Questions tagged [modular-tensor-categories]

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Vertex operator algebras and modular fusion 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 ...
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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, $T$ ...
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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 ...
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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 $M_C(D)$ ...
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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-...
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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): ...
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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 ...
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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 ...
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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{...
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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 \...
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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, ...
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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 $\mathbb C$. ...
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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.
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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 ...
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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 ...
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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 \...
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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^{\...
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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 ...
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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 ...
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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 $\...
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12 votes
1 answer
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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 ...
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5 votes
1 answer
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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 ...
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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 ...
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Is the central charge of a Drinfeld center always 0?

(If yes, is there a reference for this statement?)
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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-...
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3 votes
1 answer
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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 ...
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6 votes
2 answers
464 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 ...
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8 votes
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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 ...
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6 votes
1 answer
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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 ...
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7 votes
2 answers
519 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 ...
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8 votes
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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 ...
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3 votes
2 answers
269 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 ...
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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 $...
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8 votes
2 answers
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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 ...
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10 votes
1 answer
442 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 ...
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6 votes
1 answer
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Geometric Intuition of $P^+$ in Modular Tensor Categories

I'm currently reading through Bakalov and Kirillov's "Lectures on Tensor Categories and Modular Functors," and I am having some difficulty understanding the definition of $p^\pm$ given on page 49. ...
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18 votes
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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 ...
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11 votes
2 answers
635 views

Is "being a modular category" a universal or categorical/algebraic property?

A semisimple braided category with duals is called modular when a certain matrix $S$ is invertible. The components $S_{AB}$ are indexed by (isomorphism classes of) simple objects of the category and ...
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1 answer
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Is every premodular category the *full* subcategory of a modular category?

In Müger's article "Conformal Field Theory and Doplicher-Roberts Reconstruction", he defines the "modular closure" of a braided monoidal category. So every braided monoidal category (and therefore ...
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10 votes
5 answers
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Example for non equivalent rational full CFTs with same modular invariant (partition function)

I am looking for a counter example which shows, that a full rational 2D CFT (with respect to a given chiral subtheory) is not characterized by its modular invariant partition function. People tell me ...
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14 votes
0 answers
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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 ...
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2 votes
1 answer
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Graphical calculus in braided G crossed fusion categories: Explanation request and a question

I am trying to understand the equivalence between the 2 category of braided G crossed categories and the 2 category of braided categories containing Rep(G) as a symmetric category. The references in ...
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4 votes
2 answers
314 views

Can "premodular" be relaxed as a condition for uniqueness of Bruguieres/Mueger modularization?

Suppose that C is a ribbon monoidal category with dominant ribbon functors F_1: C->D_1 and F_2: C->D_2 such that D_1 and D_2 are modular tensor categories, does it follow that D_1 and D_2 are ...
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8 votes
1 answer
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Is tensor product exact in abelian tensor categories with duals?

Suppose we are in an abelian tensor category with duals, where all objects have finite length. Let $0 \to A \to B \to C \to 0$ be a short exact sequence and $Z$ an object of the category. Is $$0 \to ...
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4 votes
3 answers
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What's the best reference for actual formulas for RT invariants?

If one really wants to understand the formulas for how to construct the Reshetikhin-Turaev 3-manifold invariants coming from quantum groups in terms of R-matrices and such, what's the best reference ...
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