Questions tagged [modular-tensor-categories]
The modular-tensor-categories tag has no usage guidance.
62
questions
3
votes
0
answers
198
views
Is the rank of an integral MTC an upper bound for the prime factors of its FPdim?
In this post, the abbreviation "MTC" and "FPdim" stand for "Modular Tensor Category" and "Frobenius-Perron dimension".
From [DLN, Theorem II (iii)], where the ...
6
votes
1
answer
232
views
Software for working with fusion categories
One way to describe fusion categories is via a fusion system: several lists of numbers that define the fusion ring, associator, braiding (if it exists), etc. Often, these sets of numbers are quite big,...
4
votes
1
answer
179
views
Are there (non Lagrangian) algebras of Turaev-Viro TQFTs which cannot be completed to Lagrangian algebras?
Consider a 3d TQFT of the Turaev-Viro type, say TV$(\mathcal{C})$, where $\mathcal{C}$ is some fusion category. Equivalently, this is a TQFT admitting Lagrangian algebra objects $\mathcal{L}$ of the ...
2
votes
2
answers
185
views
Tensor functor between rigid tensor categories preserves $\text{Hom}$-objects
I was looking at these notes on Tannakian categories. Let me briefly recall the notion of tensor functors:
Let $(\mathcal{C},\otimes)$ and $(\mathcal{C'},\otimes')\DeclareMathOperator{\uphom}{\...
3
votes
1
answer
89
views
Relation between factor condition on von Neumann algebras and modularity condition on ribbon fusion categories
A modular tensor category is defined to be a ribbon fusion category $\mathcal{C}$ in which the only objects which commute with all other objects are multiples of the identity. That is, if we denote by ...
3
votes
2
answers
103
views
Does unitarity and modularity constrain fusion multiplicities to be 0,1?
If I have a braided tensor category that's unitary and modular, then how does the unitarity and modularity constrain the fusion multiplicities?
I know that if $a,b,c \in ob({C})$ satisfy the fusion ...
6
votes
0
answers
132
views
State-sum for 4d TQFT from fusion 2-categories and invariants of Morita equiavalence classes beyond Drinfeld center
If $\mathcal{C}$ is a fusion 1-category, the Turaev-Viro state-sum produces a 3d TQFT whose modular tensor category is the Drinfeld center of $\mathcal{C}$. In particular this means that the Turaev-...
3
votes
2
answers
201
views
Simple modular tensor category and zero entries in its S-matrix
Question 1: Is there a simple modular fusion category with a zero entry in its S-matrix?
(or equivalently, with a fusion matrix of zero determinant?)
Yes, by this answer below providing the example $\...
4
votes
2
answers
317
views
Relationship between fusion category and its Drinfel'd center
Is it true that given a fusion category $\mathcal{C}$ and its Drinfel'd center $Z(\mathcal{C})$, there is a fully faithful functor $F:\mathcal{C}\hookrightarrow Z(\mathcal{C})$? I.e. can $\mathcal{C}$ ...
2
votes
0
answers
103
views
Is there a non-pointed simple integral modular fusion category?
The weakly group-theoretical conjecture (supporting a negative answer to [ENO11, Question 2]) states as follows:
Statement 1: Every integral fusion category is weakly group-theoretical.
We wonder ...
3
votes
0
answers
204
views
Vertex operator algebras and modular tensor categories
Let $\mathcal{V}$ be a vertex operator algebra (VOA), and let $\mathcal{C}=Rep(\mathcal{V})$ be the tensor category of (ususal) $\mathcal{V}$-modules. It is a well-known open-problem whether every ...
1
vote
0
answers
94
views
Different modular data with same T-matrix
Modular data (MD) is an invariant of a modular fusion category $\mathcal{C}$. It is a couple of symmetric matrices $S, T \in M_r(\mathbb{C})$ with:
$r$ the rank of $\mathcal{C}$,
$S$ invertible,
$T$ ...
2
votes
1
answer
184
views
Relation between the modular categories SU(2)_n and Sp(n)_1
The online database [1] provides a list of some modular tensor categories classified by rank. Let us consider the two modular categories denoted kmA1_$\ell$ and kmC$\ell$_1 (i.e. Kac Moody $A_1$ level ...
3
votes
0
answers
75
views
Does a factorization of a modular fusion category imply some "factorization" of TFTs?
Mueger showed in this paper that if $C$ is a modular fusion category and $D$ is a modular fusion subcategory of $C$, then $C$ is equivalent to $D \boxtimes M_C(D)$ as ribbon categories, where $M_C(D)$ ...
5
votes
1
answer
121
views
Non-cyclotomic modular fusion categories
In a recent talk (see [1]) Richard Ng asks whether the invariants of 3-manifolds derived from any modular fusion category are cyclotomic integers. In tqft, such an invariant is computed from the F-...
