i.e. the WZW term, and the Kane–Mele invariant are equivalent, and for this rea- son the topological Z2 invariant will be called the Chern–Simons invariant in this subsection. In string theory, the topological WZW term naturally appears in the bosonization process [73], and its global SU(2) anomaly explains why it is a mod 2 index [72]. Mathematically, the Chern–Simons invariant is computed by the odd topological index, that is, the integral of the odd Chern character of a specific gauge transfor- mation, which is an application of the odd index theorem for Toeplitz operators on compact manifolds. So the topological Z2 invariant is identified as the mod 2 reduc- tion of the odd index theorem in 3d. Or equivalently, the Z2 invariant is the mod 2 version of the Witten index Tr(−1)F, where F counts the number of Majorana zero modes such as Dirac cones. In a modern language, the topological Z2 index is obtained by pairing the K-homology of Majorana zero modes with the K-theory of the Bloch bundle, for details see [47]. 4.2.1. CS/WZW correspondence Zhang et al. [61] first considered a time reversal invariant (4+1)-dimensional model generalizing the (2+1) quantum Hall system, the effective action functional was obtained by linear response theory, (4.10) (4.11) 24π2 where c2 is the second Chern number c S4d = 2 d4xdtεμνρστAμ∂νAρ∂σAτ 1 c2(f) = 32π2 d4kεijkltr(fijfkl) with non-abelian local Berry connection and curvature defined as before, ai=i⟨ψ|∂i|ψ⟩, fij=∂iaj−∂jai+i[ai,aj]. In order to get the effective action functional of a 3d topological insulator, dimensional reduction was applied to the above effective action S4d and the second Chern number as well [61]. The idea of dimensional reduction is to replace one spatial or momental dimension by an adiabatic parameter, and then integrate out this adiabatic parameter carefully. For the external gauge field, one can reduce the dimension by fixing a convenient gauge such as the Landau gauge. As for the second Chern number c2(f), one consid- ers the Chern–Simons class because of its relation with the second Chern character on the level of characteristic classes. Indeed, the Chern–Simons form cs3(a,f) is 1630003-36 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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