Abstract
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.
References
Asaeda M.: Galois groups and an obstruction to principal graphs of subfactors. Internat. J. Math. 18(2), 191–202 (2007)
Asaeda M., Haagerup U.: Exotic subfactors of finite depth with Jones indices \({(5+\sqrt{13})/2}\) and \({(5+\sqrt{17})/2}\) . Commun. Math. Phys. 202(1), 1–63 (1999)
Asaeda M., Yasuda S.: On Haagerup’s list of potential principal graphs of subfactors. Commun. Math. Phys. 286(3), 1141–1157 (2009)
Bakalov, B., Kirillov, A. Jr.: Lectures on tensor categories and modular functors. Volume 21 of University Lecture Series. Providence, RI: Amer. Math. Soc., 2001
Bigelow, S., Morrison, S., Peters, E., Snyder, N.: Constructing the extended Haagerup planar algebra. http://arxiv.org/abs/0909.4099v1 [math.OA], 2009
Bisch D.: Principal graphs of subfactors with small Jones index. Math. Ann. 311(2), 223–231 (1998)
Cassels J.W.S.: On a conjecture of R. M. Robinson about sums of roots of unity. J. Reine Angew. Math. 238, 112–131 (1969)
Conway J.H., Jones A.J.: Trigonometric Diophantine equations (On vanishing sums of roots of unity). Acta Arith. 30(3), 229–240 (1976)
Di Francesco, P., Mathieu, P., Sénéchal, D.: Conformal field theory. Graduate Texts in Contemporary Physics. New York: Springer-Verlag, 1997
Drinfeld, V., Gelaki, S., Nikshych, D., Ostrik, V.: Group-theoretical properties of nilpotent modular categories. http://arxiv.org/abs/0704.0195v2 [math.QA], 2007
Drinfeld, V., Gelaki, S., Nikshych, D., Ostrik, V.: On braided fusion categories I. http://arxiv.org/abs/0906.0620v3 [math.QA], 2010
Etingof, P., Nikshych, D., Ostrik, V.: Weakly group-theoretical and solvable fusion categories. http://arxiv.org/abs/0809.3031v2 [math.QA], 2009
Etingof P., Nikshych D., Ostrik V.: On fusion categories. Ann. of Math. (2) 162(2), 581–642 (2005)
Godsil, C., Royle, G.: Algebraic graph theory, Volume 207 of Graduate Texts in Mathematics. New York: Springer-Verlag, 2001
Gross B.H., Hironaka E., McMullen C.T.: Cyclotomic factors of Coxeter polynomials. J. Number Theory 129(5), 1034–1043 (2009)
Haagerup, U.: Principal graphs of subfactors in the index range \({4<[M:N] <3 +\sqrt2}\). In: Subfactors (Kyuzeso, 1993). River Edge, NJ: World Sci. Publ., 1994, pp. 1–38
Iwaniec H.: On the error term in the linear sieve. Acta Arith. 19, 1–30 (1971)
Izumi M.: The structure of sectors associated with Longo-Rehren inclusions. II. Examples. Rev. Math. Phys. 13(5), 603–674 (2001)
Jacobsthal, E.: Über Sequenzen ganzer Zahlen, von denen keine zu n teilerfremd ist. I, II, III. Norke Vid. Selsk. Forh. Trondheim 33, 117–124, 125–131, 132–139 (1961)
Jones A.J.: Sums of three roots of unity. Proc. Cambridge Philos. Soc. 64, 673–682 (1968)
Jones V.F.R.: Index for subfactors. Invent. Math. 72(1), 1–25 (1983)
Kanold H.-J.: Über eine zahlentheoretische Funktion von Jacobsthal. Math. Ann. 170, 314–326 (1967)
Kirillov A. Jr, Ostrik V.: On a q-analogue of the McKay correspondence and the ADE classification of \({\mathfrak{sl}_2}\) conformal field theories. Adv. Math. 171(2), 183–227 (2002)
Kronecker L.: Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten. J. Reine Angew. Math. 53, 173–175 (1857)
Loxton J.H.: On the maximum modulus of cyclotomic integers. Acta Arith. 22, 69–85 (1972)
McKee J., Smyth C.: Salem numbers, Pisot numbers, Mahler measure, and graphs. Experiment. Math. 14(2), 211–229 (2005)
Ostrik V.: On formal codegrees of fusion categories. Math. Research Letters 16(5), 895–901 (2009)
Poonen B., Rubinstein M.: The number of intersection points made by the diagonals of a regular polygon. SIAM J. Discrete Math. 11(1), 135–156 (1998) (electronic)
Siegel C.L.: The trace of totally positive and real algebraic integers. Ann. of Math. (2) 46, 302–312 (1945)
Smith J.H.: Some properties of the spectrum of a graph. In: Combinatorial Structures and their Applications (Proc. Calgary Internat. Conf., Calgary, Alta., 1969), New York: Gordon and Breach, 1970, pp. 403–406
Smyth C.J.: The mean values of totally real algebraic integers. Math. Comp. 42(166), 663–681 (1984)
Tambara D., Yamagami S.: Tensor categories with fusion rules of self-duality for finite abelian groups. J. Algebra 209(2), 692–707 (1998)
Xu, F.: Unpublished notes, 2001
Acknowledgements
We would like to thank MathOverflow where this collaboration began (see “Number theoretic spectral properties of random graphs” http://mathoverflow.net/questions/5994/). 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.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Calegari, F., Morrison, S. & Snyder, N. Cyclotomic Integers, Fusion Categories, and Subfactors. Commun. Math. Phys. 303, 845–896 (2011). https://doi.org/10.1007/s00220-010-1136-2
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DOI: https://doi.org/10.1007/s00220-010-1136-2
Keywords
- Dynkin Diagram
- Fusion Rule
- Simple Object
- Tensor Category
- Fusion Category