Abstract
Recently we introduced T-duality in the study of topological insulators, and used it to show that T-duality transforms the bulk–boundary homomorphism into a simpler restriction map in two dimensions. In this paper, we partially generalize these results to higher dimensions in both the complex and real cases, and briefly discuss the 4D quantum Hall effect.
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Avila J.C., Schulz-Baldes H., Villegas-Blas C.: Topological invariants of edge states for periodic two-dimensional models. Math. Phys. Anal. Geom. 16(2), 137–170 (2013)
Atiyah M.F.: K-Theory. Benjamin, New York (1964)
Bellissard J., van Elst A., Schulz-Baldes H.: The noncommutative geometry of the quantum Hall effect. J. Math. Phys. 35(10), 5373–5451 (1994)
Blackadar B.: K-theory for operator algebras. Mathematical Sciences Research Institute Publications, vol. 5. Cambridge University Press, Cambridge (1998)
Bourne C., Carey A.L., Rennie A.: The bulk-edge correspondence for the Quantum Hall effect in Kasparov theory. Lett. Math. Phys. 105(9), 1253–1273 (2015)
Bouwknegt, P., Evslin, J., Mathai, V.: T-duality: Topology Change from H-flux. Commun. Math. Phys. 249(2), 383–415 (2004). arXiv:hep-th/0306062
Bouwknegt, P., Evslin, J., Mathai, V.: On the Topology and Flux of T-Dual Manifolds. Phys. Rev. Lett. 92(18), 181601 (2004). arXiv:hep-th/0312052
Chang C.-Z. et al.: Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator. Science 340(6129), 167–170 (2013)
Connes A.: An analogue of the Thom isomorphism for crossed products of a C*-algebra by an action of \({\mathbb{R}}\). Adv. Math. 39(1), 31–55 (1981)
Connes A.: Non-commutative differential geometry. Publ. Math. Inst. Hautes Étude Sci. 62(1), 41–144 (1985)
Connes A.: Noncommutative Geometry. Academic Press, San Diego (1994)
Elbau P., Graf G.M.: Equality of bulk and edge Hall conductance revisited. Commun. Math. Phys. 229(3), 415–432 (2002)
Elliott, G.A.: On the K-theory of the C*-algebra generated by a projective representation of a torsion-free discrete abelian group. In: Arsene, G. et. al. (eds.) Operator Algebras and Group Representations I (Neptun, Romania 1980). In: Monographs Stud. Math., vol. 17, pp. 157–184. Pitman, Boston (1984)
Freed D.S., Moore G. W.: Twisted equivariant matter. Ann. Henri Poincaré 14(8), 1927–2023 (2013)
Fu L., Kane C.L., Mele E.J.: Topological insulators in three dimensions. Phys. Rev. Lett. 98(10), 106803 (2007)
Graf G.M., Porta M.: Bulk-edge correspondence for two-dimensional topological insulators. Commun. Math. Phys. 324(3), 851–895 (2013)
Haldane F.D.M.: Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the parity anomaly. Phys. Rev. Lett. 61(18), 2015 (1988)
Hannabuss, K.C., Mathai, V.: Noncommutative principal torus bundles via parametrised strict deformation quantization. Proc. Sympos. Pure Math. 81, 133–148 (2010). arXiv:0911.1886
Hannabuss, K.C., Mathai, V.: Parametrised strict deformation quantization of C*-bundles and Hilbert C*-modules. J. Aust. Math. Soc. 90(1), 25–38 (2011). arXiv:1007.4696
Hannabuss, K., Mathai, V., Thiang, G.C.: T-duality simplifies bulk-boundary correspondence: the parametrised case. arXiv:1510.04785
Hannabuss, K., Mathai, V., Thiang, G.C.: T-duality simplifies bulk-boundary correspondence: the general case. arXiv:1603.00116
Hatcher A.: Algebraic topology. Cambridge University Press, Cambridge (2002)
Hatsugai Y.: Chern number and edge states in the integer quantum Hall effect. Phys. Rev. Lett. 71(22), 3697 (1993)
Hori K.: D-branes, T-duality, and index theory. Adv. Theor. Math. Phys. 3, 281–342 (1999)
Hsieh D., Qian D., Wray L., Xia Y., Hor Y. S., Cava R.J., Hasan M.Z.: A topological Dirac insulator in a quantum spin Hall phase. Nature 452(7190), 970–974 (2008)
Jotzu G., Messer M., Desbuquois R., Lebrat M., Uehlinger T., Greif D., Esslinger T.: Experimental realization of the topological Haldane model with ultracold fermions. Nature 515(7526), 237–240 (2014)
Kellendonk J., Richter T., Schulz-Baldes H.: Edge current channels and Chern numbers in the integer quantum Hall effect. Rev. Math. Phys. 14(1), 87–119 (2002)
Kellendonk J., Schulz-Baldes H.: Quantization of edge currents for continuous magnetic operators. J. Funct. Anal. 209(2), 388–413 (2004)
Kellendonk J., Schulz-Baldes H.: Boundary maps for C*-crossed products with \({\mathbb{R}}\) with an application to the quantum Hall effect. Commun. Math. Phys. 249(3), 611–637 (2004)
König M., Wiedmann S., Brüne C., Roth A., Buhmann H., Molenkamp L.W., Qi X.-L., Zhang S.-C.: Quantum spin Hall insulator state in HgTe quantum wells. Science 318(5851), 766–770 (2007)
Kotani M., Schulz-Baldes H., Villegas-Blas C.: Quantization of interface currents. J. Math. Phys. 55(12), 121901 (2014)
