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Recall that the compact symplectic group Sp(n) is defined as the quaternionic unitary group, M∈Sp(n)=U(n,H), i.e.MM†=M†M=1 where (mij)† = ( mji) is the combination of taking the matrix transpose and the quaternionic conjugate. In particular, one has Sp(1) = SU(2). The Berry con- nection, which has the local form an(k) = ⟨Ψn(k)|d|Ψn(k)⟩, satisfies the identity an + J−1atnJ = 0, i.e. an ∈ sp(1). Remark 1. Instead of (3.24) the matrix coefficients of Θ as a morphism H → H ̄ are ω(φ, ψ) = ⟨φ, Θψ⟩ = ⟨φ|Θ|ψ⟩. This is actually an anti-linear 2-form and a section of ∧2(B ̄)∗. It is related to ωΘ by ω(φ, ψ) = −ωΘ(φ, ψ). (3.25) 3.3.2. Quaternionic K-theory The extra structure that Θ adds to the bundle has actually been studied by math- ematicians and is called a Quaternionic structure [28]. In this setup one considers a locally compact topological space X together with an involution ı. The relevant example here is the time reversal symmetry acting on the Brillouin torus. A Quater- nionic bundle on such a space is a bundle B together with an anti-linear morphism χ, which is compatible with ı and satisfies χ2 = −1. In our case χ = Θ and the Quaternionic bundle is simply (B, Θ). If instead χ2 = 1, then the pair is called a Real bundle, which was first considered by Atiyah [4]. The capital “Q” and “R” indicate that the morphism χ does not necessarily act as a bundle isomorphism, but rather interchanges the fibers over x and ı(x). If the involution χ is trivial, one obtains quaternionic and real bundles. Looking at classes of bundles one obtains the respective K-theories. Complex K = KU, real KO, quaternionic or symplectic KSp, Real KR and Quaternionic KH (or KQ). These are alternatively defined using homotopy theory, the Grothendieck construction or stable equivalence [4], see below. As we consider the case with Θ2 = −1, we will be mainly interested in Quaternionic structures. The others appear when there are other symmetries present, see Sec. 5.1 below. For instance the case with Θ2 = 1 represents particles with integer spin [71]. Denote KH(X) the K-group of Quaternionic vector bundles over an invo- lutive space (X,τ). Sometimes the notation KQ(X,τ) is also used [26]. If the involution τ is trivial, then K H (X ) is reduced to the quaternionic K-theory KSp(X) = KH(X) = KQ(X,idX). Similar to KR-theory, the KH-groups have Notes on topological insulators 1630003-27 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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