Questions tagged [modular-tensor-categories]

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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): ...
Sebastien Palcoux's user avatar
23 votes
5 answers
3k views

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 ...
Chris Schommer-Pries's user avatar
19 votes
4 answers
2k views

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-...
Noah Snyder's user avatar
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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 ...
André Henriques's user avatar
17 votes
1 answer
1k views

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 ...
César Galindo's user avatar
12 votes
2 answers
691 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 ...
Manuel Bärenz's user avatar
12 votes
1 answer
509 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
10 votes
5 answers
854 views

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 ...
Marcel Bischoff's user avatar
10 votes
2 answers
701 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
10 votes
1 answer
478 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 ...
Jamie Vicary's user avatar
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9 votes
1 answer
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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 ...
Manuel Bärenz's user avatar
8 votes
1 answer
917 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
8 votes
1 answer
534 views

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 ...
David E Speyer's user avatar
8 votes
2 answers
426 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 ...
Chris Schommer-Pries's user avatar
8 votes
0 answers
296 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
7 votes
1 answer
267 views

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. ...
Benjamin Horowitz's user avatar
7 votes
2 answers
329 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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7 votes
2 answers
527 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
7 votes
2 answers
570 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
6 votes
1 answer
371 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
6 votes
2 answers
157 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
539 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
174 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
6 votes
0 answers
115 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-...
Andrea Antinucci's user avatar
5 votes
1 answer
253 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
5 votes
1 answer
313 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 $...
Manuel Bärenz's user avatar
5 votes
1 answer
274 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
5 votes
1 answer
425 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
  • 471
5 votes
1 answer
120 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-...
Sebastien Palcoux's user avatar
5 votes
1 answer
289 views

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 ...
Manuel Bärenz's user avatar
5 votes
0 answers
240 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
  • 6,726
4 votes
1 answer
358 views

Is the central charge of a Drinfeld center always 0?

(If yes, is there a reference for this statement?)
Frank's user avatar
  • 41
4 votes
1 answer
422 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
4 votes
1 answer
627 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
4 votes
3 answers
455 views

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 ...
Ben Webster's user avatar
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4 votes
2 answers
273 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
  • 277
4 votes
2 answers
181 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
  • 338
4 votes
2 answers
318 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 ...
Noah Snyder's user avatar
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4 votes
0 answers
103 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 \...
Jo Mo's user avatar
  • 338
4 votes
0 answers
175 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
  • 1,369
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
  • 4,969
3 votes
2 answers
97 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 ...
pyroscepter's user avatar
3 votes
2 answers
291 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 ...
Jamie Vicary's user avatar
  • 2,423
3 votes
2 answers
190 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
3 votes
1 answer
566 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
  • 2,423
3 votes
1 answer
766 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
3 votes
1 answer
78 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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3 votes
0 answers
181 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 ...
Sebastien Palcoux's user avatar
3 votes
0 answers
74 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
  • 338
3 votes
0 answers
126 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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