On a necessary condition for unitary categorification of fusion rings
Pavel Etingof, Dmitri Nikshych, Victor Ostrik
备注：
9 pages, latex; the paper is updated with information that the first half of Question 5.8 has been answered (by other authors). Dedication to the memory of Earl J. Taft added
In arXiv:1910.12059 Liu, Palcoux and Wu proved a remarkable necessary condition for a fusion ring to admit a unitary categorification, by constructing invariants of the fusion ring that have to be positive if it is unitarily categorifiable. The main goal of this note is to provide a somewhat more direct proof of this result. In the last subsection we discuss integrality properties of the Liu-Palcoux-Wu invariants.
A fusion category of rank $4$ has either four self-dual objects or exactly two self-dual objects. We study fusion categories of rank $4$ with exactly two self-dual objects, giving nearly a complete classification of those based ring that admit pseudo-unitary categorification. More precisely, we show that if $\mathcal{C}$ is such a fusion category, then its Grothendieck ring $K(\mathcal{C})$ must be one of seven based rings, six of which have know categorifications. In doing so, we classify all based rings associated with near-group categories of the group $\mathbb{Z}/3\mathbb{Z}$.
The integral group rings $\mathbb{Z}G$ for finite groups $G$ are precisely those fusion rings whose basis elements have Frobenius-Perron dimension 1, and each is categorifiable in the sense that it arises as the Grothendieck ring of a fusion category. Here we analyze the structure and representation theory of fusion rings with a basis of elements whose Frobenius-Perron dimensions take exactly one value distinct from 1. Our goal is a set of results which assist in characterizing when such fusion rings are categorifiable. As proof of concept, we completely classify the categorifiable near-group …
In this paper, we investigate quantum Fourier analysis on subfactors and unitary fusion categories. We prove the complete positivity of the comultiplication for subfactors and derive a primary $n$-criterion of unitary categorifcation of multifusion rings. It is stronger than the Schur product criterion when $n\geq3$. The primary criterion could be transformed into various criteria which are easier to check in practice even for noncommutative, high-rank, high-multiplicity, multifusion rings. More importantly, the primary criterion could be localized on a sparse set, so that it works for multifu…
This paper classifies the Grothendieck rings of complex fusion categories of multiplicity one up to rank six. Among 72 possible fusion rings, $25$ ones are filtered out by using categorification criteria. Each of the remaining 47 fusion rings admits a unitary complex categorification. We found 6 new Grothendieck rings, categorified by applying a localization approach of the Pentagon Equation.
The Grothendieck Ring of Strictly Weakly Integral Fusion Categories of Rank 6
College of Mathematical Science, Yangzhou University（扬州大学数学科学学院）；College of Mathematics and Statistics, Nanjing University of Information Science and Technology（南京信息工程大学数学与统计学院）
We show that every unitarizable fusion category, and more generally every semisimple C*-tensor category, admits a unique unitary structure. Our proof is based on a categorified polar decomposition theorem for monoidal equivalences between such categories. We prove analogous results for unitarizable braided fusion categories and module categories.
In this paper we give a complete classification of unitary fusion categories $\otimes$-generated by an object of dimension $\frac{1 + \sqrt{5}}{2}$. We show that all such categories arise as certain wreath products of either the Fibonacci category, or of the dual even part of the $2D2$ subfactor. As a by-product of proving our main classification result we produce a classification of finite unitarizable quotients of $\operatorname{Fib}^{*N}$ satisfying a certain symmetry condition.
