Skip to main content Accessibility help
×
Home

Spectral Flow Argument Localizing an Odd Index Pairing

  • Terry A. Loring (a1) and Hermann Schulz-Baldes (a2)

Abstract

An odd Fredholm module for a given invertible operator on a Hilbert space is specified by an unbounded so-called Dirac operator with compact resolvent and bounded commutator with the given invertible. Associated with this is an index pairing in terms of a Fredholm operator with Noether index. Here it is shown by a spectral flow argument how this index can be calculated as the signature of a finite dimensional matrix called the spectral localizer.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Spectral Flow Argument Localizing an Odd Index Pairing
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Spectral Flow Argument Localizing an Odd Index Pairing
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Spectral Flow Argument Localizing an Odd Index Pairing
      Available formats
      ×

Copyright

Footnotes

Hide All

The authors thank the Simons Foundation (CGM 419432), the NSF (DMS 1700102), and the DFG (SCHU 1358/3-4) for financial support.

Footnotes

References

Hide All
[1] Atiyah, M. F., Patodi, V. K., and Singer, I. M., Spectral asymmetry and Riemannian geometry. III . Math. Proc. Cambridge Philos. Soc. 79(1976), 7199. https://doi.org/10.1017/S0305004100052105.
[2] Benameur, M.-T., Carey, A. L., Phillips, J., Rennie, A., Sukochev, F. A., and Wojciechowski, K. P., An analytic approach to spectral flow in von Neumann algebras . In: Analysis, geometry and topology of elliptic operators, World Scientific, Hackensack, NJ, 2006, pp. 297352.
[3] Carey, A. L. and Phillips, J., Spectral flow in Fredholm modules, eta invariants and the JLO cocycle . K-Theory 31(2004), 135194. https://doi.org/10.1023/B:KTHE.0000022922.68170.61.
[4] Connes, A., Noncommutative geometry. Academic Press, San Diego, CA, 1994.
[5] De Nittis, G. and Schulz-Baldes, H., Spectral flows of dilations of Fredholm operators . Canad. Math. Bull. 58(2015), 5168. https://doi.org/10.4153/CMB-2014-055-3.
[6] Gracia-Bondía, J. M., Várilly, J. C., and Figueroa, H., Elements of noncommutative geometry . Birkhäuser Advanced Texts. Birkhäuser Boston, Boston, MA, 2013.
[7] Grossmann, J. and Schulz-Baldes, H., Index pairings in presence of symmetries with applications to topological insulators . Comm. Math. Phys. 343(2016), 477513. https://doi.org/10.1007/s00220-015-2530-6.
[8] Loring, T. A., K-theory and pseudospectra for topological insulators . Ann. Physics 356(2015), 383416. https://doi.org/10.1016/j.aop.2015.02.031.
[9] Loring, T. A. and Schulz-Baldes, H., Finite volume calculations of K-theory invariants . New York J. Math. 22(2017), 11111140.
[10] Phillips, J., Self-adjoint Fredholm operators and spectral flow . Canad. Math. Bull. 39(1996), 460467. https://doi.org/10.4153/CMB-1996-054-4.
[11] Phillips, J., Spectral flow in type I and type II factors—a new approach. In: Cyclic cohomology and noncommutative geometry, Fields Inst. Commun., 17, American Mathematical Society, Proidence, RI, pp. 137–153.
[12] Prodan, E. and Schulz-Baldes, H., Bulk and boundary invariants for complex topological insulators: From K-theory to physics . Springer Int. Pub., Szwitzerland, 2016. https://doi.org/10.1007/978-3-319-29351-6.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed