Questions tagged [unitary-fusion-category]
Questions about fusion categories or tensor categories with a unitary (Hilbert space and dagger) structure on the morphisms, and dagger-preserving functors.
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Non-negative integer matrix representation of a fusion ring
Context: I am a physics grad student working on topological lines in 2D CFTs.
Let $A$ be a unital based $\mathbb{Z}_{+}$ ring with finite rank (or a Fusion ring) with the basis $B = \{b_1, b_2, \dotsc ...
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Unitary fusion category and subfactor
From a unitary fusion category $\mathcal{C}$, there are several ways to make a (hyperfinite II$_1$) subfactor.
By [Ha] there are weak Hopf algebras $H$ such that $\mathcal{C} = Rep(H)$. By unitarity (...
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Local fusion categories
A local fusion category ${\cal R}$ is a unitary fusion category equipped with a top-faithful surjective monoidal functor to the fusion category of vector spaces: $\beta: {\cal R} \to {\cal V}ec$. Here,...
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Existence of twisted metaplectic categories
The paper Classification of metaplectic modular categories by Ardonne-Cheng-Rowell-Wang (2016) mentions (in Section 3) the Grothendieck ring for the metaplectic modular categories, i.e. $SO(N)_2$, $N&...
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Is an integral fusion category pseudo-unitary over a finite field?
Here are two propositions in the book Tensor Categories:
Proposition 9.5.1. A pseudo-unitary fusion category admits a unique
spherical structure.
Proposition 9.6.5. Let $\mathcal{C}$ be ...
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Is there a fusion category not Grothendieck equivalent to a unitary one?
We refer to the book Tensor categories by Etingof-Gelaki-Nikshych-Ostrik (MR3242743) for the notion of (unitary) fusion category. Two fusion categories are Grothendieck equivalent if they have the ...
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Extended cyclotomic criterion for unitary categorification
According to this paper (Corollary 8.54) the Frobenius-Perron dimension (FPdim) of any object $a$ of a fusion category over $\mathbb{C}$ is a cyclotomic integer. Now, FPdim($a$) is the maximal ...
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The simple unitary fusion categories of multiplicity one
Here are two families of simple unitary fusion categories of multiplicity one:
$Vec(C_p)$ with $C_p$ the cyclic group of order $p$ (one or prime),
The even part of Temperley-Lieb $A_{2n}$ with $n \...
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Existence of a unitary fusion category with this relation ruled out on finite groups
In this answer, Geoff ruled out the existence of a finite group $G$ such that the fusion category $\mathrm{Rep}(G)$ has simple objects $5_1$ and $7_1$ of FPdim $5$ and $7$ resp., with (for some object ...
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What are the topological phases of quantum Hall systems?
(Fractional) quantum Hall systems are $2+1$-dimensional models which are said to possess topological order. One (maybe even complete) set of invariants of topological phases in $2+1$ dimensions is ...
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Non-trivial factor splitting from vacuum in TQFTs. F-move not unity?
Consider a unitary modular TQFT, defined by the F and R moves. More specifically, a braided tensor category relevant for anyon models in 2D topologically ordered phases of matter. I am interested in ...
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Are there irreducible multi-fusion categories that are not fusion categories?
Multi-fusion categories are a generalization of fusion categories with a non-simple unit. The direct sum of two multi-fusion categories is again a multi-fusion category. By irreducible I mean that a ...
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Why are Levin-Wen/Turaev-Viro models said to be non-chiral?
I'd like to bring together the following two notions of "non-chiral":
On the abstract algebraic side, a modular fusion category describing the anyon content of some physical system is said to be non-...
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Is there a simple argument that shows that two unitary fusion categories are Morita equivalent if their Drinfeld centers are equal?
By Morita equivalent I mean that there is an invertible bi-module between the two fusion categories. [Feel free to replace the Drinfeld centers being "equal" by an appropriate categorial notion of "...
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Can non-chiral 3D TQFTs be extended to non-orientable manifolds whereas chiral ones cannot?
As far as I know, when talking about TQFT, one usually means TQFTs on oriented manifolds with boundary (cobordisms)
It appears to me that the Turaev-Viro-Barrett-Westbury state-sum construction can ...
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What do "pivotal" and "spherical" mean for (unitary) fusion categories on the level of the $F$-symbols?
For me, a fusion category (over $\mathbb{C}$) is just a tensor $F$ (the associator, with $6$ simple-object labels and $4$ fusion space indices) and a tensor $d$ (the quantum dimensions, with one ...
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Is a unitary Hamiltonian TQFT the same as a unitary axiomatic TQFT?
Introduction
Axiomatic TQFTs
An axiomatic $n$-dimensional TQFT is a symmetric monoidal functor $\mathcal{Z}\colon \operatorname{Bord}_n \to \operatorname{Hilb}$ from $n$-dimensional oriented ...
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On the existence of a square root for a modular tensor category
The center $Z(\mathcal{C})$ of a spherical fusion category $\mathcal{C}$ (over $\mathbb{C}$) is a modular tensor category.
Question: What about the converse, i.e., can we characterize every modular ...
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Modularisation on group representations with arbitrary braiding
Applying the modularisation/deequivariantisation procedure to the representation category $\operatorname{Rep}_G$ of a finite group $G$ with trivial braiding gives the fibre functor to vector spaces. ...
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Is the modularisation of a unitary fusion category always unitary?
Suppose $\mathcal{C}$ is a unitary ribbon fusion category. Also assume that its symmetric centre has trivial twist and trivial pivotal structure, i.e. is tannakian. Thus, the Müger/Bruguières ...
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The definition of unitary fusion category
I just come across a definition of the unitary fusion category:
A fusion category $\mathcal{C}$ over the complex number is said to be unitary if we have:
We have a Hilbert space structure on each ...
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Is the representation category of quantum groups at root of unity visibly unitary?
Let $\mathfrak g$ be a simple Lie algebra.
By taking the specialization at $q^\ell=1$ of a certain integral version¹ of the quantum group $U_q(\mathfrak g)$,
and by considering a certain quotient ...
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Unitary structures on fusion categories
A unitary fusion category is a fusion category with a $C^*$-tensor structure.
Hence, in principle, a fusion category could have more than one unitary structure. Does exist a fusion category with more ...
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