12 votes
Accepted

2-bridge knots in the Rolfsen's table

Cha and Livingston's KnotInfo includes the ability to list the bridge index of knots with at most 11 crossings, and so covers all of Rolfsen's table. The 2-bridge knots are those with bridge index 2. ...
  • 3,085
12 votes
Accepted

$A \wedge A \wedge A$ in Chern-Simons

Option (1) Use the definition $(\omega \otimes S) \wedge (\eta \otimes S) = (\omega \wedge \eta) \otimes (S\otimes T)$ of the wedge product for Lie algebra valued forms. Define Lie bracket and Killing ...
11 votes
Accepted

Differential characters, Chern-Simons forms, and differential cohomology

The simple beginning of this story is that the curvature of a $\mathrm{U}(1)$ connection does not tell you the bundle it's a connection on — not even up to isomorphism. Differential cohomology ...
  • 19.8k
11 votes
Accepted

Importance of the principal bundle in Chern-Simons theory

In quantum Chern-Simons theory with gauge group $G$ (compact Lie), a field on a 3-manifold $M$ is a principal $G$-bundle with a connection $A$. The partition function/path integral associated to $M$ ...
  • 5,248
7 votes
Accepted

The existence of the extension of a non-trivial line bundle

This is a bordism problem, and as such can be answered using algebraic topology. I'll answer in the unoriented setting, then indicate how to modify things if $M$ and $W$ are required to be oriented. ...
  • 33.5k
7 votes
Accepted

Formula for the anomalies of spin Chern-Simons theories?

This is not a direct answer to your question, but I think it's relevant. One way of thinking about the anomaly for ordinary (oriented) Chern-Simons theories is that it's the evaluation of the ...
6 votes

$A \wedge A \wedge A$ in Chern-Simons

For Lie algebras of matrices (which is what you really care about in Chern-Simons theory) think of $A$ as a form with matrix coefficients $$ A=\sum_i A_i dx^i, $$ where $A_i$ are $r\times r$ ...
6 votes
Accepted

Value of the Chern-Simons functional for flat connections on $S^3/\Gamma$

For $G=SU(N)$, there is a paper SU(n)–Chern–Simons invariants of Seifert fibered 3–manifolds (Int. J. Math., 09, 295-330 (1998))
  • 106
6 votes

2-bridge knots in the Rolfsen's table

Mark Bell has a wonderful answer for the knot tables. I thought I would provide an answer for the census data. Using SnapPy, the current version of the OrientableCuspedCensus, which is a combination ...
  • 5,121
6 votes

What is the trace in the Chern-Simons action?

I will give the physicist's answer. I hope you are familiar with this notation. $A=A_\mu dx^\mu$ and in this notation the action looks like $S=\frac{k}{4\pi} \int_{\mathcal{M}} d^3x\ \epsilon^{\mu\...
6 votes

Is there a volume conjecture for closed 3-manifolds?

There's another volume conjecture formulated by Chen and Yang for Turaev-Viro invariants of closed manifolds. They present some evidence for the conjecture in the paper. In a second paper, Yang and ...
  • 62.2k
6 votes

The Precise Meaning of the Moduli Space of Flat Connections?

Let $P \to M$ be a principal $G$-bundle. The moduli space of flat connections on $P$ is, by definition, the space $\mathcal{M} = \mathcal{C}_0 / \mathcal{G}$, where $\mathcal{C}_0$ denotes the ...
  • 5,172
5 votes

Gauge invariance of Chern-Simons functional integral for a 3-manifold with boundary

It's not necessary to consider only gauge transformations that are constant on the boundary. However, you won't get that the CS invariant is well-defined, even modulo integers (which is what I ...
5 votes

How is Chern-Simons theory related to Floer homology?

I'm far from an expert, and I apologize if this is too basic / philosophical / vague. In instanton Floer homology, the functional $CS(A)$ plays the role of the potential energy function for a $4$d ...
5 votes

How do I calculate the modular fusion category from a given Lie algebra and level in Chern-Simons theory?

