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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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 ...
6
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1answer
126 views

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 ...
6
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1answer
169 views

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 ...
4
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1answer
110 views

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 ...
7
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0answers
160 views

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-...
7
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1answer
193 views

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 "...
4
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0answers
119 views

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 ...
5
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2answers
274 views

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 ...
12
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2answers
475 views

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 ...
3
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1answer
173 views

On the existence of a square root for a unitary modular tensor category

The centre $Z(\mathcal{C})$ of a fusion category $\mathcal{C}$, is a unitary modular tensor category. Question: What about the converse, i.e., can we characterize every unitary modular tensor ...
2
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1answer
129 views

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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1answer
232 views

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 ...
2
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1answer
201 views

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 ...
18
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1answer
628 views

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 ...
12
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1k views

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 ...