All Questions
Tagged with quantum-groups topological-quantum-field-theory
13 questions
8
votes
1
answer
331
views
Some fusion rings/categories I don't recognize
Recently (what I believe are) all multiplicity-free fusion categories up to rank 7 have been posted on the AnyonWiki. Most of the fusion rings belonging to these categories belong to one of the ...
8
votes
0
answers
233
views
$U_q(\mathfrak{g})$ is to knot theory as $U_q(\hat{\mathfrak{g}})$ is to $?$
Let $\mathfrak{g}$ be a finite dimensional semisimple Lie algebra over the complex numbers, e.g. $\mathfrak{sl}_n$.
Then every representation $\DeclareMathOperator\Rep{Rep}V\in \Rep U_q(\mathfrak{g})$ ...
- 12.9k
24
votes
3
answers
3k
views
How many definitions are there of the Jones polynomial?
Even with the connection to quantum groups being made clearer (I believe it was not known when the Jones polynomial was first introduced), it seems to me that still we don't have the "right" ...
- 68.9k
6
votes
1
answer
354
views
Relation between TQFT representations and factorizable sheaves
I am interested in the comparison between two different constructions which, as far as I can tell, are both supposed to produce rigorous constructions of Wess–Zumino-Witten conformal blocks.
More ...
- 8,524
7
votes
0
answers
116
views
Are there attempts to numerically finding algebraic structures over finite-dimensional vector spaces?
By "algebraic structure" I mean a finite set of linear operators between tensor products of copies of one (or more) finite-dimensional (complex or real) vector spaces, fulfilling a set of ...
- 3,001
10
votes
1
answer
806
views
DW, state sum models, and fully extended TQFTs
I am interested in state sum models and their relations with some other of TQFTs, especially the fully extended TQFTs and the Dijkgraaf-Witten TQFTs (generalized, in the sense that finite-group-...
- 28.1k
7
votes
2
answers
654
views
What is the state in the WRT TQFT associated to a handlebody?
Let $Y^3$ be a handlebody with boundary $\Sigma$. By definition, there is some associated vector $v_{WRT}(Y^3)\in Z(\Sigma)$, the (finite dimensional) Hilbert space associated to $\Sigma$ by the ...
- 492
5
votes
0
answers
210
views
Projecting GxG onto subspace with tied irreducible representations
Suppose I have a finite group $G$. With this group, I can associate an ortho-normal Hilbert space spanned by elements of the group $$\mathcal{H} = \{|g\rangle: g \in G \}$$.
I could alternatively ...
- 173
3
votes
1
answer
606
views
How to interpret sections over the $\mathrm{SU}(2)$ character variety as sections over the $\mathrm{SL}(2,\mathbb{C})$ character variety?
The motivation for this question comes from the Volume Conjecture of Kashaev-Murakami-Murakami. The Jones family of invariants of knots and $3$-manifolds can all be defined using $\mathrm{SU}(2)$ and ...
- 8,529
6
votes
1
answer
740
views
Kauffman's state model for the Alexander polynomial, via representation theory
I've been reading Oleg Viro's paper on "quantum relatives of the Alexander polynomial" (arXiv:math/0204290), which, among other more general things, derives state-sum formulas for the Alexander ...
CommunityBot
- 1
18
votes
3
answers
3k
views
How to calculate the Witten-Reshetikhin-Turaev invariants from a triangulation?
I'm interested in the Witten-Reshetikhin-Turaev invariants for 3-manifolds, and in particular, how to calculate them from a triangulation of the 3-manifold (recall that as they were first introduced, ...
CommunityBot
- 1
20
votes
1
answer
3k
views
Calculating 6j-symbols (aka Racah-Wigner coefficients) for quantum groups
Which $6j$-symbols for quantised enveloping algebras are known explicitly?
The quantum $6j$-symbols for $sl(2)$ are well-known. The references are
Masbaum and Vogel and
Frenkel and Khovanov.
What ...
CommunityBot
- 1
4
votes
3
answers
473
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 ...
- 9,132