All Questions

Filter by
Sorted by
Tagged with
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
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
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
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
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
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
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
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
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
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
0 answers
128 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
141 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
2 votes
1 answer
443 views

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 ...
math user's user avatar
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 ...
DerLoewe's user avatar