Questions tagged [modular-tensor-categories]
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62
questions
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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
230
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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
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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 ...
3
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0
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204
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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 ...
3
votes
1
answer
86
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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 ...
2
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2
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182
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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
2
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103
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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
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0
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131
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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
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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 $\...
5
votes
1
answer
121
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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-...
18
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1
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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 ...
4
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2
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316
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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 $\mathcal{C}$ ...
5
votes
1
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259
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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 ...
4
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2
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187
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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 ...
2
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0
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103
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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 ...
1
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0
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94
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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
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1
answer
184
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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
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0
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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 $M_C(D)$ ...
2
votes
1
answer
99
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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 ...
8
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2
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371
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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 ...
26
votes
3
answers
2k
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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): ...
6
votes
1
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384
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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
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0
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104
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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 \...
3
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0
answers
131
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$e^{2\pi ic_{-}/8}$ 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
1
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642
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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 $\mathbb C$.
...
10
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2
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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, ...
6
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2
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168
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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.
4
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181
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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
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1
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174
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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
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0
answers
113
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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^{\...
12
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1
answer
519
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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 ...
5
votes
1
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277
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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
769
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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 $\...
6
votes
1
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430
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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
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2
answers
100
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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 ...
23
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5
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Do all 3D TQFTs come from Reshetikhin-Turaev?
The Reshetikhin-Turaev construction take as input a Modular Tensor Category (MTC) and spits out a 3D TQFT. I've been told that the other main construction of 3D TQFTs, the Turaev-Viro State sum ...
4
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1
answer
366
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Is the central charge of a Drinfeld center always 0?
(If yes, is there a reference for this statement?)
6
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1
answer
547
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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-...
5
votes
0
answers
241
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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 ...
3
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1
answer
588
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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
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0
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130
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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
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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
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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
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2
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545
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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
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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 ...
19
votes
1
answer
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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 ...
19
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4
answers
2k
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What's the right way to think about "anomalies" in 3d TQFTs?
3d TQFTs constructed from modular tensor categories don't in general give an honest 3d TQFT, instead they have an "anomaly." My vague understanding from Kevin Walker's talks and from skimming Freed-...
8
votes
1
answer
951
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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 ...
8
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0
answers
313
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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 ...
7
votes
2
answers
577
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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 ...