# What is chern-simons: Definition and 16 Discussions

In mathematics, the Chern–Simons forms are certain secondary characteristic classes. The theory is named for Shiing-Shen Chern and James Harris Simons, co-authors of a 1974 paper entitled "Characteristic Forms and Geometric Invariants," from which the theory arose.

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1. ### A First Order in Time Derivatives + Phase Space

Hey all, I am reading David Tong's notes on Chern-Simons: https://www.damtp.cam.ac.uk/user/tong/qhe/five.pdf and he makes the following statement that doesn't make much sense to me: "Because the Chern-Simons theory is first order in time derivatives, these Wilson loops are really parameterising...
2. ### A Proving BF action is a difference of Chern-Simons actions

I believe this boils down to lack of familiarity on my part with wedge products of forms, so the answer is probably simple - but it's better to ask a stupid question than to remain ignorant! I've been looking at <https://arxiv.org/abs/hep-th/9505027>, and the idea that the BF [1] action...
3. ### A Hodge decomposition of a 1-form on a torus

I was reading Dunne's review paper on Chern-Simons theory (Les-Houches School 1998) and I don't get how he decomposes the gauge potential on the torus. My own knowledge of differential geometry is sketchy. I do know that the Hodge decomposition theorem states that a differential form can be...
4. ### I Equation of motion Chern-Simons

The Lagrangian (Maxwell Chern-Simons in Zee QFT Nutshell, p.318) has as equation of motion: Where does the 2 in front come from? Thank you very much
5. ### A Study Chern-Simons Invariant: Understanding 3-Manifold Measurement

I've been studying the Witten-Reshetikhin-Turaev (WRT) invariant of 3-manifolds but have almost zero background in physics. The WRT of a 3-manifold is closely related to the Chern-Simons (CS) invariant via the volume conjecture. My question is, what does the CS invariant of a 3-manifold...
6. ### A Pontryagin densities and Chern-Simons form

Dear All It is known that pontryagin densities are defined in even dimension space, let's say i am concerned with 4 dim space time. We also have a certain group G. What is the formula of pontryagin densities for arbitrary group? Larger group?
7. ### Chern-Simons form of the Chern character

So I'm working my way through Nakahara Geometry Topology and Physics. In exercise 11.4 of section 11.5.2 I'm tasked to show...
8. ### Solve Gauge Equiv. of Chern-Simons Action EOM

I have the Chern Simons action, and I've found the equations of motion ##\epsilon^{\mu\nu\rho}F_{\nu\rho}=0##. A problem I was looking at said show that the e.o.m. is "gauge equivalent to the trivial solution". I understand what this means. Obviously the e.o.m. is manifestly gauge invariant, and...
9. ### How to Derive the Chern-Simons Contribution to the Effective Action?

Hi PF, I'm still very much a novice when it comes to QFT, but there's a particular calculation I'd like to understand and which (I suspect) may be just within reach. In short, the result is that after coupling a system of fermions to an external U(1) gauge field, one obtains a Chern-Simons...
10. ### Chern-Simons and massive A^{\mu}

Hi, I am struggling with a problem in field theory: We are looking at a Chern-Simons Lagrangian describing a massive A field: L = -\frac{1}{4} F^{\mu\nu}F_{\mu\nu}+\frac{m}{4}\epsilon^{\mu\nu \rho}F_{\mu\nu}A_{\rho} I find those field equations: \partial_{\mu}F^{\mu\lambda}=-\frac{m}{2}...
11. ### Chern-Simons Form Explained: General Relativity

I've encountered chern-simons forms several times in papers of general relativity, such as in "actions based on the chern-simons form". I don't really understand where that form comes from. Is it simply some mathematical quantity? Can someone explain to me? or at least send me a reference?
12. ### Chern-Simons, LQG black hole entropy, and central intertwiner

Spin networks describing BH have a central node with label taken from an intertwiner Hilbertspace of high dimension (corresp. to hole entropy). Horizon area corresponds to the network links that pass through the horizon. The horizon Hilbertspace turns out to have the same high dimension (again...
13. ### Quadratic invariant in Chern-Simons Th. w/ ISO(2,1) group.

Hi dudes. I'm studying the paper of Witten: 2+1 Dimensional gravity as an exactly soluble system. Before eq (2.8) the author justifies as a way to find the inner product the fact that in this theory we have the casimir \epsilon_{abc}P^aJ^b. Then he introduce the invariant quadratic form...
14. ### Observable QG effects in Chern-Simons gravity-Stephon Alexander

Observable QG effects in Chern-Simons gravity--Stephon Alexander Important talk, http://pirsa.org/09110132/
15. ### Exploring Lorentz Scalar of Chern-Simons Lagrangian

Homework Statement Imagine a spatially 2d world. The electromagnetic field could be richer here, because you could add to the Lagrangian L an additional term (known as the Chern-Simons Lagrangian) L_{CS} = \epsilon_{0}\frac{\kappa}{2}\epsilon^{\alpha \beta...
16. ### Prove Chern-Simons Dispersion Relation | Carrol, Field, Jackiw

Hi everybody! Is there someone that can help me to prove that \omega^2E-k^2E=-ip_0k\times E+i\omega p\times E imply that the dispersion relation is (k^\mu k_\mu)^2+(k^\mu k_\mu)(p^\nu p_\nu)=(k^\mu p_\mu)^2 Thanks in advance ;) p.d. The reference for this formula is the...