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\begin{document}
\title{Classifying fusion categories}
\author{Edited by }

\urlstub{fusioncat}

\maketitle




\section{General fusion category questions}





\begin{problemblock}

\begin{problem}[.1] Is any integral fusion category unitarizable? \end{problem}




\begin{remark}
\by{Cesar Galindo, Eric Rowell}By a recent paper of Galindo, Hong and Rowell, a slightly stronger statement can be proved for weakly group-theoretical categories: they are "completely unitary."  Since weakly g.-t. categories conjecturally contain all weakly integral fusion categories, the answer is probably "yes."\end{remark}

\end{problemblock}



\begin{problemblock}

\begin{problem}[.2] Is every integral fusion category weakly group theoretical? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.3] Does pseudo-unitary imply unitarizable? \end{problem}




\begin{remark}
\by{Wang}Physicists are really interested in this question. Given a conformal field theory which is not unitary, there is a negative dimension.\end{remark}

\end{problemblock}



\begin{problemblock}

\begin{problem}[.4] Are all fusion categories pivotal? Are all fusion categories spherical? Does it depend on the ground field ($k$ v.s. $\C$)? \end{problem}




\end{problemblock}



\begin{problemblock}
\name{"Kaplansky’s sixth conjecture" for fusion categories}


\begin{problem}[.5] Is $\displaystyle \frac{\text{FPdim}(\mathcal{C})}{\text{FPdim}(X)}$ an algebraic integer for every $X\in\text{Irr}(\mathcal{C})$? \end{problem}




\end{problemblock}



\section{How many fusion categories...}





\begin{problemblock}

\begin{problem}[.1] Can we find all fusion categories with a given smallest simple object (which is not invertible)? \end{problem}




\begin{remark}
\by{Scott Morrison}For example, we know that the smallest possible fusion dimension $1/2(\sqrt{3}+\sqrt{7})$ is realized by the Izumi-Xu-Ostrik fusion category from \cite{arXiv:1004.0665}. What other fusion categories contain an object with this dimension?\end{remark}

\end{problemblock}



\begin{problemblock}

\begin{problem}[.2] Is there an effective version of Ocneanu rigidity? Is there a sub-exponential bound on the number of unitary fusion categories with respect to $N$, the sum of all the fusion multiplicities $N_{i,j}^k$? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.3] How many fusion categories have the same given fusion rules? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.4] What can you say about all fusion categories $\mathcal{C}$ for which 
$\#\{\dim(X)|X\in\mathcal{C}\}=2?$ \end{problem}




\end{problemblock}



\section{Braidings}





\begin{problemblock}
\name{Property F conjecture}
If a fusion category has property F, then can't use braiding alone for a universal quantum computer.

\begin{problem}[.1] Given a braided, weakly integral fusion category, is the image of the braid group finite?

Is braided and weakly integral fusion equivalent to finite image of the braid group?

 \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.2] Is there a physical model which gives infinite image for the braid group? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.3] For unitary theories, we can choose a gauge so the $F$ matrices formed by the $6j$ symbols are unitary, and the braiding matrices are diagonal with respect to a certain basis. Can this happen for some non-unitary fusion category?

Is the unitarity of the $F$ matrices equivalent to unitarity?  \end{problem}




\end{problemblock}



\section{Number theoretic questions}





\begin{problemblock}

\begin{problem}[.1] Is there a sense in which a randomly chosen fusion graph doesn't have cylotomic dimensions?  \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.2] Look at all spoke graphs with $N>0$ arms. Are there finitely many $N$-tuples $(\ell_1,\dots, \ell_N)$ such that the spoke graph with $N$ arms of lengths $\ell_1,\dots, \ell_N$ has cyclotomic norm squared? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[] Is there a positive real number which is a cyclotomic integer and is largest amongst its Galois conjugates, but which is not realized as the dimension of an object in a fusion category? \end{problem}




\begin{remark}
\by{Scott Morrison}The first five such numbers above 2, given in \cite{arxiv:1004.0665}, are all known to be realized.\end{remark}