26
votes
3
answers
2k
views
Sum of squares and divisibility
Consider an integer of the form $$N = 1 + \sum_{i=1}^r d_i^2$$ where $d_i \in \mathbb{N}_{\ge 3}$ and $d_i^2$ divides $N$.
Question: Must $r$ be greater than or equal to $9$?
Checking (with SageMath): ...
2
votes
1
answer
99
views
Smallest modular tensor category with a multiplicity
I wrote a 6j-inator taking the multiplication table of a based ring and calculating the equations in the 6j symbols. I successfully tested it with a small example (of the paper "On Classification ...
4
votes
2
answers
188
views
Bialgebras with rigid representation theory
Repost from math.SE since no answer after two months, but feel free to close if not appropriate:
Everything is finite-dimensional over a field $k$.
Let $B$ be a bialgebra with $B\text{-mod}$ its ...
3
votes
0
answers
131
views
$e^{2\pi ic_{-}/8}$ and $e^{2\pi ic_{-}/24}$ in unitary modular category (UMC)
Background
Unitary modular categories (UMC) do not capture the central charge $c_{-}$ of the topological quantum field theory (TQFT). However, there is a relation that fixes,
$c_{-}\bmod
8$:
\begin{...
4
votes
0
answers
105
views
Categorical interpretation of the comodulus of a Hopf algebra?
Let $H$ be a finite-dimensional Hopf algebra.
Then it has a right cointegral $\lambda \in H^*$ and a left integral $c \in H$, characterized uniquely (up to scalar) by
\begin{align}
(\lambda \...
10
votes
2
answers
745
views
Is there a non-degenerate quadratic form on every finite abelian group?
Let $G$ be a finite abelian group. A quadratic form on $G$ is a map $q: G \to \mathbb{C}^*$ such that $q(g) = q(g^{-1})$ and the symmetric function $b(g,h):= \frac{q(gh)}{q(g)q(h)}$ is a bicharacter, ...
4
votes
1
answer
644
views
Finite groups G with Rep(G) Grothendieck equivalent to a modular category
We refer to Chapter 8 of the book Tensor Categories for notions related to modular tensor categories and J.P. Serre for the basic theory of linear representations of finite groups over $\mathbb C$.
...
6
votes
2
answers
168
views
Automorphisms of a modular tensor category
I would like to ask for references on automorphisms of a modular tensor category, that do not change the objects. Some special cases, such as automorphisms of a quantum double, are also helpful.
6
votes
1
answer
387
views
Mapping class group of torus, why is $(ST)^3=S^2$?
In the context of topological quantum field theories, I am interested in the mapping class group of a torus. Here I can consider the torus as a square with identified edges and also decorated with ...
4
votes
0
answers
182
views
Quantum dimension in the Drinfeld center
Let $\mathcal{C}$ be a spherical tensor category. It is known that the Drinfeld center of $\mathcal{C}$ is modular (and therefore also spherical), see for example, Corollary 8.20.14 in [1]. Recall the ...
6
votes
1
answer
175
views
What is the etale homotopy type of the Witt group of braided fusion categories?
The Witt group $\mathcal{W}$ of braided fusion categories (see also the sequel paper) can be defined over any field; I am happy to restrict to characteristic $0$ if it matters.
Is $\mathbb k \...
4
votes
0
answers
113
views
semisimplicity of maps in braided vector spaces
Let $V$ be a finite dimensional braided vector space over $\mathbb{C}$.
This means that we have a map $$c_{V,V}:V\otimes V\to V\otimes V$$ which gives us an action of the braid group $B_n$ on $V^{\...
8
votes
2
answers
375
views
How do I calculate the modular fusion category from a given Lie algebra and level in Chern-Simons theory?
In Chern-Simons theory, one has modular fusion categories that are labelled by a Lie algebra and a "level", e.g. $SU(2)_2$ ("$SU(2)$ level $2$").
Physically this modular fusion category describes the ...
5
votes
1
answer
277
views
Modular tensor category associated to an even integral lattice and the lattice automorphism
Let $(L,\langle -,-\rangle)$ be an even integral lattice, and let $(A,q)$ be the associated discriminant form: $$
A=L^*/L, \quad q(a)=e^{\pi i \langle a,a\rangle}.
$$
We let $\hat L$ to be the ...
3
votes
1
answer
770
views
Do $G$-invariant non-degenerate quadratic forms come from $G$-invariant even lattices?
The following is a somewhat well-known fact: Given an even lattice $L$ with the pairing $\langle,\rangle: L\times L\to \mathbb{Z}$, we extend the pairing to $L\otimes \mathbb{Q}$ by tensoring with $\...
12
votes
1
answer
519
views
Is there a "killing" lemma for G-crossed braided fusion categories?
Edit: I found a serious flaw in the question and my answer, and I had to change a lot. The basic question is still there, but the details are a lot different.
Premodular categories
In braided ...
