# Tag Info

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
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 ...
• 17.8k
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
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### 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
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
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### 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 ...
• 12k

### $A \wedge A \wedge A$ 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 $r\times r$ ...
• 32.4k
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### 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

### 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

• 6,334
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### 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

### 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 ...
• 3,809
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### 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

### 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 ...
• 17.8k

### 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 ...

### 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:...

### 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?....
• 5,385
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
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 ...
• 4,444
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 ...