\end{problemblock}



\section{Objects in fusion categories}





\begin{problemblock}

\begin{problem}[.1] What values can $\dim(X)$ take in $[2,3]$ for $X\in\mathcal{C}$, a braided fusion category? \end{problem}




\begin{remark}These two dimensions come from Cyclotomic integers, fusion categories, and subfactors
Frank Calegari, Scott Morrison and Noah Snyder, Communications in Mathematical Physics Volume 303, Issue 3 (2011), pp. 845-896 \cite{arXiv:1004.0665}.\end{remark}

\end{problemblock}



\begin{problemblock}

\begin{problem}[.2] What are all the $\mathcal{C}$ generated by $X$ with $\text{FPdim}(X)\leq2$, and $X$ not self-dual? Are they group theoretical if $\dim(X)=2$?
 \end{problem}




\end{problemblock}



\begin{problemblock}
\name{A fusion category version of supertransitivity}


\begin{problem}[.3] In a fusion category, is there an upper bound on the $N$ such that $X^{\otimes N}$ is a simple object (where $\dim(X)>1$)? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.4] There are accumulation points from below for $\text{FPdim}(X)$ for an object in a fusion category or $[M\colon N]$ for finite depth subfactors. Are there any accumulation points from above?

 \end{problem}




\begin{remark}
\by{Scott Morrison}Are there non-integer accumulation points?\end{remark}

\end{problemblock}



\section{Frobenius-Schur indicators}





\begin{problemblock}

\begin{problem}[.1] Is there a way to find the Frobenius-Schur exponent of $\mathcal{C}$ without computing $Z(\mathcal{C})$? \end{problem}




\end{problemblock}



\section{Examples}





\begin{problemblock}

\begin{problem}[.1] Classify module categories and Brauer-Picard groups for known examples. \end{problem}




\end{problemblock}



\section{Subfactors}





\begin{problemblock}The supertransitivity with respect to an object $X$ with $\dim(X)>2$ is the largest $N$ such that $\text{Hom}(1,X^{\otimes n})$ is Temperley-Lieb.

Supertransitivity is the analog of transitivity of group actions. 

\begin{problem}[.1] Is there an upper bound on the supertransitivity of a subfactor planar algebra?

 \end{problem}




\begin{remark}
\by{Scott Morrison}Currently the extended Haagerup subfactor holds the record, with $n=7$. The Asaeda-Haagerup subfactor has $n=5$, and otherwise all known examples have $n\leq 4$.\end{remark}

\end{problemblock}



\begin{problemblock}

\begin{problem}[.2] Find a non-number theoretic argument to rule out the rest of the Haagerup family vine.

For example, is there a diagram that evaluates in two different ways? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.3] Is there a polymer theory of principal graphs? What graphs can appear as subgraphs of principal graphs? \end{problem}




\end{problemblock}



\section{Extensions}





\begin{problemblock}

\begin{problem}[.1] Is there an extension theory for fusion categories extended by fusion rings? (For example, near group categories) \end{problem}




\end{problemblock}



\section{Tensor functors}





\begin{problemblock}

\begin{problem}[.1] Describe functors between group-theoretical categories.

What is known for Verlinde categories? (quantum groups at roots of unity) \end{problem}




\end{problemblock}



\section{Fusion categories from subfactors}





\begin{problemblock}

\begin{problem}[.1] Can you classify all algebras in fusion categories with small Frobenius-Perron dimension (e.g., less than $3+\sqrt{3}$)? \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.2] Compute the center of the even half of the Asaeda-Haagerup and extended Haagerup subfactors. \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.2] Is there a conceptual construction of the even half of 4442? \end{problem}




\end{problemblock}



\section{Finite tensor categories}





\begin{problemblock}

\begin{problem}[.1] Suppose $\mathcal{C}$ is a finite tensor category over $\C$ with prime Frobenius-Perron dimension. Is $\mathcal{C}$ fusion?

(Hence, it would be of the form $\text{Vect}(\Z/p,\omega)$. This would be an extension of a result in Hopf algebras.) \end{problem}




\end{problemblock}



\begin{problemblock}

\begin{problem}[.2] How much from modular representations of finite groups can be carried to finite tensor categories? \end{problem}




\end{problemblock}





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