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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.

53C08 Differential geometric aspects of gerbes and differential characters
53C29 Issues of holonomy in differential geometry
Full Text: DOI
[1] Wess, J.; Zumino, B., Consequences of anomalous Ward identies, Phys. Lett. B, 37, 95-97, (1971)
[2] Witten, E., Non-abelian bosonization in two dimensions, Comm. Math. Phys., 92, 455-472, (1984) · Zbl 0536.58012
[3] Alvarez, O., Topological quantization and cohomology, Comm. Math. Phys., 100, 279-309, (1985) · Zbl 0612.55009
[4] Gawȩdzki, K., Topological actions in two-dimensional quantum field theory, (’t Hooft, G.; Jaffe, A.; Mack, G.; Mitter, P.; Stora, R., Non-Perturbative Quantum Field Theory, (1988), Plenum Press New York, London), 101-142
[5] Deligne, P., Théorie de Hodge : II, Publ. Math. de l’IHÉS, 40, 557, (1971)
[6] Murray, M. K., Bundle gerbes, J. Lond. Math. Soc., 54, 403-416, (1996) · Zbl 0867.55019
[7] Murray, M. K.; Stevenson, D., The basic bundle gerbe on unitary groups, J. Geom. Phys., 58, 1571-1590, (2008) · Zbl 1154.55011
[8] Carey, A. L.; Mickelsson, J.; Murray, M., Bundle gerbes applied to quantum field theory, Rev. Math. Phys., 12, 65-90, (2000) · Zbl 0961.81127
[9] Gawȩdzki, K.; Reis, N., WZW branes and gerbes, Rev. Math. Phys., 14, 1281-1334, (2002) · Zbl 1033.81067
[10] Carpentier, D.; Delplace, P.; Fruchart, M.; Gawȩdzki, K., Topological index for periodically driven time-reversal- invariant 2D systems, Phys. Rev. Lett., 114, 2D, 106806, (2015) · Zbl 1331.82065
[11] Carpentier, D.; Delplace, P.; Fruchart, M.; Gawȩdzki, K.; Tauber, C., Construction and properties of a topological index for periodically driven time-reversal invariant 2D crystals, Nuclear Phys. B, 896, 779-834, (2015) · Zbl 1331.82065
[12] Kane, C. L.; Mele, E. J., \(\mathbb{Z}_2\) topological order and the quantum spin Hall effect, Phys. Rev. Lett., 95, 2, 146802, (2005)
[13] Fu, L.; Kane, C. L., Time reversal polarization and a \(Z_2\) adiabatic spin pump, Phys. Rev. B, 74, 195312, (2006)
[14] Schreiber, U.; Schweigert, C.; Waldorf, K., Unoriented WZW models and holonomy of bundle gerbes, Comm. Math. Phys., 274, 31-64, (2007) · Zbl 1148.53057
[15] Gawȩdzki, K.; Suszek, R. R.; Waldorf, K., Bundle gerbes for orientifold sigma models, Adv. Theor. Math. Phys., 15, 621-688, (2011) · Zbl 1280.81089
[16] Nikolaus, T.; Schweigert, C., Equivariance in higher geometry, Adv. Math., 226, 3367-3408, (2011) · Zbl 1219.22003
[17] K. Gawȩdzki, Bundle gerbes for topological insulators, Banach Center Publications, in press. Preprint arXiv:1512.01028 [math-ph].
[18] Murray, M. K.; Stevenson, D., Bundle gerbes: stable isomorphism and local theory, J. Lond. Math. Soc., 62, 925-937, (2000) · Zbl 1019.55009
[19] Giraud, J., Cohomologie Non-Abélienne, (1971), Springer
[20] Brylinski, J.-L., Loop Spaces, Characteristic Classes and Geometric Quantization, (1993), Birkhauser Boston · Zbl 0823.55002
[21] Pachner, U., P.L. homeomorphic manifolds are equivalent by elementary shellings, European J. Combin., 12, 129-145, (1991) · Zbl 0729.52003
[22] Witten, E., Global aspects of current algebra, Nuclear Phys. B, 223, 422-432, (1983)
[23] Gomi, K., Equivariant smooth Deligne cohomology, Osaka J. Math., 42, 309-337, (2005) · Zbl 1081.14030
[24] Gomi, K., Relationship between equivariant gerbes and gerbes over the quotient space, Commun. Contemp. Math., 7, 207-226, (2005) · Zbl 1096.53016
[25] Gawȩdzki, K.; Suszek, R. R.; Waldorf, K., Global gauge anomalies in two-dimensional bosonic sigma models, Comm. Math. Phys., 302, 513-580, (2011) · Zbl 1213.81167
[26] Ben-Bassat, O., Equivariant gerbes on complex tori, J. Geom. Phys., 64, 209-221, (2013) · Zbl 1264.53034
[27] M.K. Murray, D.M. Roberts, D. Stevenson, R.F. Vozzo, Equivariant bundle gerbes. Preprint arXiv:1506.07931 [math.DG]. · Zbl 1423.55029
[28] Bonahon, F., Geometric structures on 3-manifolds, (Sher, R. B.; Daverman, R. J., Handbook of Geometric Topology, (2002), Elsevier Amsterdam), 114
[29] D. Monaco, C. Tauber, Gauge-theoretic invariants for topological insulators: A bridge between Berry, Wess-Zumino, and Fu-Kane-Mele, Lett. Math. Phys. online first. · Zbl 1370.35093
[30] Fu, L.; Kane, C. L.; Mele, E. J., Topological insulators in three dimensions, Phys. Rev. Lett., 98, 106803, (2007)
[31] Rudner, M. S.; Lindner, N. H.; Berg, E.; Levin, M., Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems, Phys. Rev. X, 3, 031005, (2013)
[32] Nathan, F.; Rudner, M. S., Topological singularities and the general classification of Floquet-Bloch systems, New J. Phys., 17, 125014, (2015)
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