Abstract
We prove the bulk-edge correspondence in K-theory for the quantum Hall effect by constructing an unbounded Kasparov module from a short exact sequence that links the bulk and boundary algebras. This approach allows us to represent bulk topological invariants explicitly as a Kasparov product of boundary invariants with the extension class linking the algebras. This paper focuses on the example of the discrete integer quantum Hall effect, though our general method potentially has much wider applications.
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Baaj S., Julg P.: Théorie bivariante de Kasparov et opérateurs non bornés dans les C*-modules hilbertiens. C. R. Acad. Sci. Paris Sér. I Math. 296(21), 875–878 (1983)
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, volume 5 of Mathematical Sciences Research Institute Publications. Cambridge Univ. Press (1998)
Brain, S., Mesland, B., van Suijlekom, W.D.: Gauge theory for spectral triples and the unbounded Kasparov product (2013). arXiv:1306.1951
Carey A.L., Neshveyev A.L., Nest R., Rennie A.: Twisted cyclic theory, equivariant KK-theory and KMS states. J. Reine Angew. Math. 650, 161–191 (2011)
Connes A.: Gravity coupled with matter and foundation of noncommutative geometry. Commun. Math. Phys. 182, 155–176 (1996)
Elbau P., Graf G.M.: Equality of bulk and edge Hall conductance revisited. Commun. Math. Phys. 229(3), 415–432 (2002)
Elgart A., Graf G.M., Schenker J.H.: Equality of the bulk and edge Hall conductances in a mobility gap. Commun. Math. Phys. 259(1), 185–221 (2005)
Kaad J., Lesch M.: Spectral flow and the unbounded Kasparov product. Adv. Math. 298, 495–530 (2013)
Kasparov G.G.: The operator K-functor and extensions of C*-algebras. Math. USSR Izv. 16, 513–572 (1981)
Kellendonk, J., Richter, T., Schulz-Baldes, H.: Simultaneous quantization of edge and bulk Hall conductivity. J. Phys. A Math. Gen. 33(2), L27 (2000)
Kellendonk J., Richter T., Schulz-Baldes H.: Edge current channels and Chern numbers in the integer quantum Hall effect. Rev. Math. Phys. 14(01), 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 an application to the quantum Hall effect. Commun. Math. Phys. 249(3), 611–637 (2004)
Kucerovsky D.: The KK-product of unbounded modules. K-Theory 11, 17–34 (1997)
McCann P., Carey A.L.: A discrete model of the integer quantum Hall effect. Publ. RIMS Kyoto Univ. 32, 117–156 (1996)
Mathai, V., Thiang, G.C.: T-duality trivializes bulk-boundary correspondence (2015). arXiv:1505.05250
Mesland B.: Unbounded bivariant K-Theory and correspondences in noncommutative geometry. J. Reine Angew. Math. 691, 101–172 (2014)
Mesland, B.: Spectral triples and KK-theory: a survey. In: Clay mathematics proceedings, volume 16: topics in noncommutative geometry, pp. 197–212 (2012)
Pask D., Rennie A.: The noncommutative geometry of graph C*-algebras I: the index theorem. J. Funct. Anal. 233(1), 92–134 (2006)
Pimsner M., Voiculescu D.: Exact sequences for K-groups and Ext-groups of certain cross-product C*-algebras. J. Oper. Theory 4(1), 93–118 (1980)
Rennie, A., Robertson, D., Sims, A.: The extension class and KMS states for Cuntz–Pimsner algebras of some bi-Hilbertian bimodules. (2015). arXiv:1501.05363
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Bourne, C., Carey, A.L. & Rennie, A. The Bulk-Edge Correspondence for the Quantum Hall Effect in Kasparov Theory. Lett Math Phys 105, 1253–1273 (2015). https://doi.org/10.1007/s11005-015-0781-y
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DOI: https://doi.org/10.1007/s11005-015-0781-y