6
votes
1
answer
431
views
Internal Hom of Deligne' tensor product
I read the following statement (equation 22) in "Monoidal 2-structure of bimodule categories" by Justin Greenough:
Let $\mathcal{C}$ be a finite tensor category (abelian k-linear rigid monoidal ...
2
votes
2
answers
100
views
Uniqueness of character for Z_+-rings
I have a question about the proof of proposition $3.3.6(3)$ in "Tensor Categories" by Etingof et al..
This part states that for $A$, transitive unital $\mathbb Z_+$-ring, there is a unique character ...
4
votes
1
answer
366
views
Is the central charge of a Drinfeld center always 0?
(If yes, is there a reference for this statement?)
5
votes
0
answers
241
views
Analogue of Reshetikhin-Turaev construction for unoriented TQFTs
The Reshetikhin-Turaev construction takes a modular tensor category $\mathcal C$ and produces a 3-2-1 oriented TQFT $Z_{\mathcal C}$ such that $Z_{\mathcal C}(S^1) = \mathcal C$.
Is there an ...
5
votes
1
answer
259
views
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 ...
3
votes
1
answer
590
views
Module categories for Fibonacci anyons
What are the module categories over the modular tensor category Fib of Fibonacci anyons?
By Ostrik's work, we know these module categories correspond to separable algebras in Fib. I do not believe ...
3
votes
0
answers
130
views
Symmetries of modular categories coming from quantum groups
This is a request for references about the computation of the braided autoequivalences of fusion categories coming from a quantum group. I could not find even the description of braided ...
2
votes
1
answer
148
views
How nontrivial can "central extensions of ribbon fusion categories" be?
In a sense, this is a follow up on this question, but one PhD programme later.
Let $\mathcal{C}$ be ribbon fusion. By $\mathcal{C}'$, we denote the symmetric centre, i.e. the full subcategory of ...
6
votes
1
answer
548
views
Do all non-degenerate quadratic forms come from positive even lattices?
Let $(G,+)$ be a finite Abelian group. We say $q\colon G\to \mathbb{T}$ is a non-degenerated quadratic form, if $q(-a)=q(a)$ and the symmetric function
$$
b(g,h) =q(g+h)q(g)^{-1}q(h)^{-1}
$$
is a non-...
6
votes
1
answer
439
views
When modular tensor categories are equivalent?
I asked this question at math stack exchange math stack exchange but I haven't got any answer yet there.
I would like to know when we say that two modular tensor categories are equivalent.
Is it ...
7
votes
2
answers
547
views
How to make a premodular category a modular tensor category?
A premodular category (also called ribbon fusion category) is roughly speaking a tensor category where fusion and braiding of the objects are defined. With an extra nondegeneracy condition for the ...
9
votes
1
answer
262
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 ...
8
votes
1
answer
954
views
Twists, balances, and ribbons in pivotal braided tensor categories
Let $\mathcal{C}$ be a pivotal tensor category. Feel free to assume finiteness, semisimplicity, fusion, sphericality, unitarity or whatever makes things interesting. Which of the following structures ...
7
votes
2
answers
578
views
Gauss-Milgram formula for fermionic topological order?
For Bosonic topological order, a very useful formula was proved to be true:
$\sum_a d_a^2 \theta_a=\mathcal{D} \exp(\frac{c_-}{8}2\pi i) $
(for more detail: $d_a$ is the quantum dimension of anyon ...
8
votes
0
answers
317
views
Structure of Lagrangian algebras in the center of a fusion category
(1) Let $\mathcal F$ be a spherical fusion tensor category. Then Müger showed that
$R=\bigoplus_{H\in\mathrm{Irr}(\mathcal F)} H\boxtimes H^\mathrm{op}$ canonically has the structure of a Frobenius ...
3
votes
2
answers
295
views
Distinct 2D RCFTs with the same underlying MTC
A 2d rational conformal field theory (RCFT) gives rise to a modular tensor category (MTC) equipped with a Frobenius algebra object (see, for example, http://arxiv.org/abs/hep-th/0204148).
Is there an ...
6
votes
1
answer
328
views
Is there a quotient or exact sequence of symmetric, premodular (ribbon fusion) and modular categories?
In Walker and Wang's article about (3+1)-TQFTs from premodular categories, they say on page 14 that you can take a quotient of a premodular category $\mathcal{C}$ by its symmetric fusion subcategory $...
9
votes
2
answers
432
views
How weird can Modular Tensor Categories be over non-algebraically closed fields?
I am trying to understand better the behaviour and character of modular tensor categories over non-algebraically closed fields. How weird can they be?
The reason I am interested in this is that my ...
10
votes
1
answer
495
views
Is every separable algebra in a modular tensor category Morita equivalent to a commutative one?
Separable algebras in modular tensor categories are interesting algebraic structures, which have received significant attention because of their connection to conformal field theories. My ...