T-duality simplifies bulk–boundary correspondence: the noncommutative case
- 139 Downloads
Abstract
We state and prove a general result establishing that T-duality, or the Connes–Thom isomorphism, simplifies the bulk–boundary correspondence, given by a boundary map in K-theory, in the sense of converting it to a simple geometric restriction map. This settles in the affirmative several earlier conjectures of the authors and provides a clear geometric picture of the correspondence. In particular, our result holds in arbitrary spatial dimension, in both the real and complex cases, and also in the presence of disorder, magnetic fields, and H-flux. These special cases are relevant both to string theory and to the study of the quantum Hall effect and topological insulators with defects in condensed matter physics.
Keywords
T-duality Topological insulators Quantum Hall effect Defects Bulk–boundary correspondence Disorder Magnetic fields H-fluxMathematics Subject Classification
Primary 58B34 Secondary 46L80 53D22 81V70Notes
Acknowledgements
Varghese Mathai and Guo Chuan Thiang were supported by the Australian Research Council via ARC Discovery Project Grants DP150100008, FL170100020, and DE170100149, respectively. The authors thank the Erwin Schrödinger Institute (ESI), Vienna, for its hospitality during the ESI Program on Higher Structures in String Theory and Quantum Field Theory, when part of this research was completed.
References
- 1.Atiyah, M.F., Donnelly, H., Singer, I.M.: Eta invariants, signature defects of cusps, and values of L-functions. Ann. Math. 118, 131–177 (1983)MathSciNetCrossRefMATHGoogle Scholar
- 2.Bellissard, J., van Elst, A., Schulz-Baldes, H.: The noncommutative geometry of the quantum Hall effect. J. Math. Phys. 35(10), 5373–5451 (1994)ADSMathSciNetCrossRefMATHGoogle Scholar
- 3.Bellissard, J.: \(K\)-theory of \(C^*\)-algebras in solid state physics. In: Statistical Mechanics and Field Theory: Mathematical Aspects (Groningen, 1985), Lecture Notes in Physics, vol. 257, pp. 99–156. Springer, Berlin (1986)Google Scholar
- 4.Bouwknegt, P., Evslin, J., Mathai, V.: T-duality: topology change from \(H\)-flux. Commun. Math. Phys. 249, 383–415 (2004)ADSMathSciNetCrossRefMATHGoogle Scholar
- 5.Bramwell, S.T., et al.: Measurement of the charge and current of magnetic monopoles in spin ice. Nature 461, 956–959 (2009)ADSCrossRefGoogle Scholar
- 6.Chang, M.-C., Niu, Q.: Berry phase, hyperorbits, and the Hofstadter spectrum: semiclassical dynamics in magnetic Bloch bands. Phys. Rev. B 53(11), 7010–7023 (1996)ADSCrossRefGoogle Scholar
- 7.Connes, A.: An analogue of the Thom isomorphism for crossed products of a \(C^*\)-algebra by an action of \({{\mathbb{R}}}\). Adv. Math. 39, 31–55 (1981)MathSciNetCrossRefMATHGoogle Scholar
- 8.Cuntz, J.: \(K\)-theory and \(C^*\)-algebras. In: Algebraic \(K\)-Theory, Number Theory, Geometry and Analysis, Lecture Notes in Mathematics, vol. 1046, pp. 55–79, Springer, Berlin (1984)Google Scholar
- 9.Cuntz, J., Meyer, R., Rosenberg, J.: Topological and bivariant \(K\)-theory. Birkhäuser, Basel (2007)MATHGoogle Scholar
- 10.Echterhoff, S.: A categorical approach to imprimitivity theorems for \(C^*\)-dynamical systems. Mem. Am. Math. Soc. 805 (2006). arXiv:math/0205322
- 11.Echterhoff, S., Nest, R., Oyono-Oyono, H.: Principal non-commutative torus bundles. Proc. Lond. Math. Soc. 99(3), 1–31 (2009)MathSciNetCrossRefMATHGoogle Scholar
