13
votes
Accepted
What do "pivotal" and "spherical" mean for (unitary) fusion categories on the level of the $F$-symbols?
Any unitary fusion category has a canonical spherical structure. See Example 2.12 of this paper and the references therein. It also follows from a more general result Prop 8.23 in this paper. So you ...
- 28.1k
12
votes
Accepted
Is there a simple argument that shows that two unitary fusion categories are Morita equivalent if their Drinfeld centers are equal?
In the non-unitary setting ENO proved that if $Z(C)$ and $Z(D)$ are equivalent as braided tensor categories, then C and D are Morita equivalent. This is Theorem 3.1 of this paper. Note that they ...
- 28.1k
11
votes
Accepted
Local fusion categories
Let $\mathcal{R}$ be a fusion category and $\beta : \mathcal{R} \to \mathrm{Vec}$ an additive monoidal functor.
I first claim that $\beta$ is automatically faithful. (I know why you use "top ...
- 54.6k
10
votes
What do "pivotal" and "spherical" mean for (unitary) fusion categories on the level of the $F$-symbols?
This is really an answer to the question in the comment of Noah's answer.
Let $F$ be a solution to the pentagon equations for some fusion category $\mathcal C$. Pivotal structures on $\mathcal C$ ...
- 1,089
9
votes
Accepted
Are there irreducible multi-fusion categories that are not fusion categories?
Matrix categories, $\mathrm{End}(\mathrm{Vec}^{\oplus n})$. (The identity on each copy of $\mathrm{Vec}$ are summands of the identity.)
(You can generalize this example by putting fusion categories ...
- 28.1k
8
votes
Accepted
Is there a fusion category not Grothendieck equivalent to a unitary one?
Yes, according to Andrew Schopieray. He just provided a categorifiable fusion ring, of rank 6 and multiplicity 2, without pseudounitary categorification (so without unitary categorification), in the ...
7
votes
Is a unitary Hamiltonian TQFT the same as a unitary axiomatic TQFT?
The answer is no:
There are unitary Hamiltonian TQFTs (ie there are gapped lattice Hamiltonian systems in physics) which are not a unitary axiomatic TQFTs.
An example can be given in 3+1-dimension ...
- 4,826
6
votes
Are there interesting examples of unitary fusion categories where a tensor product of two simple objects is simple?
Typically the tensor product of two simple objects is not simple. The smallest example is $\text{Fib}$, the Fibonacci category (also called the Yang-Lee category), that has two simple objects: $\...
- 556
6
votes
Accepted
What are the topological phases of quantum Hall systems?
Fermionic modular categories and the 16-fold way classifies the topological phases of the fractional quantum Hall effect. The Laughlin states (Abelian anyons at filling factor $1/Q$, $Q$ odd) are ...
- 188k
4
votes
Accepted
Non-trivial factor splitting from vacuum in TQFTs. F-move not unity?
Consider the case $a=b=c$. Then the fusion space $V_{aaa}$ affords a representation of $\mathbb Z/3$ via a $2\pi/3$ rotation. The F-move in your question is essentially this $2\pi/3$ rotation, and ...
- 12.8k
4
votes
Is a unitary Hamiltonian TQFT the same as a unitary axiomatic TQFT?
On the Hamiltonian level, the (axiomatic) TQFT tensors correspond to the imaginary time evolution of a microscopic system, not the real-time evolution (which would be a unitary operator). So there's ...
- 3,001
4
votes
What are the topological phases of quantum Hall systems?
We have a paper that contains lists of simple fermionic topological orders in 2+1D: https://arxiv.org/abs/1507.04673 . For fermionic topological without symmetry, there is no filling fraction. So our ...
- 4,826
3
votes
Non-negative integer matrix representation of a fusion ring
Here is a standard counterexample. Let $A$ be the ring of rank 2 with basis $1,X$ and $X^2=1+2X$ (so $X=X^*$). Then $A$ has a rank 2 module $M$ where $X$ acts via matrix $\left( \begin{array}{cc}1&...
- 3,176
2
votes
Accepted
Unitary structures on fusion categories
Reutter's recent paper "uniqueness of unitary structure for unitarizable fusion categories" answers your question in the affirmative (link: https://arxiv.org/pdf/1906.09710.pdf).
- 2,902
2
votes
Accepted
Existence of twisted metaplectic categories
In https://arxiv.org/pdf/2005.05544.pdf we describe a general procedure that accomplishes (1) and (2) which we call zesting.
In this case, let $\mathcal{C}$ be an odd metaplectic category. In the ...
- 1,639
1
vote
Non-negative integer matrix representation of a fusion ring
I think I found a counterexample if the characteristic of $M$ is nonzero:
Let $B$ be the matrices corresponding to the permutations of the alternating group of $3$ elements, denoted as $\{ 1,x,x^* \} $...
- 1,245
Only top scored, non community-wiki answers of a minimum length are eligible