I stumbled over this older question. I actually wrote a program, that takes the type of algebra (A,B,...,G), the rank, level, and appropriate root of unity as an input. It uses the associated quantum ...
5 votes

How do I calculate the modular fusion category from a given Lie algebra and level in Chern-Simons theory?

There are two constructions of the modular fusion category. The conformal field theory approach is to take representations of the affine Kac-Moody algebra of given level and define a tensor product. ...
5 votes
Accepted

Does the limit in the Volume conjecture converge?

No, this is unknown. There are heuristic arguments for convergence based on the stationary phase approximation, but as far as I know, no one has made the argument precise in general. The closest I ...
  • 62.2k
4 votes

Ground State Degeneracy of 2+1D U(1) Chern Simons Theory?

The Abeian Chern-Simons theory you write down, can be written in a very generic form with a symmetric bilinear integer matrix $K_{IJ}$ with a path integral (or partition function): $$ Z=\int DA \exp[i ...
  • 10k
4 votes
Accepted

Chern-Simons forms, characteristic numbers, and boundary terms?

You might have heard the following. I am actually referring to the second meaning of Chern-Simons classes $\tilde p(\nabla^0,\nabla^1)\in\Omega^\bullet(M)$ satisfying $d\tilde p(\nabla^0,\nabla^1)=p((\...
4 votes
Accepted

Can the Chern-Simons invariant of a cusped hyperbolic $3$-manfiold be defined mod $\mathbb Z$?

Remark 7.2 in Christian's paper suggests that for the choice of a lift of the discrete-faithful $PSL_2(\mathbb{C})$ representation to $SL_2(\mathbb{C})$, there ought to be a lift of the Chern-Simons ...
  • 62.2k
4 votes

Formula for the anomalies of spin Chern-Simons theories?

The anomaly theory (at least for the non-super level) is described in this paper: Freed, Daniel S.; Hopkins, Michael J.; Lurie, Jacob; Teleman, Constantin, Topological quantum field theories from ...
3 votes
Accepted

Why is the Chern Number Invariant under A Continuously Shrinking of the Structure Group?

I don't quite understand the quantization part of the question. However, the characteristic numbers are always given by pairing products of the (characteristic) cohomology classes with the ...
  • 17.8k
3 votes

Value of the Chern-Simons functional for flat connections on $S^3/\Gamma$

For $G = SU(2)$ and the representation given by including $\Gamma$, the value of the Chern-Simons invariant is computed by Millson (Examples of nonvanishing Chern-Simons invariants, J. Differential ...
2 votes

What is Chern-Simons theory?

I don't know about mathematicians but when physicist say that it is a TQFT, they just mean that the theory doesn't depend on the metric choice. You can easily see that the Chern-Simons action is ...
2 votes

The Chern-Simons/Wess-Zumino-Witten correspondence

I learned this correspondence from Bos and Nair's paper "Coherent State Quantization of Chern-Simons Theory" and that is what I recommend. I wrote a short review of the part that you are interested in:...
2 votes

4-dimensional TQFT with/without requiring spin structure

One of the most famous 4d TQFTs is the Crane-Yetter TQFT, or its Hamiltonian lattice formulation, the Walker-Wang model. See my question How are the Walker-Wang TQFT and the Crane-Yetter TQFT related?....
2 votes
Accepted

Chern-Simons invariants of 2-bridge knots

EDIT: the original answer had a typo the minus sign was missing from: $q^{\pm 1}=-q \mod 2p$ Here is an argument for the forward direction: Each knot in the list above appears to be amphichiral (i.e....
  • 5,121
2 votes
Accepted

The exterior derivative of a certain differential form on the space of connections of a surface

For simplicity, suppose $V$ is just a vector space (finite-dimensional, if you like). Let $\omega\in\Lambda^2 V^\vee$ (you start with a symmetric bilinear form, but combining it with the antisymmetric ...
2 votes
Accepted

Is there a combinatorial way to determine the coefficients of the universal finite-type invariant on a given knot?

First of all, however your universal finite type invariant $Z$ is given the question of computing the coefficient of a given diagram is somewhat ill-defined, since those diagrams are not linearly ...
  • 7,517

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