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

For any principal $G$-bundle $P \to M$ with principal connection $\omega$, given a $G$-invariant polynomial $p: \mathfrak{g} \to \mathbb{R}$ we can construct a form $p(F_\omega)$ on $P$ which descends ...
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### Differential characters, Chern-Simons forms, and differential cohomology

I've read through the classic Chern-Simons paper where they introduce the Chern-Simons forms. These are differential forms whose exterior derivative gives you the characteristic forms for any given ...
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### Choice of framing in Gravitational Chern Simons

I was trying to understand formula(2.21) in Witten's paper "Quantum Field Theory and Jones Polynomial"(link: https://projecteuclid.org/euclid.cmp/1104178138) (Page 360). There, it was mentioned, the ...
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I am confused with the wedging operations of Lie algebra valued differential forms. Especially, for instance, I have some problems with the Chern-Simons 3-form $$A \wedge dA + \frac{2}{3}A \wedge A \... 1answer 172 views ### Chern-Simons invariants of 2-bridge knots 2-bridge links L(p/q) are described by the continuous fraction expansion \frac{p}{q}=\left[a_1,a_2,\ldots,a_n\right], where the a_i are the numbers of twists in the boxes below: Looking at ... 2answers 236 views ### 2-bridge knots in the Rolfsen's table 2-bridge knots (aka rational knots) K(p,q) are described by a rational number \frac{p}{q} or likewise its continued fraction expansion \left[a_1,a_2,\ldots,a_k\right]. Has somebody worked out a ... 0answers 107 views ### Ground State Degeneracy of 2+1D U(1) Chern Simons Theory? I am a physics graduate student trying to understand more mathematical aspects of gauge theories. How can I understand ground state degeneracy of a simple Chern Simons Theory: 2+1D U(1) S= \int_M ... 1answer 291 views ### Does the limit in the Volume conjecture converge? The Volume conjecture says that if J_n(q) are the colored Jones polynomials of a knot K \subset S^3, then$$\lim_{N \to \infty} \frac{ 2 \pi}N \left\vert J_N(e^{2\pi i / N})\right\vert = vol(K)$$... 3answers 1k views ### What is Chern-Simons theory expected to assign to a point? Let G be a compact, connected, (simply connected?) Lie group and let k \in H^4(BG, \mathbb{Z}) be a cohomology class. Witten showed, at a physical level of rigor, that this data determines a 3-... 3answers 936 views ### Interpreting the CS/WZW correspondence It is understood that there is a correspondence between the 3d Chern-Simons topological quantum field theory (TQFT) and the 2d Wess-Zumino-Witten conformal quantum field theory (CQFT). A good summary ... 1answer 1k views ### Classical and Quantum Chern-Simons Theory Please excuse a sloppy question from an old user who hasn't been here in a long time. I think the expertise here is such that it can be answered anyway. Let \Sigma be a two-manifold and M a ... 2answers 747 views ### The trace of a wedge product of matrices I'm trying understand a computation on page 371 of Besse's book on Einstein Manifolds. I already know the curvature operator R:\bigwedge^2\to\bigwedge^2 may be written in block diagonal form ... 1answer 803 views ### Gauss linking integral and quadratic reciprocity In the setting of Mazur's "primes and knots" analogy, prime ideals \mathfrak p\subset\mathcal O_K correspond to "knots" \operatorname{Spec}\mathcal O_K/\mathfrak p inside a "3-manifold" \... 2answers 561 views ### H^d[U(1)^n,U(1)] of the Borel cohomology and Chern-Simons theory Firstly I apologize that I am a physicist, with a relatively unrigorous math training. My approach of the problem can be Feynman style. Below Z is the integer \mathbb{Z}, and U(1) Abelian group ... 2answers 2k views ### How to understand Chern-Simons action Hi all. The question I have should be a rather simple one, but I just can't think it through. So the Chern-Simons action reads S = \int_M {\rm tr} (A\wedge dA + \frac{2}{3} A\wedge A ... 4answers 592 views ### Understand Witten's “QFT and Jones Polynomials” - how does he get to the twisted Dirac operator L_{-}? Hi, this is my first post here, so I hope I am asking the question the right way. I am trying to understand to following piece of algebra: In his paper, Witten claims that \int_M Tr(B \wedge DB) + \... 7answers 4k views ### The Chern-Simons/Wess-Zumino-Witten correspondence I have often seen a relationship being alluded to between these two theories but I am unable to find any literature which proves/derives/explains this relationship. I guess in the condensed matter ... 1answer 566 views ### Analog of “Spin” Chern-Simons Theory 3-dimensional Chern-Simons theories, with compact gauge group G, are determined by H^4(BG). Looking at U(1), with generator c_1^2\in H^4(BU(1))=\mathbb{Z} for 1st Chern class c_1, there are ... 1answer 291 views ### Set of physical states of FQHE on closed Riemann surface = ? Disclaimer. One might argue that my question is off topic as it is clearly a question about physics... But I'd like a mathematically phrased answer, and I expect that only a mathematician can offer an ... 0answers 410 views ### Is the quantum dilogarithm related in any way to cohomology of quantum groups? Is the quantum dilogarithm related in any way to cohomology of quantum groups? This question is a bit vague, and I don't have any rigorous justification for such a connection, but let me explain why ... 1answer 522 views ### Is there a simplicial volume definition of Chern Simons invariants? Suppose we have some compact hyperbolic 3-manifold M=\Gamma\backslash\mathbb H^3. Now we know that the hyperbolic volume of M can be defined as (a constant times) the simplicial volume of the ... 1answer 595 views ### SL(2,C) Chern-Simons theory in genus 1 In http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1104202513, Witten claims (p. 54) that to quantize the moduli space of flat SL(2,\mathbb{C}) ... 2answers 542 views ### Deriving the Hilbert spaces for Chern-Simons TQFTs with complex gauge group One method for finding the Hilbert spaces corresponding to surfaces in Chern-Simons TQFT is by geometrically quantizing the phase space, which is just the character variety of the surface. I know that ... 1answer 440 views ### Why does the Cheeger--Chern--Simons class descend to H_3(G/B)? Cheeger and Simons here defined a family of characteristic classes for principal G-bundles (G a Lie group). I'm interested in the special case of \hat c_2:H_3(SL(2,\mathbb C);\mathbb Z)\to\... 1answer 474 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 ... 0answers 386 views ### Is there a general dilogarithm formula for the Cheeger--Chern--Simons class? I'm looking for a generalization of the calculation of the hyperbolic volume and Chern--Simons invariant for \operatorname{SL}(2,\mathbb C) representations in terms of the Rogers dilogarithm. ... 1answer 618 views ### monodromy defects and Chern-Simons In the context of string theory I recently read "The formulation of Chern-Simons theory in terms of monodromy defects can be carried through all the dualities of the present paper, leading to ... 2answers 1k views ### What is the trace in the Chern-Simons action? Warning: This is a very stupid question regarding a basic misunderstanding that I have. I realize that the question is very elementary, but I guess asking stupid questions is better than remaining ... 4answers 4k views ### Some basic questions about Chern-Simons theory Let the Chern-Simons lagrangian for a group G be,$$L= k \epsilon^{\mu \nu \rho} Tr[A_\mu \partial _ \nu A_\rho + \frac{2}{3} A_\mu A_\nu A_\rho] Then it is claimed that on "infinitesimal" ...
A typical statement of the volume conjecture, for instance in Murakami's survey 1002.0126, is Conjecture: For $K$ a knot in $S^3$, the N-th colored Jones polynomials are related to the volume of ...