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R. M. Kaufmann, D. Li & B. Wehefritz-Kaufmann Up to now, we considered the 3d Chern–Simons theory and the associated topo- logical WZW term. A similar argument can be applied to the 1d Chern–Simons the- ory, which can be viewed as a boundary theory of a 2d bulk theory. The time rever- sal symmetry also induces a specific gauge transformation, which can be plugged into the 1d WZW term, i.e. the winding number. The mod 2 version of this 1d WZW model can be used to study the continuous model of time reversal invariant Majorana chains. The physical picture behind the topological WZW term is the bosonization process, that is, two Majorana zero modes, such as two Dirac cones, pair together to become an effective composite boson. The topological WZW term as an action functional describes a bosonic theory equivalent to the fermionic theory of Majorana zero modes. The built-in global SU(2) anomaly of the WZW term explains why we have to modulo two, i.e. the parity anomaly, which will be discussed in a later subsection. To sum it up, the CS/WZW duality gives the bulk-edge correspondence in 3d topological insulators. More precisely, the Chern–Simons action of the Berry con- nection describes a 3d topological insulator in the bulk, while the topological WZW term characterizing the edge states is the action of the specific gauge transforma- tion induced by the time reversal symmetry. In other words, the bulk field theory is the 3d Chern–Simons field theory, and the boundary field theory is the 2d Wess– Zumino–Witten model. 4.2.2. Odd Chern character From the last subsection, we know that the Chern–Simons invariant can be defined by the mod 2 WZW term of a specific gauge transformation. It is well-known that the WZW term gives rise to a topological index of Toeplitz operators. In this subsection, we will review the odd Chern character of gauge transformations and its spectral flow following [37], so that the Chern–Simons invariant will be naturally interpreted as the mod 2 spectral flow through odd Chern character. For two connections A0 and A1 on some vector bundle, the relative Chern– Simons form is defined by [37] where cs(A0, A1) := (4.16) 1 0 tr(A ̇teA2t )dt At=(1−t)A0+tA1, A ̇t=A1−A0. There exists a transgression formula connecting the Chern–Simons form and Chern characters, d cs(A0, A1) = ch(A1) − ch(A0) (4.17) where the Chern character is defined as usual, ch(A) = tr(eA2 ). 1630003-38 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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