Communications in Mathematical Physics

, Volume 303, Issue 3, pp 845–896 | Cite as

Cyclotomic Integers, Fusion Categories, and Subfactors

  • Frank Calegari
  • Scott Morrison
  • Noah SnyderEmail author
Open Access


Dimensions of objects in fusion categories are cyclotomic integers, hence number theoretic results have implications in the study of fusion categories and finite depth subfactors. We give two such applications. The first application is determining a complete list of numbers in the interval (2, 76/33) which can occur as the Frobenius-Perron dimension of an object in a fusion category. The smallest number on this list is realized in a new fusion category which is constructed in the Appendix written by V. Ostrik, while the others are all realized by known examples. The second application proves that in any family of graphs obtained by adding a 2-valent tree to a fixed graph, either only finitely many graphs are principal graphs of subfactors or the family consists of the A n or D n Dynkin diagrams. This result is effective, and we apply it to several families arising in the classification of subfactors of index less than 5.


Dynkin Diagram Fusion Rule Simple Object Tensor Category Fusion Category 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We would like to thank MathOverflow where this collaboration began (see “Number theoretic spectral properties of random graphs” We would also like to thank Feng Xu for helpful conversations, and Victor Ostrik for writing the Appendix. Frank Calegari was supported by NSF Career Grant DMS-0846285, NSF Grant DMS-0701048, and a Sloan Foundation Fellowship, Scott Morrison was at the Miller Institute for Basic Research at UC Berkeley, and Noah Snyder was supported by an NSF Postdoctoral Fellowship.

Open Access

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Copyright information

© The Author(s) 2010

Authors and Affiliations

  1. 1.Department of MathematicsNorthwestern UniversityEvanstonUSA
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeleyUSA
  3. 3.Department of MathematicsColumbia UniversityNew YorkUSA

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