Kraus E., Ringel Z., Zilberberg O.: Four-dimensional quantum Hall effect in a two-dimensional quasicrystal. Phys. Rev. Lett. 111(22), 226401 (2013)
Li, D., Kaufmann, R.M., Wehefritz-Kaufmann, B.: Notes on topological insulators. arXiv:1501.02874
Li, D., Kaufmann, R.M., Wehefritz-Kaufmann, B.: Topological insulators and K-theory. arXiv:1510.08001
Marcolli, M., Mathai, V.: Twisted index theory on good orbifolds. II. Fractional quantum numbers. Commun. Math. Phys. 217(1), 55–87 (2001). arXiv:math/9911103
Mathai V.: K-theory of twisted group C*-algebras and positive scalar curvature. Contemp. Math. 231, 203–225 (1999)
Mathai V., Quillen D.: Superconnections, Thom classes, and equivariant differential forms. Topology 25(1), 85–110 (1986)
Mathai, V., Rosenberg, J.: On mysteriously missing T-duals, H-flux and the T-duality group. In: Ge, M.-L., Zhang W. (eds.) Differential Geometry and Physics. Nankai Tracts Math., vol. 10, pp. 350–358. World Scientific Publishing, Hackensack (2006). arXiv:hep-th/0409073
Mathai, V., Rosenberg, J.: T-duality for torus bundles with H-fluxes via noncommutative topology. Commun. Math. Phys. 253(3), 705–721 (2005). arXiv:hep-th/0401168
Mathai, V., Rosenberg, J.: T-duality for torus bundles with H-fluxes via noncommutative topology, II; the high-dimensional case and the T-duality group. Adv. Theor. Math. Phys. 10(1), 123–158 (2006). arXiv:hep-th/0508084
Mathai, V., Thiang, G.C.: T-duality of topological insulators. J. Phys. A Math. Theor. (Fast Track Communications) 48(42), 42FT02 (2015). arXiv:1503.01206
Mathai, V., Thiang, G.C.: T-duality simplifies bulk-boundary correspondence. Commun. Math. Phys. doi:10.1007/s00220-016-2619-6 arXiv:1505.05250 (Published online)
Nest R.: Cyclic cohomology of crossed products with \({\mathbb{Z}}\). J. Funct. Anal. 80(2), 235–283 (1988)
Nest R.: Cyclic cohomology of non-commutative tori. Can. J. Math. 40(5), 1046–1057 (1988)
de Nittis G., Gomi K.: Classification of “Quaternionic” Bloch-bundles. Commun. Math. Phys. 339(1), 1–55 (2015)
Packer J., Raeburn I.: Twisted crossed products of C*-algebras. Math. Proc. Camb. Philos. Soc. 106(2), 293–311 (1989)
Packer J., Raeburn I.: Twisted crossed products of C*-algebras. II. Math. Ann. 287(1), 595–612 (1990)
Pimsner M., Voiculescu D.: Exact sequences for K-groups and EXT-groups of certain cross-product C*-algebras. J. Oper. Theory 4, 93–118 (1980)
Prodan E.: Virtual topological insulators with real quantized physics. Phys. Rev. B 91, 245104 (2015)
Prodan E., Leung B., Bellissard J.: The non-commutative nth-Chern number (\({n \geq 1}\)). J. Phys. A 46(48), 485202 (2013)
Prodan E., Schulz-Baldes H.: Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics. Mathematical Physics Studies. Springer, Switzerland (2016)
Raeburn I., Szymański W.: Cuntz–Krieger algebras of infinite graphs and matrices. Trans. Am. Math. Soc. 356(1), 39–59 (2004)
Rieffel M.A.: Strong Morita equivalence of certain transformation group C*-algebras. Math. Ann. 222(1), 7–22 (1976)
Rieffel M.A.: C*-algebras associated with irrational rotations. Pac. J. Math. 93(2), 415–429 (1981)
Rieffel M.A.: Non-commutative tori—a case study of non-commutative differentiable manifolds. Contemp. Math. 105, 191–211 (1990)
Rieffel, M.A.: Deformation quantization for actions of \({ {\bf R}^d}\). Mem. Am. Math. Soc. 506 Providence, RI (1993)
Rieffel M.A.: Quantization and C*-algebras. Contemp. Math. 167, 67–97 (1994)
Rosenberg J.: Real Baum–Connes assembly and T-duality for torus orientifolds. J. Geom. Phys. 89, 24–31 (2015)
Shubin M.A.: Discrete magnetic Laplacian. Commun. Math. Phys. 164(2), 259–275 (1994)
Sudo T.: K-theory of continuous fields of quantum tori. Nihonkai Math. J. 15(2), 141–152 (2004)
Sunada T.: A discrete analogue of periodic magnetic Schrodinger operators. Contemp. Math. 173, 283–299 (1994)
Thiang G.C.: On the K-theoretic classification of topological phases of matter. Ann. Henri Poincaré 17(4), 757–794 (2016)
Thiang, G.C.: Topological phases: homotopy, isomorphism and K-theory. Int. J. Geom. Methods Mod. Phys. 12(9), 150098 (2015). arXiv:1412.4191
Zhang S.-C., Hu J.: A four-dimensional generalization of the quantum Hall effect. Science 294(5543), 823–828 (2001)
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Communicated by Jean Bellissard.
This work was supported by the Australian Research Council via ARC Discovery Project Grants DP110100072, DP150100008 and DP130103924.
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Mathai, V., Thiang, G.C. T-Duality Simplifies Bulk–Boundary Correspondence: Some Higher Dimensional Cases. Ann. Henri Poincaré 17, 3399–3424 (2016). https://doi.org/10.1007/s00023-016-0505-6
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DOI: https://doi.org/10.1007/s00023-016-0505-6