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Questions tagged [modular-tensor-categories]

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Is there an integral fusion category of the Ising type?

In [EGNO, Section 8.27.3], we read: Any braided fusion category ${\mathcal C}$ is obtained from a weakly anisotropic category (namely, the core of ${\mathcal C}$) using finite groups (via the ...
Sebastien Palcoux's user avatar
4 votes
1 answer
237 views

Is there a non-split super-modular positive integral fusion category?

We reference [EGNO] for the concept of a braided fusion category. Following the conventions in [JFR], let $\mathcal{C}$ denote a braided fusion category equipped with a braiding $\beta$, and let $\...
Sebastien Palcoux's user avatar
1 vote
1 answer
155 views

Are the minimal nondegenerate extensions universal?

We reference [EGNO] for the concept of a braided fusion category. Following the conventions in [JFR], let $\mathcal{C}$ denote a braided fusion category equipped with a braiding $\beta$, and let $\...
Sebastien Palcoux's user avatar
4 votes
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70 views

Ribbon fusion categories for quantum $\mathfrak{sl}_2$ at odd roots of unity

I will work over $\mathbb{C}$. Let $q=e^{2\pi i/N}$, and write $U_{q}(\mathfrak{sl}_2)$ for Lusztig's divided power quantum group for $\mathfrak{sl}_2$. One can associate to $U_{q}(\mathfrak{sl}_2)$ a ...
Thibault Décoppet's user avatar
4 votes
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223 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 ...
Sebastien Palcoux's user avatar
6 votes
1 answer
256 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,...
Gert's user avatar
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1 answer
206 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 ...
Andrea Antinucci's user avatar
2 votes
2 answers
310 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}{\...
Hajime_Saito's user avatar
3 votes
1 answer
94 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 ...
Milo Moses's user avatar
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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 ...
pyroscepter's user avatar
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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-...
Andrea Antinucci's user avatar
3 votes
2 answers
212 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 $\...
Sebastien Palcoux's user avatar
5 votes
2 answers
365 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 $\mathcal{C}$ ...
Meths's user avatar
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Is there a non-pointed simple integral modular fusion category?

The complex field $\mathbb{C}$ is assumed to be the base field. Let WGT stand for weakly group-theoretical; then [ENO11, Question 2] asks whether the following holds: Statement 1 (open): There is a ...
Sebastien Palcoux's user avatar
4 votes
0 answers
231 views

Vertex operator algebras and modular fusion categories

Let $\mathcal{V}$ be a vertex operator algebra (VOA), and let $\mathcal{C} = \text{Rep}(\mathcal{V})$ be the tensor category of $\mathcal{V}$-modules. It is a conjecture by Vaughan Jones whether every ...
Sebastien Palcoux's user avatar
1 vote
0 answers
96 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, $T$ ...
Sebastien Palcoux's user avatar
2 votes
1 answer
194 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 ...
Sebastien Palcoux's user avatar
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 $M_C(D)$ ...
Jo Mo's user avatar
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1 answer
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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-...
Sebastien Palcoux's user avatar
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): ...
Sebastien Palcoux's user avatar
2 votes
1 answer
101 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 ...
Hauke Reddmann's user avatar
4 votes
2 answers
190 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 ...
Jo Mo's user avatar
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3 votes
0 answers
137 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{...
wonderich's user avatar
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4 votes
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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 \...
Jo Mo's user avatar
  • 338
10 votes
2 answers
787 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, ...
Sebastien Palcoux's user avatar
4 votes
1 answer
660 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 $\mathbb C$. ...
Sebastien Palcoux's user avatar
6 votes
2 answers
178 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.
Xiao-Gang Wen's user avatar
6 votes
1 answer
401 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 ...
as2457's user avatar
  • 295
4 votes
0 answers
185 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 ...
Arthur's user avatar
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7 votes
1 answer
178 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 \...
Theo Johnson-Freyd's user avatar
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^{\...
Ehud Meir's user avatar
  • 5,009
8 votes
2 answers
399 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 ...
Andi Bauer's user avatar
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5 votes
1 answer
278 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 ...
Yuji Tachikawa's user avatar
3 votes
1 answer
774 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 $\...
Yuji Tachikawa's user avatar
12 votes
1 answer
522 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 ...
Manuel Bärenz's user avatar
6 votes
1 answer
434 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 ...
heller's user avatar
  • 481
2 votes
2 answers
101 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 ...
DerLoewe's user avatar
4 votes
1 answer
378 views

Is the central charge of a Drinfeld center always 0?

(If yes, is there a reference for this statement?)
Frank's user avatar
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0 answers
243 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 ...
Arun Debray's user avatar
  • 6,796
5 votes
1 answer
263 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 ...
Sebastien Palcoux's user avatar
3 votes
1 answer
612 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 ...
Jamie Vicary's user avatar
  • 2,443
3 votes
0 answers
138 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 ...
César Galindo's user avatar
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 ...
Manuel Bärenz's user avatar
6 votes
1 answer
557 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-...
Marcel Bischoff's user avatar
6 votes
1 answer
450 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 ...
user avatar
7 votes
2 answers
569 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 ...
Zitao Wang's user avatar
9 votes
1 answer
264 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 ...
Manuel Bärenz's user avatar
8 votes
1 answer
985 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 ...
Alex Turzillo's user avatar
7 votes
2 answers
598 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 ...
Yingfei Gu's user avatar
8 votes
0 answers
334 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 ...
Marcel Bischoff's user avatar