- 12.Echterhoff, S., Williams, D.P.: Locally inner actions on \(C_0(X)\)-algebras. J. Oper. Theory 45, 131–160 (2001)MathSciNetMATHGoogle Scholar
- 13.Fack, T., Skandalis, G.: Connes’ analogue of the Thom isomorphism for the Kasparov groups. Invent. Math. 64(1), 7–14 (1981)ADSMathSciNetCrossRefMATHGoogle Scholar
- 14.Freed, D.S., Moore, G.W.: Twisted equivariant matter. Ann. Henri Poincaré 14(8), 1927–2023 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
- 15.Gawedzki, K.: Bundle gerbes for topological insulators. Banach Center Publications. arXiv:1512.01028 (in press)
- 16.Green, P.: The local structure of twisted covariance algebras. Acta Math. 140, 191–250 (1978)MathSciNetCrossRefMATHGoogle Scholar
- 17.Hannabuss, K.C.: Representations of nilpotent locally compact groups. J. Funct. Anal. 34, 146–165 (1979)MathSciNetCrossRefMATHGoogle Scholar
- 18.Hannabuss, K.C., Mathai, V.: Noncommutative principal torus bundles via parametrised strict deformation quantization. AMS Proc. Symp. Pure Math. 81, 133–148 (2010). arXiv:0911.1886 MathSciNetCrossRefMATHGoogle Scholar
- 19.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 MathSciNetCrossRefMATHGoogle Scholar
- 20.Hannabuss, K.C., Mathai, V., Thiang, G.C.: T-duality simplifies bulk-boundary correspondence: the parametrised case. Adv. Theor. Math. Phys. 20(5), 1193–1226 (2016). arXiv:1510.04785 MathSciNetCrossRefMATHGoogle Scholar
- 21.Kane, C.L., Mele, E.J.: Quantum spin Hall effect in graphene. Phys. Rev. Lett. 95(22), 226801 (2005)ADSCrossRefGoogle Scholar
- 22.Kane, C.L., Mele, E.J.: \({\mathbb{Z}}_2\) topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95(14), 146802 (2005)ADSCrossRefGoogle Scholar
- 23.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)MathSciNetCrossRefMATHGoogle Scholar
- 24.Kitaev, A.: Periodic table for topological insulators and superconductors. AIP Conf. Proc. 1134(1), 22–30 (2009)ADSCrossRefMATHGoogle Scholar
- 25.Kleinert, H.: Gauge Fields in Condensed Matter, vol. 2. World Scientific, Singapore (1989)CrossRefMATHGoogle Scholar
- 26.Kleppner, A.: Multipliers on abelian groups. Math. Ann. 158, 11–34 (1965)MathSciNetCrossRefMATHGoogle Scholar
- 27.Lawson, H., Michelsohn, M.-L.: Spin Geometry, Princeton Mathematical Series, vol. 38. Princeton University Press, Princeton (1989)MATHGoogle Scholar
- 28.Lee, S.T., Packer, J.: Twisted group algebras for two-step nilpotent and generalized discrete Heisenberg groups. J. Oper. Theory 33, 91–124 (1995)MathSciNetMATHGoogle Scholar
- 29.Marcolli, M.: Solvmanifolds and noncommutative tori with real multiplication. Commun. Number Theory Phys. 2(2), 421–476 (2008)MathSciNetCrossRefMATHGoogle Scholar
- 30.Mathai, V., Rosenberg, J.: T-duality for torus bundles via noncommutative topology. Commun. Math. Phys. 253, 705–721 (2005). arXiv:hep-th/0401168 ADSMathSciNetCrossRefMATHGoogle Scholar
- 31.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, 123–158 (2006). arXiv:hep-th/0508084 MathSciNetCrossRefMATHGoogle Scholar
- 32.Mathai, V., Thiang, G.C.: T-duality of topological insulators. J. Phys. A Math. Theor. (Fast Track Commun.) 48(42), 42FT02 (2015). arXiv:1503.01206 MathSciNetCrossRefMATHGoogle Scholar
- 33.Mathai, V., Thiang, G.C.: T-duality simplifies bulk-boundary correspondence. Commun. Math. Phys. 345(2), 675–701 (2016). arXiv:1505.05250 ADSMathSciNetCrossRefMATHGoogle Scholar
