Questions tagged [reshetikhin-turaev]

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4 votes
1 answer
179 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 ...
3 votes
0 answers
117 views

Relative strength of Jones and colored Jones polynomials

this is my first post here. I've been studying some Knot Theory and I came to a question concerning invariants. We know that the Jones polynomial is related to the RT-invariant associated to the two-...
14 votes
2 answers
458 views

Lagrangian of Reshetikhin-Turaev TFT's

One of the results from the Reshetikhin-Turaev package is that given a modular tensor category $\mathscr{C}$ one can construct a TFT $Z$. In the case where $\mathscr{C}$ is the category of positive ...
21 votes
1 answer
1k views

Fully extended TQFT and lattice models

I often read that fully extended TQFTs are supposed to classify topological phases of matter. So I would like to understand the formal nature of fully extended TQFTs on a more direct physical level (...
4 votes
0 answers
183 views

Are Turaev-Viro invariants holonomic?

Consider a 3-manifold $M$ with a boundary, which is a genus $g\geq 1$ surface $\Sigma$. Fix a triangulation $T$ of $\Sigma$. Then Turaev-Viro invariants $TV_q(M)$ are functions, assigning to integer ...
4 votes
0 answers
195 views

Can non-chiral 3D TQFTs be extended to non-orientable manifolds whereas chiral ones cannot?

As far as I know, when talking about TQFT, one usually means TQFTs on oriented manifolds with boundary (cobordisms) It appears to me that the Turaev-Viro-Barrett-Westbury state-sum construction can ...
4 votes
1 answer
263 views

Can the ribbon category of f.d. reps of $\mathcal{U}_q(\mathfrak{sl}(2))$ be modified so the twist is trivial on the vector representation?

Consider the ribbon category of finite-dimensional representations of $\mathcal{U}_q(\mathfrak{sl}(2))$, with twist $\theta$. If $V$ is the vector representation, then $\theta_V$ is multiplication by $...
8 votes
1 answer
397 views

Brauer-Picard for a fusion category coming from a quantum group

In Fusion Categories and Homotopy Theory, ENO attatch a 3-groupoid to a fusion category. In the case of A graded vector spaces they further compute it's truncation as an orthogonal group $O(A \...
15 votes
1 answer
1k views

Why are Witten-Reshetikhin-Turaev invariants expected to be integral?

A Witten-Reshetikhin-Turaev (WRT) Invariant $\tau_{M,L}^G(\xi)\in\mathbb{C}$ is an invariant of closed oriented 3-manifold $M$ containing a framed link $L$, where $G$ is a simple Lie group, and $\xi$ ...
1 vote
0 answers
219 views

Categorification of WRT invariants of integral homology spheres

First, I would like to know how many definitions are there for categorification of WRT invariants. In addition, I wonder if the categorified version of WRT invariants have been explicitly computed for ...
4 votes
3 answers
460 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 ...
6 votes
2 answers
519 views

How do quantum knot invariants change when I pick a funny ribbon element?

So, there's a construction of Reshetikhin and Turaev which extracts knot invariants from ribbon monoidal categories, which are (usually) the representation category a Hopf algebra with a choice of ...