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, $KSp^0 = KO^4$.

(…)

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

References

Generalization of the Atiyah-Jänich theorem to (KO-theory and) quaternionic K-theory:

• Michael F. Atiyah, Isadore M. Singer: Index theory for skew-adjoint Fredholm operators, Publications Mathématiques de l’IHÉS 37 (1969) 5-26 [doi:10.1007/BF02684885, numdam:PMIHES_1969__37__5_0, pdf]

• Max Karoubi: Espaces Classifiants en K-Théorie, Trans. Amer. Math. Soc. 147 (1970) 75-115 [doi:10.2307/1995218, jstor:1995218]