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Square root of gerbe holonomy and invariants of time-reversal-symmetric topological insulators. (English) Zbl 1430.53027
Differential cocycles on a smooth manifolds play an important role in mathematical and physical problems. In particular, the local formula for $$2d$$ Wess-Zumino (WZ) action could be effectively described by using smooth Deligne cocycles and their cohomologies (see [\it K. Gawędzki, Non-perturbative quantum field theory, Plenum Press 101–142 (1988)]). The Feynman amplitude of the WZ action then gets an interpretation of a gerbe holonomy, which is the article under review studies. In author’s lecture notes [K. Gawędzki, Banach Cent. Publ. 114, 145–180 (2018; Zbl 1397.53044)], basics on differential geometry of bundle gerbes and their applications to topological insulators in 2- and 3-dimension as well as Floquet systems were treated.
The article under review fills the gap in literature that the square-root of the gerbe holonomy for the $$3d$$-index is independent of the choice of the simplices, maps, and the orbifold triangulation (Section 9). The technical core of this paper is Section 6 wherein the author considers the holonomy of a $$\mathbb{Z}_2$$-equivariant extension of a gerbe $$\mathcal{G}$$ over a smooth manifold endowed with an involution, and by using a local formula, he constructs a square-root and it is independent of the choice. In Section 7, he provides a non-local formula (Proposition 4), and in Section 11, the author considers Hamiltonian crystal lattices, electronic time-reversal-symmetric insulators, and their Floquet systems and states formulae relating a square-root of holonomy and the (strong) Fu-Kane-Mele invariant.
The paper is overall self-contained and easy to read.

##### MSC:
 53C08 Differential geometric aspects of gerbes and differential characters 53C29 Issues of holonomy in differential geometry
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