Square root of gerbe holonomy and invariants of time-reversal-symmetric topological insulators

Abstract

The Feynman amplitudes with the two-dimensional Wess-Zumino action functional have a geometric interpretation as bundle gerbe holonomy. We present details of the construction of a distinguished square root of such holonomy and of a related 3d-index and briefly recall the application of those to the building of topological invariants for time-reversal-symmetric two-and three-dimensional crystals, both static and periodically forced.

Topological insulators and the Kane–Mele invariant: Obstruction and localization theory

“…For the Brillouin torus X 3 = T 3 with X 2 = T 2 0 ⊔ T 2 π , the 2-dimensional localisation formulas KM E − ,τ 3 = exp π i X 2 ρ 2 = hol w * ϕ G basic , X 2 (7.45) express the definition of the 3-dimensional strong Kane-Mele invariant on T 3 as the difference of the 2-dimensional weak Kane-Mele invariants over T 2 0 and T 2 π [29]. This is also pointed out by [33] in the setting of bundle gerbe holonomy, where by using the complicated expression for the holonomy in terms of a triangulation of the surface X 2 the explicit Pfaffian formula was derived by direct, though cumbersome, calculation. Here we have applied the 2-category theory of bundle gerbes and found a remarkably simple computation of the localisation formula for the Kane-Mele invariant over the time-reversal invariant crystal momenta that is independent of all choices involved, at the price of obtaining somewhat less explicit, but more general, formulas.…”

“…Kane-Mele invariants for semimetals have also been discussed in [71]. A variety of mathematical approaches to the Kane-Mele invariant are known: they range from algebraic topology perspectives [27,59] over index theory [19,54] and C * -algebras [56,57] to geometric approaches [31][32][33]. An overview relating various different treatments of the Kane-Mele invariant is provided by [46].…”

'Real' gerbes and Dirac cones of topological insulators

“…We also mention [18], which, in d = 2, 3, regards u p = 1 − P Fermi as an equivariant field T d → U(2n) with involution u → ΘuΘ −1 on the latter (note: this involution happens to coincide with our Eq. ( 7) for u p self-adjoint).…”

Three-dimensional two-band Floquet topological insulator with Z2 index

“…(23) actually defines the winding number of the mapping T 3 × D 2 → U (3)/U (2). (Γ(g)/π) is also known as the Wess-Zumino amplitude, which has been discussed in the context of topological insulators 35 . We emphasize that the Wess-Zumino amplitude is a general topological index with its value taken from Z 2 , so the way one constructs a 3D model does not affect its value associated with the 3D model.…”