A unitary fusion category is called $\mathbb{Z}/2\mathbb{Z}$-quadratic if it has a $\mathbb{Z}/2\mathbb{Z}$ group of invertible objects and one other orbit of simple objects under the action of this group. We give a complete classification of $\mathbb{Z}/2\mathbb{Z}$-quadratic unitary fusion categories. The main tools for this classification are skein theory, a generalization of Ostrik's results on formal codegrees to analyze the induction of the group elements to the center, and a computation similar to Larson's rank-finiteness bound for $\mathbb{Z}/3\mathbb{Z}$-near group pseudounitary fusio…
We realise non-unitary fusion categories using subfactor-like methods, and compute their quantum doubles and modular data. For concreteness we focus on generalising the Haagerup-Izumi family of Q-systems. For example, we construct endomorphism realisations of the (non-unitary) Yang-Lee model, and non-unitary analogues of one of the even subsystems of the Haagerup subfactor and of the Grossman-Snyder system. We supplement Izumi's equations for identifying the half-braidings, which were incomplete even in his Q-system setting. We conjecture a remarkably simple form for the modular S and T matric…
In this note, we discuss the notion of symmetric self-duality of shaded planar algebras, which allows us to lift shadings on subfactor planar algebras to obtain Z/2Z-graded unitary fusion categories. This finishes the proof that there are unitary fusion categories with fusion graphs 4442 and 3333.
Unitary fusion categories (UFCs) have gained increased attention due to emerging connections with quantum physics. We consider a fusion rule of the form $q\otimes q \cong \mathbf{1}\oplus\bigoplus^k_{i=1}x_{i}$ in a UFC $\mathcal{C}$, and extract information using the graphical calculus. For instance, we classify all associated skein relations when $k=1,2$ and $\mathcal{C}$ is ribbon. In particular, we also consider the instances where $q$ is antisymmetrically self-dual. Our main results follow from considering the action of a rotation operator on a "canonical basis". Assuming self-duality of …
A bicommutant category is a higher categorical analog of a von Neumann algebra. We study the bicommutant categories which arise as the commutant $\mathcal{C}'$ of a fully faithful representation $\mathcal{C}\to\operatorname{Bim}(R)$ of a unitary fusion category $\mathcal{C}$. Using results of Izumi, Popa, and Tomatsu about existence and uniqueness of representations of unitary (multi)fusion categories, we prove that if $\mathcal{C}$ and $\mathcal{D}$ are Morita equivalent unitary fusion categories, then their commutant categories $\mathcal{C}'$ and $\mathcal{D}'$ are equivalent as bicommutant …
A finite bosonic or fermionic symmetry can be described uniquely by a symmetric fusion category $\mathcal{E}$. In this work, we propose that 2+1D topological/SPT orders with a fixed finite symmetry $\mathcal{E}$ are classified, up to $E_8$ quantum Hall states, by the unitary modular tensor categories $\mathcal{C}$ over $\mathcal{E}$ and the modular extensions of each $\mathcal{C}$. In the case $\mathcal{C}=\mathcal{E}$, we prove that the set $\mathcal{M}_{ext}(\mathcal{E})$ of all modular extensions of $\mathcal{E}$ has a natural structure of a finite abelian group. We also prove that the set …
Governed by locality, we explore a connection between unitary braid group representations associated to a unitary $R$-matrix and to a simple object in a unitary braided fusion category. Unitary $R$-matrices, namely unitary solutions to the Yang-Baxter equation, afford explicitly local unitary representations of braid groups. Inspired by topological quantum computation, we study whether or not it is possible to reassemble the irreducible summands appearing in the unitary braid group representations from a unitary braided fusion category with possibly different positive multiplicities to get rep…
Torsion-freeness for discrete quantum groups was introduced by R. Meyer in order to formulate a version of the Baum-Connes conjecture for discrete quantum groups. In this note, we introduce torsion-freeness for abstract fusion rings. We show that a discrete quantum group is torsion-free if its associated fusion ring is torsion-free. In the latter case, we say that the discrete quantum group is strongly torsion-free. As applications, we show that the discrete quantum group duals of the free unitary quantum groups are strongly torsion-free, and that torsion-freeness of discrete quantum groups is…
Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）；Department of Mathematical Sciences, Tsinghua University（清华大学数学科学系）；Yanqi Lake Beijing Institute of Mathematical Sciences and Applications（北京雁栖湖应用数学研究院）
A Fourier transform S is defined for the quantum double D(G) of a finite group G. Acting on characters of D(G), S and the central ribbon element of D(G) generate a unitary matrix representation of the group SL(2,Z). The characters form a ring over the integers under both the algebra multiplication and its dual, with the latter encoding the fusion rules of D(G). The Fourier transform relates the two ring structures. We use this to give a particularly short proof of the Verlinde formula for the fusion coefficients.
