nLab quaternionic K-theory

Idea

What is called symplectic K-theory (denoted $K Sp$) or, equivalently, quaternionic K-theory (denoted $K Q$) is the topological K-theory of quaternionic vector bundles.

This is in direct analogy with how complex K-theory (KU) is the topological K-theory of complex vector bundles and KO that of real vector bundles.

In fact, quaternionic K-theory is equivalently KO-theory in degrees which are multiples of 4.

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Applications

To topological phases of matter

In solid state physics, under the K-theory classification of topological phases of matter, quaternionic K-theory is thought to classify crystalline topological insulator-phases of materials whose electron-dynamics respects time-reversal symmetry (this makes the classification be by $KO^\bullet$) but no parity symmetry? (this restricts the classification further to $KO^{4 n} \simeq K Q^n$). Then spin-less such phases are classified by $K Q^0 \simeq K O^0$, while spin-ful such phases are classified by $K Q^1$.

In this context, the Kane-Mele invariant is a projection $K Q^1(\mathbb{T}^d) \to \mathbb{Z}/2$ from the quaternionic K-theory of the Brillouin torus of a material, which detects (in particular, for $d = 2$) the non-triviality of the graphene-phase (which is a time-reversal-symmetric topological insulator when its spin-orbit coupling is taken into account).

Related concepts

• KR-theory, KU-theory, KO-theory

• quaternionic oriented cohomology theory