Cyclotomic integers, fusion categories, and subfactors

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 then 5.

47 pages, with an appendix by Victor Ostrik

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Cited by in corpus (24)

• The classification of subfactors of index at most 5
• Subfactors of index less than 5, part 2: triple points
• Subfactors of index less than 5, part 3: quadruple points
• Higher Gauss sums of modular categories
• Subfactors of index less than 5, part 1: the principal graph odometer
• Cyclic extensions of fusion categories via the Brauer-Picard groupoid
• A Remark on CFT Realization of Quantum Doubles of Subfactors. Case Index < 4
• Pivotal fusion categories of rank 3 (with an Appendix written jointly with Dmitri Nikshych)
• The little desert? Some subfactors with index in the interval (5,3+\sqrt{5})
• The F-Symbols for the H3 Fusion Category
• Constructing spoke subfactors using the jellyfish algorithm
• Subfactors of index less than 5, part 4: vines
• On braided near-group categories
• A Construction of the "2221" Planar Algebra
• Quantum Cluster State Model with Haagerup Fusion Category Symmetry
• Quantum fields for unitary representations of Thompson's groups F and T
• Spectral Measures for $G_2$
• Lifting shadings on symmetrically self-dual subfactor planar algebras
• An obstruction to subfactor principal graphs from the graph planar algebra embedding theorem
• Subfactors of index exactly 5
• Quadratic $d$-numbers
• Arithmetic of the canonical component of the Knot $7_4$
• On the magnitudes of some small cyclotomic integers
• Cyclotomic Points and Algebraic Properties of Polygon Diagonals