- 34.Mathai, V., Thiang, G.C.: T-duality simplifies bulk-boundary correspondence: some higher dimensional cases. Ann. Henri Poincaré 17(12), 3399–3424 (2016). arXiv:1506.04492 ADSMathSciNetCrossRefMATHGoogle Scholar
- 35.Mathai, V., Thiang, G.C.: Differential topology of semimetals. Commun. Math. Phys. 355(2), 561–602 (2017)ADSMathSciNetCrossRefMATHGoogle Scholar
- 36.Packer, J., Raeburn, I.: Twisted crossed products of \(C^*\)-algebras. Math. Proc. Cambridge Philos. Soc. 106, 293–311 (1989)ADSMathSciNetCrossRefMATHGoogle Scholar
- 37.Paschke, W.: On the mapping torus of an automorphism. Proc. Am. Math. Soc. 88, 481–485 (1983)MathSciNetCrossRefMATHGoogle Scholar
- 38.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)MathSciNetMATHGoogle Scholar
- 39.Prodan, E., Schulz-Baldes, H.: Bulk and Boundary Invariants for Complex Topological Insulators: From \(K\)-Theory to Physics. Mathematical Physics Studies. Springer, Cham (2016)CrossRefMATHGoogle Scholar
- 40.Raeburn, I., Rosenberg, J.: Crossed products of continuous-trace \(C^*\) algebras by smooth actions. Trans. Am. Math. Soc. 305, 1–45 (1988)MathSciNetMATHGoogle Scholar
- 41.Raeburn, I., Williams, D.: Morita Equivalence and Continuous-Trace \(C^*\)-Algebras. Mathematical Surveys and Monographs. American Mathematical Society, Providence (1998)MATHGoogle Scholar
- 42.Ran, Y., Zhang, Y., Vishwanath, A.: One-dimensional topologically protected modes in topological insulators with lattice dislocations. Nat. Phys. 5, 298–303 (2009)CrossRefGoogle Scholar
- 43.Ray, M.W., Ruokokoski, E., Kandel, S., Möttönen, M., Hall, D.S.: Observation of Dirac monopoles in a synthetic magnetic field. Nature 505, 657–660 (2014)ADSCrossRefGoogle Scholar
- 44.Rieffel, M.A.: Connes’ analogue for crossed products of the Thom isomorphism. Contemp. Math. 10, 143–154 (1981)MathSciNetCrossRefMATHGoogle Scholar
- 45.Rieffel, M.A.: Strong Morita equivalence of certain transformation group \(C^*\)-algebras. Math. Ann. 222(1), 7–22 (1976)MathSciNetCrossRefMATHGoogle Scholar
- 46.Rørdam, M., Larsen, M., Laustsen, M.: An Introduction to \(K\)-theory for \(C^*\)-algebras. London. Math. Soc. Student Texts 19. Cambridge Univ. Press, Cambridge (2000)Google Scholar
- 47.Rosenberg, J.: Some results on cohomology with Borel cochains, with applications to group actions on operator algebras. Oper. Theory Adv. Appl. 17, 301–330 (1986)MathSciNetMATHGoogle Scholar
- 48.Rosenberg, J.: \(C^*\)-algebras, positive scalar curvature, and the Novikov conjecture–III. Topology 25(3), 319–336 (1986)MathSciNetCrossRefMATHGoogle Scholar
- 49.Schröder, H.: \(K\)-Theory for Real \(C^*\)-Algebra and Applications. Pitman Research Notes in Mathemathical Series. Longman, Harlow (1993)Google Scholar
- 50.Scott, P.: The geometries of 3-manifolds. Bull. Lond. Math. Soc. 15(5), 401–487 (1983)MathSciNetCrossRefMATHGoogle Scholar
- 51.Takai, H.: On a duality for crossed product algebras. J. Funct. Anal. 19, 25–39 (1975)MathSciNetCrossRefMATHGoogle Scholar
- 52.Thiang, G.C.: On the \(K\)-theoretic classification of topological phases of matter. Ann. Henri Poincaré 17(4), 757–794 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
- 53.Wegge-Olsen, N.E.: \(K\)-Theory and \(C^*\)-Algebras. Oxford University Press, Oxford (1993)MATHGoogle Scholar
- 54.Wu, Y.-S., Zee, A.: Cocycles and magnetic monopoles. Phys. Lett. B 152, 98–102 (1985)ADSMathSciNetCrossRefGoogle Scholar