13
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 ...
- 21.5k
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 $M$ ...
- 5,550
9
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 ...
- 12.3k
8
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.
...
- 35k
7
votes
Defining extended TQFTs *with point, line, surface, … operators*
Yes there's a very natural way to incorporate defects of arbitrary dimension into the formalism of extended topological field theory, and this is a vital part of the structure of TQFT. For example in ...
- 23.2k
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
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 ...
- 66.8k
6
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 ...
- 361
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 ...
- 5,472
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 ...
- 51
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 ...
- 19.2k
5
votes
Define the 3d Chern-Simons TQFT on a discrete simplicial complex
This question has partial answers in Lattice Gauge Theory. In general, the Abelian theory is pretty well under control and the non-Abelian theory is wide open. I wrote a fair bit about the ...
- 5,879
5
votes
Define the 3d Chern-Simons TQFT on a discrete simplicial complex
For compact $U^k(1)$ Abelian group, we recently have a paper https://arxiv.org/abs/1906.08270 to define its Chern-Simons theory on spacetime discrete simplicial complex. The quantized coefficient of ...
- 4,716
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. ...
- 9,052
5
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 ...
- 4,304
4
votes
Accepted
Explicit examples of Classical, Flat $U(2)$-connections on a torus link complement with non-trivial holonomy
For torus knots, all of the representations into $SU(2)$ were rather explicitly worked out by Eric Klassen (Representations of knot groups in $SU(2)$. Trans. Amer. Math. Soc. 326 (1991), no. 2, 795–...
- 19.2k
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 ...
- 66.8k
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 ...
- 10.3k
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((\...
- 6,666
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 ...
- 19.2k
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 ...
- 18.6k
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 ...
- 4,659
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?....
- 5,535
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 ...
- 8,244
1
vote
Derive how the level quantization for 3d quantum Chern-Simons theory path integrals?
Our recent paper https://arxiv.org/abs/2008.02613 provides an answer. It reveals that how quantization of chiral central charge and Hall conductance depend on the groundstate degeneracy on Riemannian ...
- 4,716
1
vote
2-bridge knots in the Rolfsen's table
Rolfsen's diagrams of 2-bridge knots are easy to see (from Conway's tangle notation(. I had no problem listing them first in my 1974 table of 10-crossing knots, although those diagrams were Tait's. ...
- 23
1
vote
Is there a general dilogarithm formula for the Cheeger–Chern–Simons class?
As noted in the comments, a paper [1] of Garoufalidis, D. Thurston, and Zickert answers this question for $\operatorname{SL}_n(\mathbb{C})$.
[1]
Garoufalidis, Stavros; Thurston, Dylan P.; Zickert, ...
- 2,483
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