Quantitative K-Theory Related to Spin Chern Numbers
We examine the various indices defined on pairs of almost commuting unitary matrices that can detect pairs that are far from commuting pairs. We do this in two symmetry classes, that of general unitary matrices and that of self-dual matrices, with an emphasis on quantitative results. We determine wh...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2014 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2014
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/146609 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Quantitative K-Theory Related to Spin Chern Numbers / T.A. Loring // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 22 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862732773126569984 |
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| author | Loring, T.A. |
| author_facet | Loring, T.A. |
| citation_txt | Quantitative K-Theory Related to Spin Chern Numbers / T.A. Loring // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 22 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We examine the various indices defined on pairs of almost commuting unitary matrices that can detect pairs that are far from commuting pairs. We do this in two symmetry classes, that of general unitary matrices and that of self-dual matrices, with an emphasis on quantitative results. We determine which values of the norm of the commutator guarantee that the indices are defined, where they are equal, and what quantitative results on the distance to a pair with a different index are possible. We validate a method of computing spin Chern numbers that was developed with Hastings and only conjectured to be correct. Specifically, the Pfaffian-Bott index can be computed by the ''log method'' for commutator norms up to a specific constant.
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| first_indexed | 2025-12-07T19:33:57Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-146609 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T19:33:57Z |
| publishDate | 2014 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Loring, T.A. 2019-02-10T09:58:49Z 2019-02-10T09:58:49Z 2014 Quantitative K-Theory Related to Spin Chern Numbers / T.A. Loring // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 22 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 19M05; 46L60; 46L80 DOI:10.3842/SIGMA.2014.077 https://nasplib.isofts.kiev.ua/handle/123456789/146609 We examine the various indices defined on pairs of almost commuting unitary matrices that can detect pairs that are far from commuting pairs. We do this in two symmetry classes, that of general unitary matrices and that of self-dual matrices, with an emphasis on quantitative results. We determine which values of the norm of the commutator guarantee that the indices are defined, where they are equal, and what quantitative results on the distance to a pair with a different index are possible. We validate a method of computing spin Chern numbers that was developed with Hastings and only conjectured to be correct. Specifically, the Pfaffian-Bott index can be computed by the ''log method'' for commutator norms up to a specific constant. This paper is a contribution to the Special Issue on Noncommutative Geometry and Quantum Groups in
 honor of Marc A. Rief fel. The full collection is available at http://www.emis.de/journals/SIGMA/Rieffel.html. 
 The author wishes to thank Matt Hastings and Fredy Vides for discussions, both useful and entertaining.
 Also he wishes to thank Robert Israel and Nick Weaver for help via MathOverflow.
 Finally, thanks are due to the anonymous referees, whose suggestions improved the paper, especially
 Sections 3 and 4. This work was partially supported by a grant from the Simons
 Foundation (208723 to Loring). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Quantitative K-Theory Related to Spin Chern Numbers Article published earlier |
| spellingShingle | Quantitative K-Theory Related to Spin Chern Numbers Loring, T.A. |
| title | Quantitative K-Theory Related to Spin Chern Numbers |
| title_full | Quantitative K-Theory Related to Spin Chern Numbers |
| title_fullStr | Quantitative K-Theory Related to Spin Chern Numbers |
| title_full_unstemmed | Quantitative K-Theory Related to Spin Chern Numbers |
| title_short | Quantitative K-Theory Related to Spin Chern Numbers |
| title_sort | quantitative k-theory related to spin chern numbers |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/146609 |
| work_keys_str_mv | AT loringta quantitativektheoryrelatedtospinchernnumbers |