Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Fusion Categories Associated to Subfactors with Index $3+\sqrt{5}$   We classify fusion categories which are Morita equivalent to even parts of
subfactors with index $3+\sqrt{5} $, and module categories over these fusion
categories. For the fusion category $\mathcal{C} $ which is the even part of
the self-dual $3^{\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} } $
subfactor, we show that there are $30$ simple module categories over $
\mathcal{C}$; there are no other fusion categories in the Morita equivalence
class; and the order of the Brauer-Picard group is $360$. The proof proceeds
indirectly by first describing the Brauer-Picard groupoid of a $
\mathbb{Z}/3\mathbb{Z} $-equivariantization
$\mathcal{C}^{\mathbb{Z}/3\mathbb{Z} } $ (which is the even part of the $4442$
subfactor). We show that that there are exactly three other fusion categories
in the Morita equivalence class of $\mathcal{C}^{\mathbb{Z}/3\mathbb{Z} } $,
which are all $ \mathbb{Z}/3\mathbb{Z} $-graded extensions of $\mathcal{C} $.
Each of these fusion categories admits $20$ simple module categories, and their
Brauer-Picard group is $\mathcal{S}_3 $. We also show that there are exactly
five fusion categories in the Morita equivalence class of the even parts of the
$3^{\mathbb{Z}/4\mathbb{Z} }$ subfactor; each admits $7$ simple module
categories; and the Brauer-Picard group is $\mathbb{Z}/2\mathbb{Z} $.

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The categories of this paper are: atom-ph
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The categories of this paper are: funct-an
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The categories of this paper are: patt-sol
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The categories of this paper are: cmp-lg
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The categories of this paper are: chao-dyn
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-ex
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The categories of this paper are: nlin
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The categories of this paper are: q-bio
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The categories of this paper are: cs
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The categories of this paper are: comp-gas
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The categories of this paper are: plasm-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: stat
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The categories of this paper are: dg-ga
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The categories of this paper are: alg-geom