The little desert? Some subfactors with index in the interval (5,3+\sqrt{5})
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Morrison, Scott
Peters, Emily
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http://arxiv.org/
Abstract
Progress on classifying small index subfactors has revealed an almost
empty landscape. In this paper we give some evidence that this desert continues up to index 3 + \sqrt{5}. There are two known quantum-group subfactors with index in this interval, and we show that these subfactors are the only way to realize the corresponding principal graphs. One of these subfactors is 1-supertransitive, and we demonstrate that it is the only 1-supertransitive subfactor with index between 5 and 3 +\sqrt{5}. Computer evidence shows that any other subfactor in this interval would need to have rank at least 38. We prove our uniqueness results by showing that there is a unique flat connection on each graph. The result on 1-supertransitive subfactors is proved by an argument using intermediate subfactors, running the `odometer' from the FusionAtlas` Mathematica package and paying careful attention to dimensions.
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Keywords
subfactors, planar algebras, fusion categories
Citation
Morrison, S. & Peters, E. (2012) The little desert? Some subfactors with index in the interval (5,3+\sqrt{5}). arXiv:1205.2742 [math.OA]
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Working/Technical Paper
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Open Access
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